Journal of the Pennsylvania Governor’s School for the Sciences
Class of 2007
Volume 26, 2008
Mellon College of Science Carnegie Mellon University, Pittsburgh, PA 15213
Journal of the Pennsylvania Governor's School for the Sciences
Class of 2007
Volume 26, 2008
Copyright (c) 2008, by The Pennsylvania Governor's School for the Sciences and Carnegie Mellon University
Permission is granted to quote from this journal with the customary acknowledgement of the source. To reprint a figure, table or other excerpt, requires, in addition, the consent of one of the original authors and notification of the PGSS.
Journal of the PGSS Page iii
Table of Contents
Preface ...... 1
Biology Team Projects
Detection and Identification of Genetically Modified Soy (Glycine max) in Assorted Chocolate Food Products Bhaskar, Hanzok, Kedar, Lu, Oniskey, Punati, Qin, Reddy, Shoemaker, Singh, Viswanathan, Young ...... 5
Determination of the Midpoint of Adolescent Myelin Maturation as Measured by Change in Radial Diffusivity in Diffusion Tensor Imaging Baybars, Chang, Jiang, Onofrey, Ponnappa, Ponte, Ramgopal, Shen, Trevisan, Sheng, ...... 35
Biophysics Team Projects
The Isotope Effects of Deuterium on the Enzyme Kinetics of Alcohol Dehydrogenase Alla, Bharathan, Comerci, Desai, Field, Gillis-Buck, Hsu, Jain, Kessler, Kim, Kozminsky, Magge, Negrin, Owunna, Rogi, Tadavarthy, Vin, Wang, Wu, Zimmerman ...... 69
Chemistry Team Projects
Computational Predictions of Chromatographic Selectivity of Stationary Phases Badgley, Carpenter, Rastogi, Widenor, Zhang...... 95
Effects on Resistance and Lattice Structure Using Substitution-Based Superconductors Cockerham, Frey, Herrold, Johnson, Ko, Lucas, Phan, Rosario, Ru...... 117
Computer Science Team Projects
LexerEvolution Bruggeman, Chu, Jiang, McGuier, Shotter, Triano ...... 159
Project LPX: A Functional Web Design Language in SML Elfont, Guenin, Kamani, McDowell, Ruberg ...... 169
Mathematics Team Projects
Isomorphisms of Three-Dimensional Lie Algebras Chan, Falk, Radomy, Wang, Warner...... 181
Journal of the PGSS Page iv
Physics Team Projects
Gambling for the Truth: A Monte Carlo Simulation of Magnetic Phase Transitions Helbers, Jordan, Petkun ...... 199
An Investigation of Chaos in One and Two Dimensions Das, Goyal, Manolache, Morris, Paul, Sampath...... 225
A Verification of Wave-Particle Duality in Light Alexakos, Harder, Kapelewski, Long, Manolache, Miller, Rahman, Rastogi, Reichardt, Rhinehart ...... 241
Wil-ber-force Be With You: An Analysis of the Wilberforce Pendulum Cook, Han, Jin, Menendez, Nataraj, Stein, Strom, Sunday, Yang ...... 283
Appendix
The Students, Faculty, Teaching Assistants/Counselors of the Pennsylvania Governor's School for the Sciences for 2007...... 325
Journal of the PGSS Page 1
Preface
The Pennsylvania Governor's School for the Sciences (PGSS) is a five-week residential summer program held on the campus of Carnegie Mellon University for talented Pennsylvania high school students. The PGSS was initiated and is supported by the Pennsylvania Department of Education (PDE). It has been continuously operating since 1982. The 2007 class consisted of 100 students, 56 male students and 44 female students.
The PGSS was in session from June 24 to July 28. The program is very intense; students take formal courses in biotechnology of HIV and AIDS, organic chemistry, concepts of modern physics, discrete mathematics and computer science. They also have the opportunity to take a number of elective courses and a laboratory course in biology, chemistry, computer science or physics.
A major objective of the PGSS is to provide an opportunity for students to participate in college-level courses, laboratories and research experiences designed to enrich their background in science and to encourage them to pursue technological careers. One of the many student activities during the PGSS is the participation in a team research project in biology, chemistry, computer science, mathematics or physics. Team research projects involve the investigation of an original problem or the solution of a problem by techniques original to the investigators. The students choose the area of science for their project and faculty suggest specific possible topics in these areas. Interested students form teams. While faculty is available for initial direction and advice, most of the accomplishments come from the students' own initiative.
The Journal of the Pennsylvania Governor's School for the Sciences reports the results of the students' efforts and is the official record of these team research projects. Each team investigated their chosen problem using resources and techniques appropriate for that topic. The following reports were written by the student team members. The faculty in each respective area reviewed the final reports and made necessary minor corrections.
This journal was edited by Maria Wilkin, who assumed the responsibility of converting all the reports to a similar format, reproducing figures, and dealing with other publication issues. Otherwise, the original character of the student-authored papers was maintained to the maximum feasible extent.
The PGSS gratefully acknowledges the continuing support of the Mellon College of Science at Carnegie Mellon University and generous contributions from the parents and guardians of the PGSS Class of 2007.
Barry B. Luokkala, Ph.D. University Director of the Pennsylvania Governor's School for the Sciences
April 2008
BIOLOGY TEAM PROJECTS
Detection and Identification of Genetically Modified Soy (Glycine max) in Assorted Chocolate Food Products
Ashok Bhaskar, Jackie Hanzok, Vijay Kedar, Louis Lu, Taylor Oniskey, Ramya Punati, Frank Qin, Varun Reddy, Katie Shoemaker, Rohini Singh, Varun Viswanathan, Matthew Young
Abstract
Soybean (Glycine max) is a major commercial GM crop found in many products including various chocolate products (Theobroma cacao derivatives) and was detected and identified through the utilization of techniques in DNA and protein isolation, PCR analysis, and ELISA. The purpose of the experiment was to detect GM soy in assorted chocolates, to determine whether products corroborate or conflict with the “organic”/“non-GMO” claim made on product labels, and to compare the prevalence of genetic modification in small samples of American and European chocolates. The PCR analysis yielded inconclusive results for most samples; there was some evidence for GM in a few samples, but the sequences could not be conclusively linked to soy. A kanamycin selectable marker was identified, but further studies are required for a definitive conclusion. In the ELISA analysis, five chocolate samples tested slightly positive for Roundup Ready® protein indicating the possible presence of CP4-EPSPS. Improving DNA extraction is necessary for further work.
I. Background
A. An Introduction to Genetically Modified Organisms
A gene is a section of DNA that encodes for a specific inherited trait. Whereas genes can be altered through natural mutations, scientists can also create a genetically modified organism (GMO) by experimental means such as by inserting a gene from one species into another species, inducing the organism to express traits that it would not usually have1. Scientists have several methods of making recombinant DNA. One method is using a transformed bacterium as a vector to transform other organisms2. A second commonly-used method is to insert DNA-coated particles via a gene gun into a cell that will uptake the foreign DNA and incorporate it into its chromosome. A third method involves cross-breeding GM plants through cross-pollination, encouraging a certain phenotype to be acquired by more organisms1.
In general, genetic modification can be useful in developing organisms that overcome the obstacles they face in the environment. Introducing herbicide-resistant genes into plants, for example, enable farmers to use herbicides more liberally with less damage to crops and to plant more crops per acre3. Pest-resistant plants reduce the need to spray pesticides, decreasing side effects on non-target insects. As a result, crop yield may improve.
Page 6 Bhaskar, Hanzok, Kedar, Lu, Oniskey, Punati, Qin, Reddy, Shoemaker, Singh, Viswanathan, Young
Alternatively, the taste, hardiness, and nutritional value of certain foods can be improved via genetic modification3. For example, introducing pectin into tomatoes increases water retention, making the tomatoes plumper and juicier4. Golden rice is healthier rice that has been genetically engineered to produce beta-carotene, a precursor to vitamin A5. Vitamin A deficiency leads to increased susceptibility to disease worldwide, especially in impoverished societies which are often rice-dependent. That is why rice has been genetically engineered for this purpose6.
However, genetic modification is controversial because of the risks it presents to consumers and the environment. For example, while herbicide-resistant plants are advantageous, the consequently more liberal spraying of pesticides presents a threat to the environment. Genetically modifying organisms can cause genetic drift and artificial selection in which organisms pass advantageous GM traits, such as antibiotic resistance, onto their offspring, thereby perpetuating the expression of this genetically modified trait. Subsequent cross-contamination of the genome of the recipient organism and incorporation of this gene into other unintended organisms is also a risk. Reproductive mechanisms such as pollination can allow other organisms to acquire the modified gene3. Selection may favor this gene and thus reduce the biodiversity in a population. The implications of this have prompted biotechnology companies to prevent cross-pollination and inheritance of GM traits by making GM crops sterile7.
Several food producers label their food as GMO-free, such as the Dolfin® chocolates8, and these labels may be inaccurate. Additionally, genetically modified organisms could have the capacity to be unexpectedly allergenic or incite sensitivities in consumers9. Horizontal gene transfer helps promote this because this is a mechanism by which the DNA of one species passes on to the DNA of another species9. The ramifications of the genetically modified organisms controversy are important because they can impact a large variety of food crops such as canola, maize, and soy1 as well as the foods which utilize these products.
B. Organic Food Labeling Regulations
Part of the purpose of this project is to determine whether “GMO-free” and “organic” labels placed on some of the chocolates are corroborated by the data from the experiment. Organic food should not be genetically modified or treated with such materials as growth hormone or antibiotics10. In order for food to be labeled “organic”, the United States Department of Agriculture (USDA) requires that food contain at least 95% organically produced ingredients by weight. Food labeled as “made with organic ingredients” must contain 70% organic ingredients by weight. Foods with less than 50% organic content can only indicate which ingredients are actually organic10.
Journal of the PGSS Detection and Identification of Genetically Modified Soy Page 7 (Glycine max) in Assorted Chocolate Food Products
European organic food labeling guidelines differ from those of the United States because of a higher anti-GM sentiments across the Atlantic11. In 2007, European Union agricultural leaders agreed upon a new set of labeling regulations for organic foods. Like the United States, in order to be labeled “organic,” the food must contain at least 95% organic material by weight. However, the intentional use of genetically modified organisms in food products is strictly prohibited. The remaining 5% of percent of the food content can consist of non-GM inorganic materials. There is a tolerance of 0.9% by weight for accidentally introduced GMOs12.
C. Soybeans and the Roundup Ready® Gene
Grown as a commercial crop in 84 countries in 2004 with a combined harvest of 206 million metric tonnes, soy (Glycine max) and its derivatives (e.g. soy lecithin) appear in many products that uninformed consumers are not aware of13. Monsanto Roundup Ready® is a commercially available, herbicide-tolerant GM soybean line. In order to kill most weeds, multiple herbicide applications are required in a soy field. Using one herbicide is not an option because, for most herbicides, specificity is too high (kills only one weed type) or too low (kills all vegetation)13. For instance, the herbicide Roundup® has low specificity because its active ingredient, glyphosate, inhibits the enzyme 5-enolpyruvylshikimate-3-phosphate synthase (EPSPS), which is required in aromatic amino acid synthesis; this pathway is found in a wide variety of plants. The GM soybean line MON89788, also known as Monsanto Roundup Ready® Soy, produces CP4-EPSPS, a form of the enzyme EPSPS which is not inhibited by glyphosate and is therefore unaffected by Roundup®13. Through recombinant DNA technology and bacterial transformation, the soy plant is infected by the common soil bacterium Agrobacterium tumefaciens (called CP4) that introduces a Ti (tumor inducing) plasmid into the plant, and inserts a gene cassette, including the transgene encoding CP4-EPSPS, into the chromosome. Thus, the new form of EPSPS, tolerant of glyphosate, is expressed in the GM plant (Figure 1)13.
D. Origin of Chocolate Samples and Manufacturing Process
Many consumers may not realize that many chocolates actually utilize soy derivatives, usually soy lecithin, a phospholipid. As the cheapest and most widely used lecithin, soy lecithin is frequently used in chocolates. Chocolate is such a commonly consumed commodity that any changes in production will impact a vast number of consumers. In fact, the United States is the eleventh highest chocolate consuming country, consuming three billion pounds of chocolate in 200114. Although soy is a minor ingredient in chocolate, its impact can accumulate because of the enormity of chocolate consumption in the United States.
Page 8 Bhaskar, Hanzok, Kedar, Lu, Oniskey, Punati, Qin, Reddy, Shoemaker, Singh, Viswanathan, Young
Sugar
Shikimate-3-phosphate + Phosphoenolpyruvate CP4- RR EPSPS RR EPSPS
5-enolpyruvylshikimate-3-phosphate
Aromatic Amino Acids
Figure 1. The Shikimate Pathway in for Aromatic Amino Acid Synthesis in Vegetation
Chocolate typically includes soy lecithin, milk, cocoa butter, sugar, vanillin, and artificial flavor15. Cacao beans, the basis of chocolate, originate from the tropical cacao tree grown in Central America, the Caribbean, Brazil, and Africa. These Theobroma cacao trees produce football- shaped pods with lavender seeds which are fermented at 40-50°C to become dark brown and develop the chocolate flavor. Dried cacao beans are then cleaned, roasted at 121°C16, and shelled to release the characteristic aroma of chocolate. The meat of the cacao bean, called the "nib," contains the chocolate flavor and is ground into a liquid. This liquid is then mixed with other ingredients such as sugar and milk and processed through steel rolls and large mixers to fully develop the flavor and ensure smooth texture. During the mixing stage, soy lecithin is introduced into the chocolate mix. Commonly derived from soybean oil, lecithin is a phospholipid that acts as an emulsifier and stabilizer often found in processed foods17. By breaking lipids and fats into very small pieces and allowing them to mix with the rest of the ingredients, lecithin helps to homogenize, smoothen, and solidify a variety of processed foods like margarine and chocolate17. As lecithin is generally safe for humans and is biodegradable, lecithin is one of the most common emulsifiers present in processed foods18. Because soy lecithin is an inexpensive and stable emulsifier, it is the most widely used emulsifier in chocolate. After the lecithin is added, the chocolate is then warmed in the conche: a large vat in which beads are used help grind the Journal of the PGSS Detection and Identification of Genetically Modified Soy Page 9 (Glycine max) in Assorted Chocolate Food Products chocolate and warm it using frictional heat. Finally, chocolates are poured into molds, cooled into solid bars, and wrapped15.
Hershey's® makes no claims regarding the genetically modified or organic content of its milk chocolate bars19. The sole ingredient in Ghirardelli® Premium Baking Cocoa is unsweetened cocoa. Ghirardelli makes no claims regarding the genetically modified or organic content of its milk chocolate bars20. Scharffen Berger® is one of 12 companies to import raw cacao beans. Its cacao beans are cleaned and roasted in the factory itself. Scharffen Berger® claims that it uses non-GMO soy lecithin15. Lindt® makes no claims about the genetically modified or organic nature of its chocolate bars, but it may contain traces of soybean. Product labeld do not indicate the presence of soy lecithin21. Dove® follows the same general process for producing its chocolates. The Dove® chocolate bars tested do contain soy lecithin22. Dolfin® claims that its product is GMO- free8. Green and Black’s® (G&B) chocolates contain soy, and also claim to be organic23. Nestle Crunch® does contain soy lecithin, but does not make any claims about its genetically modified or organic nature24. Michel Cluizel® is unique because Cluizel banned the use of soy lecithin in his chocolates. In fact, the company does not use emulsifiers at all to create the creamy texture of chocolate. Instead Michel Cluizel® grounds the cacao beans to a finer texture so as to better create the cream. The company, however, does follow the same general production and processing procedures as the other companies listed above25 (Table 1).
Sample Origin Label Michel Cluizel Europe GMO Free Dolfin Europe GMO Free Lindt Dark Europe GMO Free Ghirardelli Domestic No claim Scharffen Berger Domestic No claim G&B White Europe Organic G&B Milk Europe Organic Hershey's Milk Domestic No claim Lindt White Truffle Europe No claim Nestle Domestic No claim Dove Domestic No claim Hershey Cacao Reserve Domestic No claim
Table 1: Regional Origins and Label Claims of Chocolate Products
E. Purpose
The basic purpose of this experiment is to detect genetically modified organisms in chocolate samples or the lack thereof. A secondary purpose of the experiment is to determine whether or Page 10 Bhaskar, Hanzok, Kedar, Lu, Oniskey, Punati, Qin, Reddy, Shoemaker, Singh, Viswanathan, Young not the data corroborate the “non-GMO” and “organic” claims made on some of the chocolate samples being tested. On a much larger scale, the project aims to compare the prevalence of GM soy in American and European chocolates.
II. Materials and Methods
A. Introduction
This project investigated chocolate for genetically modified soy on the DNA and protein level. Chocolate samples were manufactured by Michel Cluizel®, Ghirardelli®, Lindt®, Scharffen Berger®, G&B®, Hershey’s®, Nestle®, and Dove®. Products used were chosen based on origin (domestic or European), labeling (non-GMO, organic, or no classification), and major ingredients. Gloves were worn at all times to avoid contamination of samples.. Aprons and gloves were worn while handling ethidium bromide (EtBr). Each method utilized several appliances that performed different functions such as mixing (vortex) and separating (centrifuge). After isolation various methods were used to analyze the DNA in the chocolate products, including DNA isolation, Polymerase Chain Reaction (PCR), agarose gel electrophoresis, and spectrophotometry. To detect Roundup Ready protein, Enzyme-Linked ImmunoSorbent Assay (ELISA) and spectrophotometry were used.
B. DNA Isolation
The purpose of DNA isolation was to obtain a sample of DNA from each chocolate product, which was performed with a kit. The protocol was derived from the NucleoSpin® Food Kit (BD Biosciences) User Manual. This method isolated all the DNA present from all ingredients, and possible contaminants, from each product. The process used ground chocolate processed with a progression of different buffers to break down food particles, wash contaminants and elute DNA from a separating column. The resulting sample of isolated DNA was analyzed by spectrophotometry.
NucleoSpin® Food Kit (BD Biosciences ClontechTM): • NucleoSpin® polypropylene columns with resin • Buffer CF (polysaccharide removal buffer) • Buffer C2 (lysis buffer) • Buffer C3 (lysis buffer) • Buffer C4 (lysis buffer) • Buffer CQW (lysis buffer) Journal of the PGSS Detection and Identification of Genetically Modified Soy Page 11 (Glycine max) in Assorted Chocolate Food Products
• Buffer C5 (wash buffer) • Buffer CE (elution buffer) • 20 mg/mL Proteinase K (degrades proteins) • 96-100% Ethanol • 2-mL collection tubes Other materials: • Porcelein mortar and pestle • VWR® Scientific Parts Mini Vortexer • Eppendorf® MiniSpin Centrifuge • Labsystems 4500 Micropipettes • 1.5 mL microcentrifuge tubes • 2.0 mL microcentrifuge tubes • Distilled water • Liquid Nitrogen • OHAUS Adventurer Electronic Balance
Of the buffers used, Buffers C4, C5, and CQW were wash buffers; Buffer CE was an elution buffer; and Buffer CF was a lysis buffer. These buffers played instrumental roles in the isolation process. This process began when the samples were ground with a mortar and pestle. The chocolate was easier to suspend in water; this suspension, after an amount of vortexing, would yield a liquid solution. The solution for each chocolate product was put into microcentrifuge tubes and 550 µL Buffer CF was added. Buffer CF, as a lysing buffer, functioned to break down the material in the sample in order to access DNA. This step was further implemented when the lysate was incubated at 65ºC for 30 minutes. The lysate was then centrifuged for 10 minutes at 10,000 rpm to separate the supernatant from the cellular and product debris. Then 300 µL of the DNA-containing supernatant was transferred to an unused collection tube, and 300 µL of Buffer C4 and 200 µL of ethanol were added. Buffer C4, a wash buffer, and the ethanol inhibited enzyme activity. This solution, after being mixed in the vortex for 30 seconds, was transferred to a NucleoSpin column with collection tube. The NucleoSpin column has an activated silica resin that separates the DNA from fats, proteins, and other unwanted molecules. The entire ensemble was then centrifuged for 1 minute at 11,000 rpm, and the flow through collected in the bottom of the collection tube. The silica resin in the NucleoSpin column trapped DNA molecules and kept them out of the flowthrough, while the extraneous debris was washed out with the buffers when centrifuged, collected in the bottom of the tube, and then discarded. Next, 400 µL of Buffer CQW was added, centrifuged at 11,000 rpm for 1 minute, and the flow through was removed. Then 700 µL of Buffer C5 was added to wash the solution, which was centrifuged for 2 minutes at 11,000 rpm. Again, the flow through was discarded. This step was repeated with 200 µL of Buffer C5 for Page 12 Bhaskar, Hanzok, Kedar, Lu, Oniskey, Punati, Qin, Reddy, Shoemaker, Singh, Viswanathan, Young
2 minutes at 11,000 rpm to ensure that all Buffer C5 had been processed through the resin. The flow through was discarded. The NucleoSpin column was transferred to a 1.5 mL microcentrifuge tube. Then 50 µL of Buffer CE, the elution buffer, warmed to 70ºC, was added to deactivate the resin that bound the DNA in the sample. Because the elution buffer was more negative than the DNA, the DNA was released and was free to flow through the resin. After a 5-minute incubation to ensure that the DNA dissolved in Buffer CE, the column was centrifuged for 1 minute at 11,000 rpm to force the DNA out of the resin. The sample was thus transferred into a clean microcentrifuge tube and stored at 4ºC. Spectrophotometry was then performed to determine the DNA yield of each sample as well as DNA concentration and purity.
C. Spectrophotometry
A Spectronic Instruments Genesys 5 Spectrophotometer was used in DNA isolation to determine DNA yield. The spectrophotometer employs ultraviolet light, which is passed through a sample. DNA absorbance peaks at 260 nm and 280 nm. The optimal absorption ratio at 260: 280 nm, 26 should be between 1.4-1.7 . An ELX800 plate reader (BIOTEK Instruments, Inc) was used to analyze the ELISA samples for absorbance at 450 nm.
Polymerase Chain Reaction (PCR)
The purpose of PCR is to test for of specific genetic sequences in a DNA sample (See Table 2, Primer Chart, in Appendix), such as the DNA isolated from the food samples. Primers, short sequences of DNA that are complementary to parts of a gene, bind regions surrounding the desired sequence. If present, the primer binding allows the gene sequence to be amplified. Thus samples can be tested for sequences common to GM, as well as endogenous controls27. Materials • DNA isolated from soy and chocolate samples • Primers (Integrated DNA Technologies, Inc.) • Distilled water • Microcentrifuge tubes (Eppendorf®) • PCR beads containing polymerase, buffer salts, magnesium, and dNTPs (BD Biosciences) • Thermocycler (Perkin Elmer DNA Thermal Cycler 480) • Mineral oil • Vortex (VWR Scientific Parts Minivortexer) • Centrifuge (Eppendorf MiniSpin Centrifuge)
Journal of the PGSS Detection and Identification of Genetically Modified Soy Page 13 (Glycine max) in Assorted Chocolate Food Products
5 µL of sample DNA was added to the PCR bead and tube, along with 18 µL of distilled water and 2.5 µL each of forward and reverse DNA primers. (Table 2.) The solution was mixed in the vortex. Then one drop of mineral oil was added to the top of the solution in the tube. The purpose of the mineral oil was to prevent any sample from evaporating in the heat of the thermocycler, and to prevent the sample from getting trapped in the top of the tube. Once the tubes were placed to the thermocycler well with mineral oil, the PCR program began. In the initial step of the first cycle, DNA melted and thus separated in the thermocycler. This temperature was selected to break the hydrogen bonds present in the DNA, and the process was activated. This results in mass of single stranded DNA. At a cooler temperature, the next step of the thermocycler process annealed the primers to each complementary region of the single strand of DNA, so that DNA polymerase could elongate the strand to produce double stranded DNA. This cycle was repeated many times.
Thermocycler Program: 1. First step: a. First cycle: 95ºC for 5 minutes b. Following cycles: 94ºC for 20 seconds 2. Second step: 54ºC for 40 seconds 3. Third step: 72ºC for 60 seconds 4. Repeat cycle 20-35 times
D. Agarose Gel Electrophoresis
The purpose of agarose gel electrophoresis is to separate DNA molecules based on molecular weight. Molecular weight markers (MWM) are used as a ruler to estimate the size of DNA fragments. To perform gel electrophoresis, the gel was prepared, DNA samples were mixed with a visible dye and inserted wells of the gel, and an electric current was passed through the gel to force the DNA fragments to separate.
Materials • TBE 6X loading buffer with marker dye • Molecular Weight Marker: 1Kb and 100 bp (Promega) • Microcentrifuge tube • DNA sample • Distilled water • 2% agarose TBE gel • 2 grams of agarose Page 14 Bhaskar, Hanzok, Kedar, Lu, Oniskey, Punati, Qin, Reddy, Shoemaker, Singh, Viswanathan, Young
• 100 mL 1X TBE buffer • 100 µL ethidium bromide (EtBr) • Molecular weight marker • Thermo Electron Corporation EC105 Power supply • KODAK 1D3.6.1 software • KODAK DC290 digital camera • KODAK Model MPH Multipurpose Gel Electrophoresis Gel Box (Cat. #-52000)
The gel was made by combining 90 mL of distilled water, 1g agarose, and 10 mL of 10X warm TBE buffer. The solution was heated until it boiled, then 100 µL EtBr was added to make DNA visible, and the solution was mixed thoroughly. After it had cooled to manageable temperature, the mixture was poured into the gel plate, and a comb was inserted. When the gel had polymerized and become translucent, the comb was gently removed and the plate placed in the electrophoresis apparatus. The gel was then covered with 1X TBE Buffer, that had been diluted 1:9 from 10X TBE Buffer in distilled water.
The DNA samples were mixed with dye in loading buffer before loading in the wells: 2 µL of loading buffer, 2 µL distilled water, and 8 µL of DNA sample were combined in a microcentrifuge tube and centrifuged for 30 seconds at 11,000 rpm at room temperature (to collect all of the sample at the bottom of the tube). The samples were then loaded into the wells of the gel. A molecular weight marker was added as a reference point to estimate fragment size. The gel was run at 100 volts for 2-3 hours and the gel was imaged using Kodak® software and cameras.
E. Enzyme-Linked ImmunoSorbent Assays
To determine the presence of the CP4-EPSPS protein present in the food samples, a sandwich Enzyme-Linked ImmunoSorbent Assay (ELISA) test was performed. The ELISA utilizes protein- specific antibodies and color-producing enzyme/substrate interactions to reveal the presence of an antigen. A sandwich ELISA, as its name implies, layers two different target-specific antibodies on either side of the antigen, as opposed to a competitive binding ELISA, in which the enzyme conjugates bind the same antibody as the protein. This experiment utilized an antibody-coated well plate targeted to CP4-EPSPS (EnviroLogix QuantiPlate Kit for Roundup Ready Soybean and Soy Flour (Cat.#-AP 032)). The kit has a 0.02% limit of detection and a 0.2% limit of quantification for Roundup Ready soy. A second antibody, also targeted to CP4-EPSPS conjugated to a color-producing enzyme, completes the sandwhich. The enzyme component, of the enzyme-antibody conjugate, horseradish-peroxidase, acts upon the added substrate, Journal of the PGSS Detection and Identification of Genetically Modified Soy Page 15 (Glycine max) in Assorted Chocolate Food Products producing a yellow color (positive signal) as measured quantitatively by spectrophotometry at 450 nm. If no color is produced, a negative signal indicates little or no protein detected.
EnviroLogix QuantiPlate Kit for Roundup Ready Soybean and Soy Flour: • 0% Soy Powder (negative control) • 0.1% Roundup Ready Soy Powder • 1% Roundup Ready Soy Powder • 2% Roundup Ready Soy Powder • Wash buffer salts • CP4-enzyme conjugate • Substrate • Stop solution (1.0 N HCl).
The soy powders and buffer salts were combined with distilled water to produce solutions, according to kit directions.
Other materials: • DNA samples from isolations • Centrifuge (Eppendorf MiniSpin Centrifuge) • Vortex (VWR Scientific Parts Minivortexer) • Electronic balance (OHAUS Adventurer Electronic Balance) • Porcelain mortar and pestle • Beakers, flasks, graduated cylinders • 1.5-mL microcentrifuge tubes • Distilled water • Pechiney Plastic Packaging Parafilm M* Laboratory Wrapping Film
• Spectrophotometric plate reader (BIOTEK Instruments, Inc, ELX800)
Chocolate samples consisted of 10 g of shaved chocolate. The chocolate was then dissolved in 40 mL of distilled water with vortexing, and heated in a warm water bath if needed to dissolve. The suspension was then incubated at room temperature for 1 hour, vortexed again, and put on a for 3-10 days. After releasing pressure from fermentation, each sample was then centrifuged for 5 minutes at 5,000 rpm at room temperature to separate out the solid cellular debris and lipids. To prepare the ELISA samples, 20 µL of the water-based fraction was removed with a micropipette and combined with 980 µL of Wash buffer. The soy samples (soybean and soy flour) were prepared by grinding up soybeans by mortar and pestle and diluting with water. For assay Page 16 Bhaskar, Hanzok, Kedar, Lu, Oniskey, Punati, Qin, Reddy, Shoemaker, Singh, Viswanathan, Young standards, 100 µL of each standard extract (0.0%, 0.1%, 1%, and 2% Roundup Ready soy powder) was diluted with 400 µL of wash buffer.
A well was selected for each sample, to which 50 µL of sample and 50 µL of Roundup Ready enzyme conjugate, were added. The solutions were mixed, covered with Parafilm, and incubated for 45 minutes at the ambient room temperature. After the elapsed time, the solution inside was discarded. (At this step, any proteins that did not bind, along with the unbound enzyme conjugate, were removed). The wells were washed out four more times with wash buffer to remove all of the unbound components, then 100 µL of substrate was added to each well. The wells were again covered with Parafilm and incubated at room temperature for 15 minutes. After the elapsed time, 100 µL of stop solution was added. Wells with a positive signal produced a yellow color, while samples without detectable CP4-EPSPS did not. All samples were analyzed by a spectrophotometer to determine the optical density (OD) of each sample at 450 nm of light.
III. Results
A. Introduction to Results
Two tests were used to detect GM in chocolate products: Polymerase Chain Reaction (PCR) and Enzyme Linked ImmunoSorbent Assay (ELISA). PCR is a method used to amplify a specific gene sequence. Primers were designed to frame the gene sequences of interest. Gel electrophoresis was then used to analyze the results of PCR. ELISA, which uses specific antibodies to detect proteins of interest, was then used to see if the proteins corresponding to the genes of interest were produced.
B. PCR Results
PCR tested for the presence of specific genes, some of which can be found in GM soy and others that served as controls. The endogenous soy lectin and -tubulin primers were the positive controls for DNA quality. The -tubulin was more general than the endogenous soy lectin; the - tubulin primer tested for plant DNA, for which soy samples or other sources of plant DNA would test positive, while the endogenous soy lectin was specific to soy plants.
To test for GM sequences, CAMV35S, NPTII, CP4-EPSPS, NOS terminus, and the NOS/EPSPS junction primers were used. CAMV35S is a promoter sequence commonly used in conjunction with transgenes, such as CP4-EPSPS which encodes for Roundup Ready. NOS is a terminator sequence also used with transgenes while the NOS/EPSPS junction primers frame the region Journal of the PGSS Detection and Identification of Genetically Modified Soy Page 17 (Glycine max) in Assorted Chocolate Food Products between the Roundup Ready gene and the terminator. NPTII, which encodes a gene for kanamycin resistance, is commonly used as a selection marker when developing GM products.
1. PCR Results for Control Soy Samples
The soy controls were tested to ensure that the PCR was, in fact, performed correctly (Table 3). Every positive control (Sigma® soy flour, field sample soybean, and the unknown soybean) tested positive for endogenous soy lectin which shows that the samples had sufficient DNA for PCR. Hodgson Mill® organic soy flour, a negative control, also tested positive for soy lectin. Sigma® soy flour and unknown soybean were also tested for the -tubulin gene and tested positive, further indicating that the DNA yield was sufficient.
The control soy samples were also tested with the GM primers. The positive controls, Sigma soy flour, field sample soybean, and the unknown soybean, all tested positive for the CAMV35S promoter, and CP4-EPSPS (Table 3).
Sigma soy flour and the unknown soybean also tested positive for NPTII; the field sample, however, was not tested for that specific primer. The field sample soybean tested positive for the NOS terminus and the NOS/EPSPS junction. Sigma soy flour and the unknown soybean were not tested for either of the two primers. The negative controls, Hodgson Mill Organic soy flour and the non-GMO soybean, both tested negative for CAMV35S. Hodgson Mill tested negative for CP4-EPSPS. Hodgson Mill Organic soy flour originally tested negative for CAMV34S, but after further testing, had different results. The second results showed a positive reading for CAMV, but still had some indefinable bands, which can only be explained as contamination. (Figure 2)
NOS/ α Soy CP4/ Sample CAMV35S NPTII NOS EPSPS Tubulin Lectin EPSPS
Sigma® + + + + + ND ND Soy Flour Field Sample ND + + ND + + + Soybean Unknown + + +,+ + + ND ND Soybean Hodgson Mill® ND + -,-,+* ND - ND ND Soy Flour Non-GMO ND ND -,-,+ ND ND ND ND Soybean + = signal fragment identified, - = no signal, * later results positive Table 3: PCR Results of Soy Controls
Page 18 Bhaskar, Hanzok, Kedar, Lu, Oniskey, Punati, Qin, Reddy, Shoemaker, Singh, Viswanathan, Young
2. PCR Results for Chocolate Samples
PCR results for the chocolate samples did not turn out as expected. All lanes on the gels, except for the ladder, had primer dimers (Figures 2, 3). A primer dimer is created when the primers anneal to each other, rather than producing signal fragments. Nearly every chocolate sample, Michel Cluizel®, Dolfin®, Ghirardelli®, Lindt® Dark Chocolate, Green & Black’s® white chocolate, Green & Black’s® milk chocolate, Lindt® White Truffle, Nestle®, Dove®, and Hershey’s® Cacao Reserve, tested negative for the CAMV35S promoter, EPSPS, the NOS terminator, the NOS/EPSPS junction, and -tubulin (Table 4). Scharffen Berger® and Hershey’s® milk chocolate tested positive for NPTII.
500 bp Æ
Å Hershey’s Milk (NPTII +)
100 bp Æ Å primer dimers
Figure 2: PCR Analysis shows NPTII Signal for Hershey’s Milk Chocolate.
Journal of the PGSS Detection and Identification of Genetically Modified Soy Page 19 (Glycine max) in Assorted Chocolate Food Products
Figure 3: PCR Analysis shows NPTII Signal for Scharffen Berger Chocolate.
NOS/ α Soy CP4/ Sample CAMV35S NPTII NOS EPSPS Tubulin Lectin EPSPS
Michel Cluizel ------Dolfin ------Ghirardelli ------Lindt Dark ------Scharffen Berger - - - + - - - G&B White ------G&B Milk ------Hershey’s Milk - - - + - - - Lindt White ------Nestle ------Dove ------Hershey’s Cacao Reserve ------+ = signal fragment identified, - = no signal, * later results positive
Table 4: PCR Results of Chocolate Samples.
Page 20 Bhaskar, Hanzok, Kedar, Lu, Oniskey, Punati, Qin, Reddy, Shoemaker, Singh, Viswanathan, Young
C. ELISA Results
The ELISA was used to detect EPSPS (Roundup Ready) protein from soy in twelve chocolate samples. This was determined by absorbance as measured by a spectrophotometer. (Figure 4.) Absorbance is a measure of how much light is absorbed by (and therefore not transmitted by) a sample. In this assay, the absorbance of the substrate product (yellow) is measured at an optical wavelength of 450 nm. The intensity of yellow color in the sample is determined by the quantity of substrate catalyzed by the enzyme (horseradish peroxidase). The amount of horseradish peroxidase in the sample well is in turn determined by the quantity of protein bound to the antibodies adherent to the plate. Therefore, the optical density reading output by the spectrophotometer indirectly indicates the concentration of Roundup Ready protein in each sample.
To provide both baseline and controls, pre-manufactured standard extracts containing different concentrations of Roundup Ready soy powder were assayed. The 0% standard extract was made entirely with non-GMO soy beans. The 0.1%, 1% and 2% standard extracts were made with the specified percentages of Roundup Ready soy beans and the remainder was non-GMO soy. One purpose of the four standard extracts was to ensure that all parts of the ELISA procedure were carried out properly and that the spectrophotometer worked. In addition, conducting the ELISA with the standard extracts served as a positive control to demonstrate the sensitivity of ELISA, in that it can detect Roundup Ready soy protein in soy flour samples made from as little as 0.1% GM soy. It also showed that the detector would not saturate at these levels. Finally, they served as a control to guarantee that optical density does in fact increase with increasing concentrations of Roundup Ready soy protein. (Table 5 and Figure 5.) The optical densities of the chocolate samples could then be analyzed relative to those of the standard extracts for concentration of Roundup Ready soy powder.
Sample OD OD, rescaled 0% Standard Extract 0.058 0.000 0.1% Standard Extract 0.223 0.165 1% Standard Extract 1.114 1.056 2% Standard Extract 1.792 1.734
Table 5: Optical Density (450 nm) of Standard Extracts in ELISA for CP4-EPSPS Protein
Journal of the PGSS Detection and Identification of Genetically Modified Soy Page 21 (Glycine max) in Assorted Chocolate Food Products
Optical Densities of Standard Extracts
2
1.8 1.6 1.4 y = 0.863x + 0.0699 sit 1.2 2
e R = 0.9878 1
cal i0.8 D n y pt O 0.6 0.4 0.2
0
00.511.522.5
Percentage of Roundup Ready Soy Powder in Standard Extract
Figure 5: OD at 450 nm of Standard Extracts in ELISA for CP4-EPSPS Protein
The optical densities of all samples (standard extracts, soy, and chocolate) were then rescaled to the baseline by subtracting the optical density of the 0% Standard Extract from every optical density value (Table 5). Once rescaled, positive optical densities may indicate the presence of Roundup Ready soy in the sample, while negative optical densities do not. However, due to the margin of error, very small optical density values (both positive and negative) can be considered inconclusive (”noise”). The greater the magnitude of positive optical density, the more Roundup Ready soy protein the sample contains, and the more definitively it can be claimed that it the sample is indeed genetically modified for CP4-EPSPS expression.
ELISA analyses were also conducted for the five control samples of soybeans and soy flour. It was already known that Sigma® soy flour, unknown soybeans, and the field sample soybeans were positive controls containing Roundup Ready soy, and it was also known that Hodgson Mill® soy flour and the non-GMO soybeans had no genetic modification and thus served as negative controls. These samples were therefore tested to confirm test accuracy. (Table 6.) As expected, Page 22 Bhaskar, Hanzok, Kedar, Lu, Oniskey, Punati, Qin, Reddy, Shoemaker, Singh, Viswanathan, Young the three positive controls had positive signals, which fell approximately halfway between the optical density values of the 0.1% Standard Extract and the 1% Standard Extract, suggesting that they contained less Roundup Ready soy protein than did the 1% standards. The Hodgson Mill® (Organic) soy flour had a negative signal, and the non-GMO soybean had a signal near zero (within the margin of error).
Sample OD OD, rescaled Sigma ® Soy Flour (GMO +) 0.666 0.586 Dekalb ® Unknown (GMO +) 0.732 0.652 Field Sample (GMO +) 0.513 0.433 Hodgson Mill ® Soy Flour (GMO -) 0.066 -0.014 Non-GMO Soybean (GMO -) 0.084 0.004
Table 6: Optical Density at 450 nm of Soy Controls in ELISA for CP4-EPSPS Protein
The twelve chocolate samples were also assayed by ELISA. Time and resources also allowed a second trial to be conducted on eight of the twelve samples. (Table 7.) The samples for which a second trial was conducted included a variety of domestic and foreign chocolates, as well as milk, dark, and white chocolates. Samples with multiple trials were rescaled to baseline and averaged. Relative to the standard extracts and positive soy controls, all chocolate samples had very low optical densities (Figure 6). The chocolate sample with the highest optical density, Michel Cluizel®, had an optical density less than half of that of the 0.1% Standard Extract, (indicating very low levels of CP4-EPSPS protein). While all samples had low signal, five chocolate samples had optical densities noticeably greater than the others: Michel Cluizel®, Dolfin®, Scharffen Berger®, Ghirardelli®, and Lindt® Dark (ranging from 0.0155 to 0.061) (Figure 2).
Journal of the PGSS Detection and Identification of Genetically Modified Soy Page 23 (Glycine max) in Assorted Chocolate Food Products
Trial 1 Trial 2 Average Sample OD, rescaled OD, rescaled OD, rescaled Michel Cluizel® 0.075 0.047 0.061 Dolfin® 0.027 0.024 0.0255
Lindt® Dark 0.051 0.051
Ghirardelli® 0.012 0.019 0.0155 Scharffen Berger® 0.022 0.022 G&B® White 0.010 0.005 0.0075 G&B® Milk 0.005 0.003 0.004 Hershey’s® Milk 0.006 0.011 0.0085
Lindt® White Truffle 0.008 0.008
Nestle® 0.009 0.002 0.0055 Dove® 0.005 0.004 0.0045 Cacao Reserve® 0.004 0.004
Table 7: OD (450 nm) of Chocolate Samples in ELISA for CP4-EPSPS Protein
Optical Densities of Standard Extracts, Soy Controls, and Chocolate Samples
2.000 1.800 1.600 1.400 1.200 1.000 Trial 1 0.800 Trial 2 0.600
Optical Density 0.400 0.200 0.000 -0.200
Dove® Dolfin® Nestle®
Milk® G&B Ghirardelli® Lindt® Dark Lindt® G&B White®
Michel Cluizel® Hershey's® Milk Hershey's® Cacao Reserve® Scharffen Berger® Scharffen
Truffle White Lindt® 0% Standard Extract Standard 0% Extract Standard 1% Extract Standard 2% 0.1% Standard Extract Standard 0.1% +) Field Sample (GMO HodgsonMill® Soy Flour Sigma® Soy Flour (GMO +) Sigma® (GMO Soy Flour Dekalb® Unknown (GMO +) Dekalb® Unknown(GMO
-) (GMO Soybean Non-GMO
Figure 6: OD at 450 nm of all Samples in ELISA for CP4-EPSPS Protein Page 24 Bhaskar, Hanzok, Kedar, Lu, Oniskey, Punati, Qin, Reddy, Shoemaker, Singh, Viswanathan, Young
Optical Densities of Chocolate Samples
0.080
0.070
0.060
y 0.050
nsit Trial 1 0.040
l De Trial 2 tica 0.030
Op 0.020 0.010 0.000
® k ilk fle e® r ite® M lli® le® v ruf e st Da rd Dolfin T a Ne Do t® ite r d &B Wh G&B Milk® Reserve®in G Wh Ghi o L ershey's® Michel Cluizel® H ® harffen Berger® dt c Caca in S L
Figure 2: OD at 450 nm of Chocolate Samples in ELISA for CP4-EPSPS Protein
IV. Discussion
This project analyzed soy and chocolate controls for GM sequences and proteins. Polymerase Chain Reaction (PCR) analysis was conducted on both the control soy samples and on the chocolate samples using several primers: endogenous soy lectin, -tubulin, the positive controls, and CaMV35S, NOS terminus, CP4-EPSPS, NOS/EPSPS junction, and NPTII. Endogenous soy lectin and -tubulin served as positive controls, to check for the quality of the DNA sample, while CaMV35S, CP4-EPSPS, NOS/EPSPS junction, NOS terminus, and NPTII as experimental primers, designed to check for GM sequences.
Starting with sample DNA, the PCR was run with the NPTII primer, endogenous soy lectin, or CaMV35S primer. As a control procedure, the DNA is tested for endogenous soy lectin. Lectins Journal of the PGSS Detection and Identification of Genetically Modified Soy Page 25 (Glycine max) in Assorted Chocolate Food Products are a class of proteins found in plants and animals, especially in legumes like soy that bind to carbohydrates28, 29. The sequence for endogenous soy lectin is specific for soybeans, thus is utilized as a positive control to confirm that the DNA sample contains soybean DNA. If the tests are positive, then it can be said that the food sample contains some form of soy, genetically modified or not; if negative, the sample is examined with the -tubulin primer to make sure that there is plant DNA. -tubulin is a globular protein together with -tubulin form microtubules in eukaryotic cells30. The -tubulin primers that are being used in the experiment are plant specific, and thus indicate that the DNA samples contain plant-derived DNA.
The experimental primers utilized include the promoter CaMV35S, transgene CP4-EPSPS, junction NOS/EPSPS, NOS terminus, and NPTII selectable marker. CaMV35S, a promoter sequence, is the most widely used promoter for genetic modification and was the first primer to be checked with all the samples. The CaMV35S primer is a non-specific promoter used by the Monsanto Roundup Ready gene. A positive test for CaMV35S promoter indicates further testing for the NOS terminus and the CP4-EPSPS primers is justified. The CP4-EPSPS primer set is specific for the Roundup Ready transgene in soy. The NOS terminus is a non-specific terminator sequence. The confirmation of the two primer sequences would strongly support a GMO positive for soybean. Separate sequences from the CaMV35S promoter, the NOS terminus, the EPSPS (CP4-EPSPS) series, and the NOS/EPSPS junction are all part of the most common genetically modified sequence for soy, Roundup Ready31. Positive signals for both the NOS terminus and the CP4-EPSPS primer warranted a final test for NOS/EPSPS junction, a series of codons that join the NOS terminus and the CP4-EPSPS sequence. However, a sample may not have all three segments of the Roundup Ready modification; so it may only test positive for one of the three segments. A negative test for CaMV35S promoter led to PCR on the NPTII kanamycin resistance gene. The NPTII gene codes for kanamycin (antibiotic) resistance and is used as a selectable marker that helps to ensure that a foreign gene is inserted into a sample’s genome. The soybean genome does not normally contain the kanamycin resistance gene so its presence indicates genetic modification. If neither soy lectin nor -tubulin is present, it is clear that there is no workable DNA in the sample and that DNA isolation techniques may need improvement. It is worth noting that even if the sample was negative for the CaMV35S promoter, there may still be the Roundup Ready gene in the DNA as long as there is soy or plant DNA present in the sample. This is possible because a promoter other than the CaMV35S promoter may have been used or due to false negatives. Lastly, samples confirmed to contain the EPSPS sequence with the CP4- EPSPS primer set have ELISA performed to test for the EPSPS gene-produced protein.
At the beginning of the experiment, the soy control samples were tested to ensure that the PCR methods and procedures were correct. The soy control samples included field sample soybean, Page 26 Bhaskar, Hanzok, Kedar, Lu, Oniskey, Punati, Qin, Reddy, Shoemaker, Singh, Viswanathan, Young non-GMO soybean, the unknown soybean, Hodgson Mill® organic soy flour and the Sigma® soy flour.
Soy Sample Positive Control GM Sequences GMO +/- CaMV35S, EPSPS, NOS terminus, Field Sample Soybean lectin GMO + NOS junction Sigma® Soy flour -tubulin, lectin CaMV35S, NPTII, EPSPS GMO +
Dekalb® Unknown Soybean -tubulin, lectin CaMV35S, NPTII, EPSPS GMO +
Hodgson Mill® Organic flour lectin CaMV35S, EPSPS GMO -
Table 8: PCR results for Control Soy Samples
As shown in Table 8, the field sample soybean tested positive by PCR for the endogenous soy lectin confirming that it contained soy-derived DNA. It also tested positive for CaMV35S, EPSPS, the NOS terminus, and the NOS/EPSPS junction. These results give clear evidence that the field sample is genetically modified as was expected from a positive control. The non-GMO soybean tested slightly positive for the CaMV35S promoter. This was the only positive signal for the non- GMO soybean, and it is not clear whether the non-GMO soybean is genetically modified or not. Considering the previous results by other researchers and the weak signal, these results are probably due to contamination during DNA isolation or loading errors in gel electrophoresis analysis. The unknown soybean sample tested positive for endogenous soy lectin, -tubulin, CaMV35S promoter, NPTII, and Roundup Ready EPSPS. Because the unknown sample had positive signal for these sequences, it can be said with confidence that the unknown soybean is genetically modified. The Hodgson Mill® organic flour tested slightly positive for CaMV35S promoter and it is actually unclear from the gel run whether or not the organic flour has a band that corresponds to the CaMV35S promoter. This is most likely an error in the procedure for DNA isolation or loading error, because the Hodgson Mill® flour is known to be organic. The Sigma® soy flour tested positive for endogenous soy lectin, -tubulin, CaMV35S promoter, NPTII, and EPSPS, suggesting the hypothesis that the sample contained GM soy. The ELISA analysis supported these results for the control samples.
The ELISA tests for CP4-EPSPS protein were also performed on the soy control samples to detect GM at the protein level and complement the PCR data.
Sample ELISA result GMO +/- Sigma® Soy Flour positive GMO + Dekalb® Unknown positive GMO + Journal of the PGSS Detection and Identification of Genetically Modified Soy Page 27 (Glycine max) in Assorted Chocolate Food Products
Field Sample positive GMO + Hodgson Mill® Soy Flour negative GMO - Table 9: ELISA results for Control Soy Samples
The field sample soybean had a positive signal for CP4-EPSPS indicating the sample was consistent with the PCR data, as shown in Table 9. The optical density absorbance (OD) levels were significant and strengthen the argument for the field sample soybean being GMO positive. The Sigma® soy flour also had positive signal for the CP4-EPSPS of GM soy, as did the unknown soybean, and were similarly consistent with the PCR data. The Hodgson Mill® organic soy flour, which tested positive for the CaMV35S DNA in PCR tests, tested negative for CP4- EPSPS protein in the ELISA test. The non-GMO soybean had questionable results for the PCR tests, and tested negative for CP4-EPSPS in the ELISA test. This suggests that the positive results for CaMV35S promoter in the PCR tests was most likely due to contamination during DNA isolation or loading errors.
The control soy samples were confirmed to be either genetically modified or not by both PCR and ELISA. The chocolate samples were then investigated, first using PCR to amplify target DNA sequences. Four primers were used for each type of chocolate: endogenous soy lectin, CaMV35S promoter, NPTII, and CP4-EPSPS.
Chocolate Sample Positive control Sequences GMO +/- Michel Cluizel® Lectin CaMV35S, NPTII, EPSPS GMO - Dolfin® Lectin CaMV35S, NPTII, EPSPS GMO - Ghirardelli® Lectin CaMV35S, NPTII, EPSPS GMO - Lindt® Dark Lectin CaMV35S, NPTII, EPSPS GMO - Scharffen Berger® Lectin CaMV35S, NPTII, EPSPS GMO +? G&B® White Lectin CaMV35S, NPTII, EPSPS GMO - G&B® White Lectin CaMV35S, NPTII, EPSPS GMO - Hershey's® Milk Lectin CaMV35S, NPTII, EPSPS GMO +? Lindt® White Truffle Lectin CaMV35S, NPTII, EPSPS GMO - Nestle® Lectin CaMV35S, NPTII, EPSPS GMO - Dove® Lectin CaMV35S, NPTII, EPSPS GMO - Hershey’s® Cacao Reserve Lectin CaMV35S, NPTII, EPSPS GMO - Table 10: PCR results for Chocolate Samples
Page 28 Bhaskar, Hanzok, Kedar, Lu, Oniskey, Punati, Qin, Reddy, Shoemaker, Singh, Viswanathan, Young
The PCR results from the chocolate samples (Table 10) show for both Hershey’s® milk chocolate and Scharffen Berger® chocolate, the sample DNA was negative for lectin, CaMV35S, and CP4- EPSPS, but was positive for NPTII indicating that both chocolates had the kanamycin resistance gene in an ingredient. However, Michel Cluizel®, Dolfin®, Lindt® Dark, Ghirardelli®, Green & Black’s® White, Green & Black’s® Milk, Lindt® White Truffle, Nestle®, Dove®, and Hershey’s® Cacao Reserve had negative signals for all primers tested. While genetically modified sequences were detected, the results are questionable because although most chocolate samples are known to contain soy, the PCR results instead failed to amplify enough soy DNA to make any conclusions about the presence of soy in the chocolate.
The most plausible explanation is that insufficient DNA was isolated from the chocolate samples during DNA extraction, as evidenced by spectrophotometer analysis and the PCR. Spectrophotometer readings of the DNA isolated showed minimal or undetectable quantities of DNA. In an ideal DNA yield there are spike peaks at 260 and 280 nanometer wavelength since DNA has peak absorbance at 260 and protein absorbs at 280 nm as measured by spectrophotometer. The samples isolated from chocolate had no peak at these wavelengths; rather, the absorbance is actually less than that observed for the baseline, leading to the conclusion that not enough DNA was isolated.
The hypothesis that almost no DNA was isolated was strengthened when PCR results from the chocolate samples were analzyed. Most of the samples produced only primer dimers, small fragments that migrate ahead of the molecular weight marker. Primer dimers form when there is no complementary sequence for the primers to bind to, as is the case when there is a insufficient DNA. Thus, these results lead to the conclusion that insufficient DNA was isolated, particularly for the soy DNA from chocolate products.
ELISA is a procedure that tests for a specific target protein in an extract from sample. Like the PCR procedure, the ELISA results were as expected for the control soy samples.
Journal of the PGSS Detection and Identification of Genetically Modified Soy Page 29 (Glycine max) in Assorted Chocolate Food Products
Sample ELISA Result GMO +/- Michel Cluizel® Slightly Positive GMO +? ® Dolfin Slightly Positive GMO +? Lindt® Dark Slightly Positive GMO +? Ghirardelli® Slightly Positive GMO +? Scharffen Berger® Slightly Positive GMO +? G & B® White Negative GMO - ® G & B Milk Negative GMO - Hershey’s® Milk Negative GMO - Lindt® White Truffle Negative GMO - ® Nestle Negative GMO - Dove® Negative GMO - ® Hershey’s Cacao Reserve Negative GMO -
Table 11: ELISA results for Chocolate Samples
When targeting EPSPS proteins with ELISA, five samples of chocolate, Michel Cluizel®, Dolfin®, Lindt® Dark, Ghirardelli®, and Scharffen Berger® were determined to have slightly positive signals; thus, these five samples may contain genetically modified soy, as shown in Table 11. However, the evidence is not strong enough to confirm GM in any of the chocolate samples with positive signals. Alternatively, Green & Black’s White®, Green & Black’s Milk®, Hershey’s® Milk, Lindt® White Truffle, Nestle®, Dove®, and Hershey’s® Cacao Reserve all had negative signals, but again, it is unclear whether these are truly non-GM or not because of the nature of chocolate processing. Chocolate is highly processed and the DNA goes through many serious heating and grinding, which can destroy DNA and protein.
There are several possible sources of error in our experimental results. The results for all the chocolate samples in the PCR test could have been affected by the human error during the DNA isolation process. Insufficient cleaning of the mortars and pestles used to grind the samples may have led to cross-contamination. The integration of genetically modified non-soy ingredients in the chocolate may have caused the PCR to indicate a GM sequence which was not linked to soybeans. This suspicion is exemplified by the NPTII positive signal in Hershey’s® Milk chocolate and Scharffen Berger®. (Figure 2) This band which signifies the presence of the kanamycin selection marker, but the source may very well be from a genetically modified substance other than soy. Hershey’s® Milk tested negative for CP4-EPSPS protein in the ELISA tests and all other GM genes in PCR, yet this kanamycin band was present. The most likely cause for the insufficient DNA can be attributed to how the chocolate itself is manufactured and processed. There was most likely insufficient DNA because a majority of the DNA was lost or denatured during the process of chocolate making. The fermenting, roasting, and grinding can potentially denature both DNA and protein in the chocolate samples, which can lead to poor PCR and ELISA Page 30 Bhaskar, Hanzok, Kedar, Lu, Oniskey, Punati, Qin, Reddy, Shoemaker, Singh, Viswanathan, Young results. Finally the time constraints on the research do not allow retesting of many of the trials and hypotheses and there is also little time to modify protocols. Additional time to conduct modified PCR and ELISA trials in the laboratory may clarify these results.
In future investigations, the most important procedural change would be to isolate more DNA, either by using more sample or by increasing reaction volume. In the DNA isolation procedure instead of preparing only 0.2 grams of sample for centrifuging into the spin column, 0.5 grams or even 1 gram of sample can be prepared to be loaded onto the same spin column and thus collect more DNA. Alternatively, more lysis buffer than indicated by the procedure could be added in order to more thoroughly dissolve the sample, and possibly allow for more DNA to be extracted. In order to improve ELISA results, the EPSPS test can be repeated; hundreds of trials could reliably determine whether the five samples of chocolate were in fact GMO positive or if the signals were experimental noise. There are also many different ELISA kits for various GM- associated proteins other than CP4-EPSPS-protein available. In addition to optimization and repetition, other known genetically modified sequences can be tested, such as -glucuronidase (GUS), a non-specific plant marker, and cowpea trypsin inhibitor (CpTI), an insect resistance gene32. Another option is using different primers of another part of the DNA sequence for the same CaMV35S promoter or even the EPSPS Roundup Ready gene.
In summary most of the results proved inconclusive because of insufficient DNA yield from the DNA isolations. This was due to not only the manufacturing process of chocolate where a majority of the DNA and proteins are denatured through heating, grinding and other methods, but also the low quantities of soy in the chocolate samples. Many chocolates do contain soy, but it is not the main ingredient, which not only lowers soy DNA yield but also soy proteins. The minimal amounts of DNA isolated was confirmed by spectrophotometry and PCR. However, Hershey’s® Milk and Scharffen Berger®, tested positive for NPTII, the kanamycin selectable marker; but because NPTII is non-specific, it is not clear whether the genetic modification is from the soy or from some other genetically modified ingredient such as cocoa, or a contaminating material during manufacturing. In the ELISA, five chocolates tested slightly positive for the CP4-EPSPS protein. Of these, Michel Cluizel®, Dolfin®, and Lindt® Dark were all European and labeled GMO free, yet tested weakly positive for GM protein. Ghirardelli®, Scharffen Berger®, and Hershey’s® Milk were manufactured in the United States and tested weakly positive for genetic modification. These results indicate that while European chocolate is almost exclusively labeled GMO-free, there are products that may contain GM ingredients. While domestic chocolates make few claims about genetic modifications, several products may also contain GM ingredients. In the end, while these results are not conclusive, there is sufficient evidence that further testing can clarify the presence of genetically modified soy in chocolate products. Journal of the PGSS Detection and Identification of Genetically Modified Soy Page 31 (Glycine max) in Assorted Chocolate Food Products
Appendix A: Tables and Figures
Primer BP# Sequence Fragment Description Length
-tubulinF 20 GAG AGT GCA TCT CCA TCC AT endogenous
-tubulinR 19 CTG CTC AGG GTG GAA GAG T plant -tubulin
SOYBEAN LEC3119F 21 GGG TGA GGA TAG GGT TCT CTG 210 bp endogenous SOYBEAN 21 GCG ATC GAG TAG TGA GAG TCG 210 bp soy lectin LEC319R
CaMV35SF 19 GCT CCT ACA AAT GCC ATC A 195 bp CaMV CaMV35SR 20 GAT AGT GGG ATT GTG CGT CA 195 bp promoter
GCC AAT GAA CTG CAG GAC GAG NPTIF EK 26 411bp kanamycin GC NPTII R EK 26 411bp resistance GCA GGC ATC GGT CAC GAC GA
CP4- EPSPSF356BP 22 TGG CGC CCA AAG CTT GCA TGG C 356 bp Roundup CP4- 24 CCC CAA GTT CCT AAA TCT TCA AGT 356 bp Ready EPSPSR356BP
NOSterF 20 GAA TCC TGT TGC CGG TGT TG terminator 125 bp NOSterR 23 GCG GGC CTC TAA TCA TAA AAA CC sequence
NOS/CP4-EPSPSF 20 GCG CGG TGT CAT CTA TGT TA RR/terminator 291 bp NOS/CP4-EPSPSR 19 AAT CGT AGA CCC CGA CGA G junction
Table 2: Chart of Primers Utilized in PCR
This chart shows the primers used in the experiment. The first column shows the primer, where “forward” and “reverse” are abbreviated as “F” and “R”. The second column states the length of the primer in base pairs (bp). The rest of the chart describes the nucleotide sequence, length of strand, and function of the primer27, 33.
Page 32 Bhaskar, Hanzok, Kedar, Lu, Oniskey, Punati, Qin, Reddy, Shoemaker, Singh, Viswanathan, Young
1: Molecular Weight Marker 2: Field Sample- Endogenous Lectin (+) 3: Field Sample- 145 bp soybean EPSPS (+) 4: Field Sample- 356 bp for Roundup Ready (+) 5: Field Sample - Endogenous Lectin (+) 6: Organic Flour- Endogenous Lectin (NO SIGNAL) 7: Organic Flour- 145 bp soybean EPSPS (-) 8: Organic Flour- 356 bp for Roundup Ready (-) 9: Organic Flour- Endogenous Lectin (+)
Figure 6: PCR Gel Photo of Field Sample Soybean and Organic Soy Flour
1 Adamson, Anne E. “Genetically Modified Foods and Organisms.” Human Genome Project. 20 July 2007. http://www.ornl.gov/sci/techresources/Human_Genome/elsi/gmfood.shtml 2 Chen, Ines and David Dubnau. “DNA Uptake during Bacterial Transformation.” Nature Reviews Microbiology. March 2004. 3 Watkinson, A.R. et. al. "Predictions of Biodiversity Response to Genetically Modified Herbicide-Tolerant Crops." Science 1 September 2000. Vol. 289 no. 5484, pp.1554-1557. 4 Takada, Norisha, and Philip E. Nelson. “Pectin-Protein Interaction in Tomato Products” Journal of Food Science. September 1983, pp.1408-1411. 5 Paine et al. 2005. Improving the nutritional value of Golden Rice through increased pro-vitamin A content. Nature Biotechnology doi:10.1038/nbt1082 Journal of the PGSS Detection and Identification of Genetically Modified Soy Page 33 (Glycine max) in Assorted Chocolate Food Products
6 Welch, R.M. and Graham, R.D. “Breeding for Micronutrients in Staple Food Crops from a Human Nutrition Perspective.” 2007. http://www.goldenrice.org 7 Feil, B., U. Weingartner and P. Stamp. “Controlling the release of pollen from genetically modified maize and increasing its grain yield by growing mixtures of male-sterile and male-fertile plants.” Euphytica March 2003. 163-165. 8 “Dolfin Chocolate – Belgium.” Chocosphere 2007. 20 July 2007. http://www.chocosphere.com/Html/Products/dolfin.html 9“Adoption of Genetically Engineered Crops Grows Steadily in the U.S.” 5 July 2007. United States Department of Agriculture
26 Seidman, Lisa, and Jeanette Mowery. "Spectrophotometry." *The Biotechnology Project*. 25 Sep. 2006. National Science Foundation. 27 July 2007
29 “Plant Lectins.” 4 Oct 2001. Animal Science at Cornell University. http://www.ansci.cornell.edu/plant/toxicagents/lectins/lectins.html#lectfood. 30 Cooper, Geoffrey M. *The Cell: a Molecular Approach*. 2nd ed. Sunderland: Sinauer Associates, Inc., 2000. *National Center for Biotechnology Information*. 21 July 2007 < http://www.ncbi.nlm.nih.gov/books/bv. fcgi?db=Books&rid=cooper.section.1820>. 31 Windels, Pieter, Isabel Taverniers, Ann Depicker, Erik van Bockstaele, and Marc de Loose. “Characterisation of the Roundup Ready soybean insert.” Eur Food Res Techol 2001. pp.107-112. 32 Tao, Zhen, Xing-feng Cai, Sheng-Li Yang, and Yi Gong. "Detection of Exogenous Genes in Genetically Modified Plants With Multiplex Polymerase Chain Reaction." Plant Molecular Biology Reporter 19 (2001): 289-98. 23 July 2007
Determination of the midpoint of adolescent myelin maturation as measured by change in radial diffusivity in diffusion tensor imaging
Ayse Baybars, Jessica Chang, Yihui (Connie) Jiang, Lauren Onofrey, Naveen Ponnappa, Elena Ponte, Archana Ramgopal, Christine Shen, Alexandra Trevisan, Stanley G. Zheng, Sera Thornton and Robert Terwilliger
Abstract
The purpose of this investigation was to determine the midpoint of adolescent myelin maturation through a technique known as diffusion tensor imaging (DTI), which is a specific type of MRI. An indirect correlation between radial diffusivity in axons and myelination has been proposed and supported in scientific literature. The DTI has the unique ability to provide details on the physiology of the brain in terms of radial diffusion and diffusion direction, which made it possible to compare the differences in these properties between the brains of many individual subjects in this study. The data retrieved during this experiment, along with the compilation of data from the Laboratory of Neurocognitive Development (LNCD), will help to create a template for tracing normal development of the brain in terms of myelin maturation.
The processes utilized in this experiment were mainly computational. DTI scans of a group of brains were combined and analyzed using FSL (FMRIB Software Library) and MedINRIA (Medical Image Navigation and Research Tool). The brains was regenerated and depicted as a mass of voxels (volumetric pixels), and the different distributions and intensities of these voxels were compared between individuals to generate histograms. Statistical analysis was then utilized to compare the different subjects’ histograms by three different methods, each exhibiting the same result.
This experiment suggests that the thirteen year olds are at the midpoint of adolescent myelin maturation. If the assumption of the correlation between radial diffusivity and myelination holds true, then the pre-adolescence to the thirteen-year-old age group had similar amounts of the difference in myelination as the thirteen-year-olds to adults. With this data, a more complete developmental curve of the brain can be plotted.
I. Introduction
A. Brain Physiology
1. Structure of the Neuron
The neuron consists of four components: the soma, the dendrites, the axon, and the terminal buttons. Similar to the cell bodies of other eukaryotic cells, the soma controls the life processes and the basic functions of neurons. Branching off of the soma, the dendrites receive the electrochemical signals from the synapse, or the space between the neurons. After the dendrites receive a nerve impulse, they transmit the signal to the soma, which in turn carries it to the axon. The axon then carries the signal to the terminal endings of the neuron. Finally, these small, knob-like structures release chemicals, called neurotransmitters, into the synapse. The dendrites of the postsynaptic Page 36 Baybars, Chang, Jiang, Onofrey, Ponnappa, Ponte, Ramgopal, Shen, Trevisan, Zheng
neuron receive the neurotransmitters, and the process begins again (Wakely 25-26).
Figure 1: Neuron Diagram (Young)
2. Gray And White Matter
Although the neuron is the core of signal transmission, other structures in the brain also affect cortical function. Tissues in the brain primarily fit into two categories: gray matter and white matter. Gray matter refers to the dendrites, soma, axons, and terminal endings, while white matter refers to blood vessels, glial cells, and myelin. Although gray matter is responsible for basic neural functioning, white matter is also essential for cortical processes and actually accounts for 40-50% of the cerebral volume. The blood vessels of white matter function like the blood vessels in the rest of the body: they provide oxygen and nutrients to cells. Specifically, cortical arteries branch from the cerebral and lenticulostriate arteries, then extend upward into the brain. Four different types of glial cells exist in the central nervous system: oligodendrocytes, astrocytes, ependymal cells, and microglia. The oligodendrocytes and astrocytes control the structure and function of white matter. Oligodendrocytes deposit myelin and wrap it around axons, while the astrocytes provide structural support. In addition, astrocytes make contact with other neurons at unmyelinated portions of the axon, or Nodes of Ranvier. Finally, myelin is a structure composed of lipids (70% composition) and proteins (30% composition). It amplifies the neural impulse and affects the speed, efficiency, and function of the signal (Filley 19-20). Journal of the PGSS Adolescent Myelin Maturation Page 37
3. Unmyelinated and Myelinated Neurons
In an unmyelinated neuron, the dendrites receive nerve impulses and conduct the signal to the soma. This impulse causes the neuron's potential state to increase from the resting potential of -65 mV. On the molecular level, the stimulation causes voltage-controlled sodium gates to open and sodium cations to permeate the membrane, thus, increasing membrane voltage. If the membrane potential exceeds the threshold of -55 mV, enough sodium gates open to cause the permeability of the membrane to become greater for sodium cations than for potassium cations. Due to this increased permeability and the negative potential inside the membrane, sodium ions undergo a rapid influx into the axon. The axon, consequently, becomes fully depolarized and has a potential between + 40 mV for depolarization and +55 mV for the equilibrium potential for sodium. After the axon has achieved the action potential, the sodium gates de-activate, leaving the sodium cations inside the membrane. Next, the voltage-gated potassium channels open and the potassium cations exit the cell membrane, creating once again a negative potential within the axon. In actuality, the axon briefly hyperpolarizes, falling below the resting potential of -65 mV; however, the membrane quickly restores itself to its resting potential (Lerner). This refractory period is a combination of two distinct processes: during the absolute refractory period, sodium cations enter the cell; during relative refractory period, potassium cations leave the cell. Once the resting potential ion concentrations are restored, the neuron is again ready to conduct an impulse (Long). This process is repeated along the axon from the soma to the terminal buttons because the depolarization of one segment of the axon propagates depolarization in the next segment. In this way, the signal travels down the entire length of the axon (Buehler).
4. Myelinated Neurons
In a myelinated neuron, the neuron signaling process is essentially the same; however, the myelin increases the rate of impulse transmission. The specific form of myelinated axons is correlated to its function. Myelin does not cover the entire axon, but instead, covers 1-2 mm segments which are interrupted by 2 mm Nodes of Ranvier. In the covered portions, the myelin acts as a layer of insulation against the ions in the extracellular fluid; consequently, sodium and potassium cations do not cross the internodal axonal membrane. Regardless of these inactive regions along the axon, the signal must travel from the cell body to the terminal buttons. As a result, the signal “jumps” from one node to the next. Specifically, when an impulse begins to travel down the axon of a myelinated neuron, the transfer of ions will occur at the first node, and the change in potential there will offset a change at the next node instead of at an adjacent region. Thus, the myelinated axon is analogous to a row of dominoes spaced farther apart yet still able to hit each other at the same rate: because fewer dominoes are used, less time is required to knock down the same length of domino chain. In the axon, because the signal can travel from node to node rather than by passing every segment of the axon, the rate of transmission is greatly increased. Furthermore, myelin reduces the overall amount of energy in the form of adenosine triphosphate (ATP) required for transmission of the impulse. Since ion pumps only function at the Nodes of Ranvier and not at the internodal axon, the propagation of a signal down a myelinated axon requires less ATP. The effects of myelination are quite significant: signals travel up to 100 times faster through a myelinated axon than they travel through an unmyelinated axon (Halici). Page 38 Baybars, Chang, Jiang, Onofrey, Ponnappa, Ponte, Ramgopal, Shen, Trevisan, Zheng
5. Development of Myelin
In contrast with gray matter, white matter maturation occurs throughout life. Surprisingly, although gray matter is fully developed at birth, white matter is still only 90% developed by the age of two years. Recent evidence even suggests that myelination continues into the sixth decade of life (Filley 33). White matter development follows a natural course in evolutionary terms: the brainstem and the cerebellum were the first structures to develop, and only later did the cortex come to comprise the largest portion of the human brain. Accordingly, myelin development begins in the caudal region, near the spine, and continues in the rostral direction towards the forehead. Recent research has shown that myelination correlates strongly with reasoning, impulse control, and judgment. Understandably, the frontal cortex controls these processes. Myelination in the cortex also seems to be related to maturation, as suggested by the observation that children with delayed myelination in the frontal lobe often suffer cognitive impairment (Filley 33).
6. White Matter Tracts
Axons throughout the cortical region can be myelinated. However, there exist regions where myelinated axons extend through the cortex together, in bundles or tracts. These tracts connect cortical and subcortical structures, ranging from 1mm to 1m in length and from 0.2 to 20 micrometers in width. The three different types of white matter tracts are characterized by their different functions: projection, commissural, and association respectively. Projection tracts include the corticopetal and the corticofugal tracts. The corticopetal tracts connect the spinal cord to the cortex while the corticofugal connects the brain to other structures. Commissural tracts link the two hemispheres; for instance, the corpus callosum, the major bundle of white matter between the right and left hemispheres, is a commissural tract. The association tracts connect cerebral areas within the hemispheres (Gleitman).
7. The Promise of White Matter Tracts in Research
These white matter tracts might be a kind of window into the development and function of white matter in the brain. In the recent past, the field of neurology has understood very little about white matter. Limited information came from the study of monkey cortexes, and only very recently have researchers employed neuroanatomic, neuroimaging, electrophysiological, and computational techniques to investigate white matter. These methods can utilize the data on white matter tracts because the bundles of axons reveal properties very different from those found in gray matter. (Filley 27).
B. Radial Diffusivity
In any biological system, water diffusion serves as one of the primary methods of transport for chemical signals. Specifically, in the brain, water diffusion transmits electrochemical signals that are essential for cerebral functions. Evidence for this assertion comes from the substantial size of the ventricles, or bodies of cerebro-spinal fluid, in the cortex. The ventricles foster isotropic diffusion, meaning that diffusion takes place in a uniform manner in all directions (see Figure 1). However, the entire cortex requires water diffusion, and thus the ventricles are not the only reservoir of fluid in the Journal of the PGSS Adolescent Myelin Maturation Page 39
brain. Diffusion also occurs throughout the myelinated axon and glial cell portions of the white matter, and throughout the gray matter. In the white matter tracts, particularly, the presence of myelin greatly restricts water movement. The presence of the axons forces the water to diffuse in a primarily anisotropic manner because fluid encounters far greater resistance moving radially in reference to a white matter tract than it encounters moving lengthwise along the axon bundles (see Figure 2).
Figure 2: Isotropic diffusion
Figure 3: Anisotropic diffusion
There are two scalar measures of the diffusion of water that are correlated to neural maturation. The first, fractional anisotropy (FA), is a measure of what proportion of the total diffusion is anisotropic. In general, the degree of anisotropy is related to age, for the FA has been observed to increase as the brain matures. In the ventricles, the FA is approximately zero because the movement of water is isotropic, whereas the FA in the axons is close to 1 because the movement of water is highly directed. However, a second method is often preferred because it demonstrates a more significant difference with age than the FA exhibits. Radial diffusivity (RD), measured in meters2 / second, is a gauge of the amount of diffusion that occurs perpendicular to the axon. Therefore, a low radial diffusivity indicates that a higher proportion of the diffusion is along the length of the axon as opposed to perpendicular to it. In tensor imaging, each ellipsoid has three dimensions, the lengths of which are called Eigen values (λx).
Error! Page 40 Baybars, Chang, Jiang, Onofrey, Ponnappa, Ponte, Ramgopal, Shen, Trevisan, Zheng
λ2 λ1
λ3
Figure 4: Eigen values
2 2 2 3 ()λ1 − λ + ()λ2 − λ + ()λ3 − λ FA = 2 2 2 2 λ1 + λ2 + λ3
Where:
λ1 is the length of the major axis,
λ2 and λ3 are the lengths of the two minor axes, and
λ is the mean axis length, (λ1 + λ2+ λ3)/3
RD ≡ (λ2 + λ3)/2
See Appendix B for FA and RD brain spatial normalizations.
C. MRI vs. Histology and the Reasons for DTI
Both histology and MRI aid in the analysis of brain structure and function, and each method has unique strengths as well as shortcomings. Histology utilizes a microscope to reveal highly specific physiological processes in the brain on the two-dimensional level. However, by nature, histology is an invasive, ex vivo technique which destroys functioning tissue and necessitates a labor-intensive dyeing process. Conversely, MRI renders three-dimensional images without destroying living tissue. Unfortunately, MRI images sacrifice specificity and resolution in favor of speed and convenience. Resolution improves slightly if the scans are taken ex vivo, but the major concerns regarding loss of detail due to image condensation remain problematic (Mori 2006).
An MRI machine creates an image by detecting signals from hydrogen protons in water molecules. The machine applies a magnetic field to the subject and causes the protons to orient themselves along the axis of the field, either parallel or anti parallel. They also precess around the axis of the field. Journal of the PGSS Adolescent Myelin Maturation Page 41
Figure 5: Parallel / Antiparallel oriented protons in a magnetic field
Next, the machine emits a radio frequency pulse which causes a 90 degree flip onto the transverse plane, which is perpendicular to the magnetic field. Previously, the protons had the same longitudinal component of motion, but the transverse component varied. When the proton is flipped ninety degrees, the “new” transverse component is the same for all of the protons and the longitudinal component varies. This does not matter, as it is the transverse component that eventually determines the strength of the radio signal that is measured. This shift to the transverse plane makes the protons precess in phase, since they were already precessing at the same frequency. To sum up, the protons in the magnetic field after the 90 degree radio frequency pulse are all in phase. Then, a magnetic field gradient is applied (in addition to the previous magnetic field). This causes the magnet to become nonuniform, as a strong and weak side of the magnet are created due to the gradient. Protons closer to the strong side of the magnet begin to spin faster and the protons closer to the weak side of the magnet begin to spin slower. The proton precession is now out of phase. When the magnetic gradient is removed, the protons all return to the original uniform speed but they do so at different times. Thus, they precess at uniform speed but not in phase with each other. Then, a 180 degree radio frequency pulse is applied, which flips the entire transverse axis 180 degrees. This places the protons in a position opposite of their original location, so that if the same magnetic field gradient is applied, protons that were originally closer to the strong side of the magnet are now closer to the weak side and vice versa. The purpose of this second gradient application is to realign all of the protons to their original uniform orientation. Thus, the “slow” protons will precess faster the second time because they are now closer to the strong side of the magnet, and the “fast” protons will spin slower the second time because they are closer to the weak side of the magnet. When the second gradient stops, all protons should precess in phase just like the original state after the 90 degree flip (assuming no water diffusion). The magnetic field gradient should be applied on all axes – x, y, z, - in opposite directions in order to obtain six different gradients. The protons at the end of this process emit their own radio frequency that, in an idealistic situation, equals the amount of radio frequency put into the system. However, water diffusion must be taken into account as this greatly affects the output radiofrequency of the protons. If protons move due to water diffusion between application of the first and second gradient application, then when the 180 degree radio frequency is Page 42 Baybars, Chang, Jiang, Onofrey, Ponnappa, Ponte, Ramgopal, Shen, Trevisan, Zheng
applied, the protons will not be in the exact opposite position of their original location. This will lead to under and overcompensation of proton phasing in the second magnetic field gradient and the protons will not realign in phase. The slight differences in the final spin states of the protons lowers the radio frequency emitted by them in the end. Thus, the radio frequency detected is much lower than the input radio frequency. The greater the water diffusion, the lower the output radio frequency. Comparing these outputs of the six different gradients allows the formation of diffusion weighted images (DWI), which can explain the physiological aspects of the brain.
Diffusion tensor imaging (DTI), a form of MRI, shows the physiology of the brain, instead of the anatomical structure. DTI can apply various magnetic fields in order to calculate a tensor, or an ellipsoid, for every three-dimensional volumetric element, or voxel, of the brain. Seven measurements describe each tensor: measurements along six axes and along a base axis without a specific gradient. The measurements define the ellipsoid once they are converted to parameters by a three-by-three matrix, also called a tensor. DTI specifically identifies diffusion patterns, and therefore, provides spatial descriptions of the medium under study (Kubicki 2002). These diffusion patterns, largely anisotropic, in the white matter of the brain are caused by from the presence of bundles of myelinated axonal fibers. The anisotropy occurs due to the restricted mobility of water in the white matter: movement is restricted perpendicular to the axon as a result of the tightly bound white matter tracts. Though myelin is not necessarily the only cause of anisotropic diffusion, it is generally assumed to have the greatest influence over the phenomenon (Kubicki 2002). The ability to detect, characterize, and map this diffusion anisotropy is a powerful feature of diffusion tensor imaging. In addition, a process called tractography creates the ellipsoids based on the measurements. This process is sensitive enough to be thrown off by minor interference; therefore, the process must be used with care. Since DTI has a low signal-to-noise ratio (SNR), signal averaging (the scanning time) must be increased or image resolution must be reduced in order to reduce the interference and noise (Mori 2006). Journal of the PGSS Adolescent Myelin Maturation Page 43
Figure 6: Diffusion Tensor Image of the Brain
D. Correlation between Radial Diffusivity and Myelination
Fluctuations in levels of radial diffusivity, defined as diffusion that occurs perpendicular to the axon (while axial diffusivity is diffusion that “goes with” the axon) has been shown to be correlated to the amount of myelination present in a brain. Myelin, which is primarily composed of lipids, is a substance that coats the axons of neurons, enabling electrochemical signals to be transmitted throughout the brain more efficiently. The more myelinated the axons are, the lower the radial diffusivity is expected to be. Thus, the water diffusion in the axons occurs as axial diffusivity. It is thought that the radial diffusivity will be lower as more myelin is present because the myelin’s nonpolar nature inhibits the highly polar water molecules from diffusing efficiently; the water then prefers to diffuse by following the path of the axons (axially), rather than going against the current of electrical flow (Beaulieu, 2002).
Studies have been performed in model organisms where myelination did indeed indicate lower levels of radial diffusivity. In an experiment performed on shiverer mice (those with a mutation that inhibits full central nervous system myelination) and wild-type mice, results showed that radial diffusivity was greater in the less-myelinated axons of the shiverer mice (Song 2002). Another experiment confirmed that the relatively unmyelinated axons of embryonic and neonatal mice exhibited very high levels of radial diffusivity as compared to the levels of radial diffusivity in adult (myelinated) mice. It was also Page 44 Baybars, Chang, Jiang, Onofrey, Ponnappa, Ponte, Ramgopal, Shen, Trevisan, Zheng
found that the level of radial diffusivity decreased as the mice matured and myelin development sky- rocketed (Mori 2006).
Experiments have confirmed that what held true for the anisotropic water diffusion of mice also held true for humans: radial diffusivity increases as humans mature. It was found that levels of radial diffusivity in newborns are much higher than in adults. In other words, the radial diffusion is much higher in those subjects that tend to have unmyelinated axons (children), while radial diffusivity occurs in lower levels in adult humans, just like in adult mice (Neil 1998).
E. The Purpose of Statistical Analysis
When given a limited amount of data, one needs to be able to determine whether or not the results are significant in the scientific realm or if they are simply due to random variability. Statistical analysis covers this purpose. The following three values are the cornerstones as used in scientific analysis: the p-value, the z-statistic or score, and the t-test and its value, the t-statistic. Furthermore, there exists both parametric and nonparametric data analysis. Parametric analysis assumes an equal distribution of data and bases the data on a normal curve. Nonparametric analysis, which was utilized in this study, uses the data itself, which is not equally distributed, to find the curve of the distribution.
Figure 7: Nonparametric Distributions
1. P-Value
The p-value represents the probability that the results obtained were due to chance alone. An alpha level is a measure often used in an analysis to denote relative statistical significance. If the p-value is Journal of the PGSS Adolescent Myelin Maturation Page 45
less than the alpha level, in our case = 0.05, then the results are deemed significant. The alpha value is chosen based on the subject being studied and the values that have been shown to have relevance in similar studies in literature. The null hypothesis, usually denoted by H0, is used to establish an initial statement, which is tested against an alternate hypothesis, denoted as HA, to be disproved. The null hypothesis can not be proven unless it is under specific conditions, as lurking and confounding variables must first be taken into consideration.
The following example highlights the purpose of the p-value. The given null hypothesis states that the rate of growth is the same for all races. The alternative hypothesis states that the rate is not the same across all races. Two events can occur: the first is that after passing a statistical test of significance, the p-value will be less than 0.05, meaning that the probability that the results are due to chance alone is less than 5%. Thus, the results are considered significant at the alpha 0.05 level, and the null hypothesis is disproved in favor of the alternative hypothesis. The other possible outcome is that the p-value will be great, for example: 0.78, meaning that there is a 78% chance that the outcome of the test is due to chance alone. In this case there is not enough evidence to disprove the null hypothesis. The null hypothesis cannot be proven, as confounding variables not considered may be the cause of the outcome. In the two groups tested, one may contain only females and the other only males and the difference in rates might be due to this rather than just the rate of growth.
In the study presented, the p-value is used to represent the probability that the specific voxel has been paired with another voxel by chance.
2. Z-Statistic
The z-statistic represents the number of standard deviations over or below the mean that a value is located with respect to the entire data set. Standard deviation is defined by sigma, , and measures the spread of data. The following example illustrates the usage of the z-statistic in analysis. Suppose a normal distribution for IQ scores has a mean of 100 units, and the standard deviation has a value of 15 units. If it is concluded that an individual has an IQ score of 145 then to find the z-stat,
145 −100 z = 15
z = 3
The z-stat is found to be 3, meaning that an IQ of 145 is three standard deviations above the mean.
The z-statistic is a special case of the t-statistic. For infinite degrees of freedom, the t-stat values are z-statistics because the degrees of freedom no longer play a part in the values. The study conducted does not specifically use the z-stat, as the t-stat in this study has 10 degrees of freedom. It is, however, used to simplify the explanation of the t-statistic. Page 46 Baybars, Chang, Jiang, Onofrey, Ponnappa, Ponte, Ramgopal, Shen, Trevisan, Zheng
3. T-Test
The t-test is a method to organize raw data in order to convert it into a p-value. The t-test:
x − x t = 1 2 4 s N R
The t-test can be utilized by both parametric and nonparametric data. In the case of this study, as it requires a nonparametric analysis, the t-test is employed using the Wilcoxon Mann-Whitney rank sum test, a method developed in recent neuroimaging literature (NeuroImage 1531-1537). Wilcoxon Mann-Whitney organizes data by means of ranking it, so that it is less influenced by values beyond the average data range and thus uses the mean t-statistic to measure how greatly the results were affected. The Wilcoxon Mann-Whitney rank test does have its disadvantages. The test assumes that the groups have the same range and shape, and its results are only accurate for large experimental groups, thereby requiring extensive calculations. In the study conducted, the computer performed these calculations. In this study, we utilized the t-test as an intermediate step between the raw data and the p-value to find the probability that the experimental results occurred due to chance. For the case of an individual voxel, the data for all experimental subjects is compared in order to yield a distribution of their values, and the t-statistic uses this raw-data to translate the information into the syntax of the p-value.
F. Project Overview and Goals
The purpose of this study is to identify the age group that represents the midpoint of human myelin maturation. Two extreme groups were formed: a group of six kids aged eight, nine, and ten, and a group of six adults aged twenty-five, twenty-six, twenty-seven, and twenty-eight. These two groups were then compared to seven specific intermediate age groups ranging from eleven to seventeen year olds of six subjects each. By applying a statistical analysis of the radial diffusivities, comparisons were formed between each of the experimental groups and the extreme groups. To find the age group that represents the midpoint of white matter development, the absolute difference of t-stat values between a particular age group and kids, and that age group versus adults needed to be the minimum difference of all the variable age groups. In other words, this group has the most similar differences to the endpoint groups.
Three independent tests were conducted to achieve this goal, and each test added credence to the results determined in the other tests. The first of these tests analyzed the t-stat values of the radial diffusivities of individual voxels. The second test, however, involved cluster-based thresholding, holding the concept that the probability that a value occurred by chance alone decreases as the number of adjacent voxels analyzed increases. Finally, the median t-stat values in a cluster were analyzed, thereby eliminating any extraneous values that may occur in the second test. Journal of the PGSS Adolescent Myelin Maturation Page 47
G. Data Collection: The Bigger Study
The study being presented is in actuality a branch of a much larger study, dealing with the changes in cognitive development as a result of myelination. The goal of the larger study, conducted by the Laboratory of Neurocognitive Development (LNCD), was to compile a template to trace the pattern of normal development. The purpose of finding the optimal age of myelin growth was to see at which maturation stage an individual is more susceptible to the onset of neuropsychiatric disorders. During the critical period of myelination, and the onset of maturation as defined by the study, individuals are more prone to the allures of alcohol, drug abuse, etc.; thus, obtaining such information is important to establishing a proficient environment.
Furthermore, the LNCD study incorporated a comparison between the individuals with a normal pattern of myelination and the myelination patterns of autistic individuals. The study being presented is a culminating analysis of the database utilized by the LNCD study. The LNCD study, however, took a further step by accurately testing the cognitive development of primates in comparison to that of human beings, as a result of myelination. The study being presented takes into consideration the radial diffusivity, which is augmented as an onset of increased myelination. It also utilizes previous experimental data, and generally infers that increased myelination results in an increased maturation pattern.
It is therefore important to note that the study presented utilizes the collected data and samples gathered by the LNCD study, and the inferences and conclusions of this smaller study are directly correlated to the larger study.
II. Methods
A. Hardware and Software
A computational analysis of the sample brain images collected was conducted in this study. This experiment utilized Power PC-based Mac computers running O.S. 10.4, and the two programs used were FSL (FMRIB Software Library, developed by a group at Oxford University in June 2000) and MedINRIA (Medical Image Navigation and Research Tool). FSL is an interactive tool that projects data in the form of images and statistics through the command line software, Unix. It is a package that includes several functions, including FSL View, Randomise, and Avwutils, which were used in this study.
The FSLView was a critical component to comparing the brains. This program contains images in 3D and 4D form, where the fourth dimension of an image represents the various subject samples. The program can mask brain images, which means that images of two different brains can be stacked together, and only the voxels that are in common are displayed. Once masked, FSLView computes a t-stat and a p-stat and generates a histogram to show the differences between the two subjects. The most active areas of the brain can be observed using the three different view angles (axial, sagittal, and coronal); the user can view the brain in slices in all three dimensions. In addition, the tools can rotate, zoom, compare, and contrast specific sections of the brain to depict specific Page 48 Baybars, Chang, Jiang, Onofrey, Ponnappa, Ponte, Ramgopal, Shen, Trevisan, Zheng
voxels and to display the myelin. They also allow for manipulation of the color scale and brightness level of the image to highlight the structures and content of the brain. Entering specific coordinates for a voxel returns intensity values of that voxel.
The program FSL Randomise allows users to work with data without a null distribution. This program is essentially a simple permutation test that allows the user to obtain voxel and cluster-based values from tests in this case. Using the Monte Carlo permutation test, the Randomise program does not have to run through exhaustive permutation tests. Additionally, the Randomise program has a smoothing option to fix nonparametric graphs, so it is well suited to this study.
The AVWUtils are commands that process and convert data in both 3D and 4D data. The commands utilized for this project included: avwstats++, avwmerge, and avwmaths. Avwstats++ gives statistical summaries of 3D and 4D images by calculating the t-stat values for certain percentiles. Avwmerge takes several inputs and groups these values into one. For instance, the avwmerge can take multiple 2D inputs and convert these values into a single 3D output. This program was used in combining different sets of brain images within a certain age group. Avwmaths returns the maximum voxel count and a median t-stat value for a cluster of voxels.
The MedINRIA programs were not used exclusively in this study, but they provided helpful visuals for tensor imaging and tractography.
B. Preliminary image manipulation
For this study, a total of fifty-four brains were scanned and then grouped by age. In order to conduct the experiment, a group of kids and a group of adults needed to be obtained to create the two extreme age groups. These were then used as comparisons for each group of teenaged brains ranging from eleven year olds to seventeen year olds. These groups were comprised of:
Two Comparison groups: - Kids (08F, 08F, 08F, 08M, 09M, 10M) - Adults (25F, 25F, 26F, 25M, 27M, 28M) Teenage groups: - 11 year olds (3 females, 3 males) - 12 year olds (3 females, 3 males) - 13 year olds (3 females, 3 males) - 14 year olds (3 females, 3 males) - 15 year olds (3 females, 3 males) - 16 year olds (3 females, 3 males) - 17 year olds (3 females, 3 males)
In order to compare the brains to each other, they needed to be equalized. Each individual brain has a unique shape that is difficult to match closely to another brain. First, each scanned brain was regenerated into a mass of voxels. For the statistical comparisons to be possible, the brains needed Journal of the PGSS Adolescent Myelin Maturation Page 49
to have the same number of voxels in the same place. To achieve this, the FA spatial normalizations were produced, and of these, one subject was identified to be the closest in structure to all other sample brains. A template was derived from this brain on which to base the fifty-three other brains. The particular brain chosen was that of a sixteen-year-old female, but age and gender had no effect on the choice. It was chosen simply because it made the transfer of brains easiest. The fifty-three other brains were then transformed onto the FA spatial normalization by projecting each sample onto the template. From the fifty-four FA images generated, a mean FA spatial normalization was found. This mean was skeletonized to get the mean FA skeleton.
Once all the brains were equalized, the mean FA skeleton was set as a second template. Each brain was then skeletonized using this template, and at this point, the white matter tracks of all fifty-four brains were aligned. The result was that all the brains had the same number of voxels in both the FA spatial normalization and the FA skeleton, and the voxels all had the same placement within the structure. This was essential because the tests following this relied on comparing each voxel to its respective counterpart in another brain. However, the voxels continued to have unique t and p values that would be used in the statistical comparisons. Only the skeletons were statistically analyzed for t- stat values.
C. Statistical Testing
Once the groups had been compiled, tests for radial diffusivity on the individual voxel level could commence. Each teenage group, ranging from eleven to seventeen year olds, had to be compared to both of the extreme groups, kids and adults. Three tests were conducted in order to verify or falsify the results of each of the other tests.
The first method compared the radial diffusivities between brains on the individual voxel level. Each age group was merged with either kids or adults, forming a total of twelve subjects for each test. These were then randomized into 924 permutations to create all possible groupings for comparisons. The t-statistic values were found for each age group in combination with both the kid group and the adult group. The object was to find the t-stat differences that were closest to each other, meaning that the teenage group in question was the same level of difference from both the kids and the adults. Once the 924 permutations were generated, t-stat values were calculated for percentiles ranging from 5 to 95 (increasing in increments of 10). For each percentile, two absolute differences in t-stats were computed: between a particular age group and kids, and the same age group and adults. These differences were added and compared among all age groups in order to find the minimum value. The significance of the minimum value is that it corresponds to the smallest difference between the extremes, meaning that the age group lied in the middle of the maturation process.
A second test was held to verify the results of the first test. It involved cluster-based thresholding, which aims to lower the probability of collecting results due to chance by analyzing groups of contiguous voxels. It is more likely for an MRI to read one individual voxel incorrectly than for it to read a whole cluster of adjacent clusters incorrectly. A higher extent of clustering decreases the Page 50 Baybars, Chang, Jiang, Onofrey, Ponnappa, Ponte, Ramgopal, Shen, Trevisan, Zheng
probability that the values of the voxels are all due to chance. By thresholding, low values can be discarded while higher values can be more closely examined. The thresholding values in this case were a t-stat of 3.0 and a 1-p-stat of 0.7. Each voxel was compared to these thresholds, and new values were assigned as a function of this comparison. First, t-stats were recalculated. If the t-stat of a particular voxel was lower than 3.0, the value of that voxel would be set at 0.0 instead. If the value of the voxel was greater than or equal to 3.0, the voxel would retain its original value. Consequently, only values above 3.0 were compared in this calculation. Next, p-stats, or the figures representative of the absolute difference between 1 and the p-value, were recalculated. If a voxel had a p-stat less than 0.7, the value was set to 0.0; if the voxel had a p-stat greater than or equal to 0.7, it maintained its original value. Each cluster formed from all adjacent voxels of t-stat greater than or equal to 3.0. A procedure very similar to the one held after the first test was followed after this test. This time, the cluster size was considered. Once again, the minimum value of voxels per cluster was determined to be the middle age group of maturation.
A final test further confirmed previous results. Thresholding was again utilized; low t-stats and p-stats were essentially discarded. However, for the third method, median t-stats were compared. These t- stats were calculated for each cluster. The minimum t-stat emerged from a calculation of the absolute difference between the value of the age group compared to kids and the value of the age group compared to adults. Once again, this value was correlated to the median age group of myelin maturation.
See Appendix 1 for complete program commands. Journal of the PGSS Adolescent Myelin Maturation Page 51
III. Results
A. Method 1:
Method 1 analyzed the t-stat values of the radial diffusivities of individual voxels. To determine what age group was the middle point of white matter maturation, the absolute differences between a particular age group and kids, and that same age group and adults were calculated for percentiles ranging from 5 to 95 in increments of 10. The minimum difference would determine this median age group, for it has the closest differences between the two extremes.
Table 1: Test Method 1 - Comparison of individual voxel T-stat values
Percentiles Age Groups’ T-Stat Values Absolute Sum of Differences of Absolute
Kids Vs 17 17 Vs Adults 5 -0.24 -0.96 0.72 15 0.66 -0.42 1.08 25 1.21 -0.03 1.24 35 1.65 0.26 1.39 45 2.04 0.5 1.54 55 2.43 0.73 1.7 65 2.82 0.96 1.86 75 3.25 1.2 2.05 85 3.79 1.5 2.29 95 4.75 2.06 2.69 16.56
Page 52 Baybars, Chang, Jiang, Onofrey, Ponnappa, Ponte, Ramgopal, Shen, Trevisan, Zheng
Kids Vs 16 16 Vs Adults 5 -0.35 -1.16 0.81 15 0.54 -0.55 1.09 25 1.13 -0.15 1.28 35 1.58 0.22 1.36 45 2.02 0.54 1.48 55 2.43 0.86 1.57 65 2.88 1.13 1.75 75 3.36 1.43 1.93 85 3.99 1.84 2.15 95 5.13 2.63 2.5 15.92
Kids Vs 15 15 Vs Adults 5 -0.09 -1.12 1.03 15 0.78 -0.51 1.29 25 1.24 -0.08 1.32 35 1.58 0.28 1.3 45 1.89 0.64 1.25 55 2.2 0.98 1.22 65 2.55 1.3 1.25 75 2.94 1.63 1.31 85 3.47 2.03 1.44 95 4.45 2.75 1.7 13.11
Journal of the PGSS Adolescent Myelin Maturation Page 53
Kids Vs 14 14 Vs Adults 5 -0.21 -1.19 0.98 15 0.69 -0.63 1.32 25 1.25 -0.22 1.47 35 1.69 0.16 1.53 45 2.08 0.5 1.58 55 2.45 0.84 1.61 65 2.83 1.19 1.64 75 3.26 1.54 1.72 85 3.84 1.94 1.9 95 5.05 2.61 2.44 16.19
Kids Vs 13 13 Vs Adults 5 -0.61 -0.73 0.12 15 0.05 0.045 0.005 25 0.41 0.58 0.17 35 0.68 0.96 0.28 45 0.93 1.25 0.32 55 1.18 1.53 0.35 65 1.41 1.8 0.39 75 1.69 2.17 0.48 85 2.05 2.69 0.64 95 2.75 3.54 0.79 3.54
Page 54 Baybars, Chang, Jiang, Onofrey, Ponnappa, Ponte, Ramgopal, Shen, Trevisan, Zheng
Kids Vs 12 12 Vs Adults 5 -1.342 -1.05 0.292 15 -0.559 -0.432 0.127 25 -0.003 0.0066 0.0096 35 -0.001 0.378 0.379 45 -0.0003 0.737 0.7373 55 0 1.09 1.09 65 0.0003 1.442 1.4417 75 0.003 1.782 1.779 85 0.542 2.25 1.708 95 1.379 2.954 1.575 9.14
Kids Vs 11 11 Vs Adults 5 -0.65 -0.65 0 15 0.06 0.13 0.07 25 0.46 0.67 0.21 35 0.76 1.07 0.31 45 1.04 1.44 0.4 55 1.3 1.8 0.5 65 1.56 2.18 0.62 75 1.85 2.6 0.75 85 2.2 3.11 0.91 95 2.84 4.11 1.27 5.04
Journal of the PGSS Adolescent Myelin Maturation Page 55
Kids Vs Adults Vs Adults Kids 5 -0.55 -5.71 5.16 15 0.45 -4.47 4.92 25 1.15 -3.76 4.91 35 1.7 -3.2 4.9 45 2.21 -2.71 4.92 55 2.71 -2.21 4.92 65 3.2 -1.7 4.9 75 3.76 -1.15 4.91 85 4.47 -0.45 4.92 95 5.71 0.55 5.16 49.62
Page 56 Baybars, Chang, Jiang, Onofrey, Ponnappa, Ponte, Ramgopal, Shen, Trevisan, Zheng
B. Method 2:
Method 2 analyzed the number of voxels in clusters after thresholding the t-stat value at 3.0 or greater, and after thresholding the p-stat value at 0.3 or lower. Once again, the median age group of white matter development was determined by the minimum difference between that particular age group and the extremes.
Table 2: Test Method 2 - Comparison of Cluster Based Thresholding Values
Voxels of Members of Clusters with a T Abs Diff. of Value of > 3.0 Voxels
Kids vs. 11 2210 11173 11 vs. Adults 13383 11173 Kids vs. 12 18636 16463 12 vs. Adults 2173 16463 Kids vs. 13 1954 6032 13 Vs Adults 7986 6032 Kids Vs 14 26333 24278 14 Vs Adults 2055 24278 Kids vs. 15 19629 17644 15 Vs Adults 1985 17644 Kids vs. 16 27544 26154 16 vs. Adults 1390 26154 Kids vs. 17 25492 25409 17 Vs Adults 83 25409
Journal of the PGSS Adolescent Myelin Maturation Page 57
C. Method 3:
Method 3 thresholds t-stat values at 3.0 or greater, and thresholds p-stat values at 0.3 or lower. The median t-stat that survived after thresholding was determined, and the minimum difference between a particular age group and the extremes would provide the median age group of white matter development.
Table 3: Test Method 3 - Comparison of Median T-stats
Abs Diff. of Median Median t-stat
Kids vs. 11 3.48 0.25 11 vs. Adults 3.73 0.25 Kids vs. 12 3.74 0.38 12 vs. Adults 3.36 0.38 Kids vs. 13 3.44 0.12 13 Vs Adults 3.56 0.12 Kids Vs 14 3.84 0.47 14 Vs Adults 3.37 0.47 Kids vs. 15 3.73 0.24 15 Vs Adults 3.49 0.24 Kids vs. 16 3.93 0.55 16 vs. Adults 3.38 0.55 Kids vs. 17 3.80 0.18 17 Vs Adults 3.62 0.18
Page 58 Baybars, Chang, Jiang, Onofrey, Ponnappa, Ponte, Ramgopal, Shen, Trevisan, Zheng
IV. Discussions
A. Method I Discussion
The data shown in the methods section represent the differences between the histograms comparing different subject groups in both shift and shape. T-stat, in this case, represents the difference in radial diffusivity between any two given subject groups. For each t-stat, the number of voxels exhibiting that t-stat is plotted. A set of points needed to be plotted on each curve in equivalent positions because this would allow a comparison between shapes of any two histograms to take place. This was accomplished by picking ten points, each of whose position was determined by the area underneath the curve lying to the left of the coordinate (x=t-stat). The sum of the absolute differences of the t-stat values of a particular age group with kids versus that age group with adults shows the degree of contrast or similarity between the shape and shift of any two different histograms.
The method I calculations suggest that age thirteen is the midpoint of myelin development during adolescence. This conclusion follows logically from the difference of two differences: first, the difference between the t-stat values at a particular age and the t-stat values during childhood; and, second, the difference between the t-stat values at the age and the t-stat values during adulthood. The first difference measures the degree to which the age group differs from the childhood figure, and the second difference measures the degree to which the age group differs from the adulthood figure. This overall difference, calculated for all age groups, reaches its minimum for the 13-year-old group. In practice, a minimum value for this figure signifies that the group was closest to the middle of myelination development. The figure indicates that the amount of difference between the age group and childhood and the amount of difference between the age group and adulthood was almost exactly the same. Ideally, this value would be zero: a value of zero would indicate that the difference in maturation since childhood and the difference in maturation until adulthood were absolutely equivalent in magnitude. Realistically, instead of searching for a value of zero, a minimum gives enough information to determine a midpoint age.
B. Method 2 Discussion
The second test required similar calculations. However, in the second test, clusters of voxels replaced individual voxels. Criteria for a cluster included a degree of myelination in addition to a location adjacent to a qualifying voxel. Using clusters in place of voxels can potentially reveal more than merely using voxels, because groups of highly myelinated voxels indicate either tracts of white matter or concentrated regions of white matter. The results show that the thirteen year old age group possessed the minimum value for the absolute difference of the number of voxels per cluster between the age group and kids, and between that same age group and adults. Because the clustered calculations drew the same conclusion as the individual voxel calculations, a greater level of confidence that the first results were not due to chance is attained. Groups of voxels with a high level of myelination strongly correlated with individual voxels. Journal of the PGSS Adolescent Myelin Maturation Page 59
C. Method 3 Discussion
The third method implemented a system of clustering, thresholding, and calculating a median t-stat value of each age group. The difference between method 2 and method 3 lay in the thresholding procedure: all voxels with low p-values or t-stats were ignored, and only the statistically significant values remained in the calculations. Thus, method 3 calculations took into account only the statistically significant voxels when determining the median value. In method 3 calculations, the age group with the lowest absolute difference of two differences determined the midpoint of adolescent myelin maturation. The first difference compared the age group thresholded median to the kids’ thresholded median, while the second difference compared the age group thresholded median to the adults’ thresholded median. The absolute difference took the difference between these two differences. Again, the thirteen-year-olds minimized this difference (see Table 3).
V. Conclusion
In general, this study utilized the indirect correlation between radial diffusivities, myelination, and brain development. Lower radial diffusivities correlate with increased myelin, which in turn correlates with further brain development. This study endeavored to determine the midpoint of myelin development during adolescence by examining changes in radial diffusivity as a function of age. All three methods utilized found the midpoint age to be thirteen years. The first method examined the difference in the t distribution of radial diffusivities between each age group against each extreme (adults and children). The procedure returned the thirteen-year-olds because the change in radial diffusivity from childhood to thirteen years was similar in magnitude to the same measure from thirteen years to adulthood. The second method attempted to replicate results through a different procedure: calculations only considered clustered voxels above a certain t threshold and ignored clusters that showed an insignificant change in radial diffusivity. The difference between each extreme was reexamined, this time considering the total number of voxels in the significant clusters. Again, analysis found the thirteen-year age group to be the midpoint. The third and final method further confirmed this result in another calculation. After thresholding and discounting insignificant clusters, method three found the median value for the t distribution in the significant clusters for each age group, and found the difference between this value and each extreme. Once more, the investigation discovered the midpoint in the thirteen-year-olds. Thus, each of three independent analyses found the thirteen-year- olds to be the midpoint of adolescent myelin maturation as inferred by the radial diffusivity. Page 60 Baybars, Chang, Jiang, Onofrey, Ponnappa, Ponte, Ramgopal, Shen, Trevisan, Zheng
VI. Acknowledgements
We would like to acknowledge all the several entities which helped make this paper and the accompanying presentation possible. First of all, the state of Pennsylvania should be thanked in collaboration with Carnegie Mellon University for funding and making PGSS possible. In this vein, Dr. Barry Luokkala is to be thanked for his leadership and his part in making this opportunity possible. The Laboratory of Neurocognitive Development is to be thanked for their contribution to our collection of data. On a more team-specific note, Robert Terwilliger should be thanked for his help in offering his time and understanding to the project. And last, but not least, Sera Thornton is to be thanked immensely for her leadership and all around amazing performance in this study. Journal of the PGSS Adolescent Myelin Maturation Page 61
VI. Appendix A
A. UNIX commands
1. Method 1 - Individual voxel analysis cd pgss enter pgss directory ls open directory enter kids_vs_16 directory (this can be any existing cd kids_vs_16 file) cd skels enter skels (scans) of kids_vs_16 copy files of subjects needed for new group cp -rv kids_vs_16 kids_vs_12 kids_vs_12 cd kids_vs_12 open new file enter skels (scans) of kids_vs_12 (these has the cd skels same files as kids_vs_16 remove 16 year old scans from file because they rm 16* are extraneous cp ../../allskels/12* . import 12 year old scans into new file avwmerge -t ../stats/all_RD_skeletonised *.gz merge 12 year olds and kids into kids_vs_12 cd ../stats open statistics for kids_vs_12 cat ../../randomize_example.txt randomize (copy line & change name of tbss to kids_vs_12) run permutations avwstats++ kids_vs_12_tstat1 -P 5 (15,25,35,..,95) calculate percentiles
2. Method 2 - Clustering analysis cd pgss enter pgss directory ls open directory cd kids_vs_12 enter kids_vs_12 directory cd stats open stats folder of kids_vs_12 cat ../../randomize_example.txt randomize (copy line & change name of tbss to 3.0 -c 3.0) run permutations avwstats++ 3.0_maxc_tstat1 -R returns upper bound of number of voxels per cluster
3. Method 3 - Median analysis cd pgss enter pgss directory ls open directory cd kids_vs_12 enter kids_vs_12 directory cd stats open stats folder of kids_vs_12 thresholds clusters to tstat values of 3.0 or greater avwmaths 3.0_maxc_tstat1 -thr 0.7 pstat_3.0 and pstat values of 0.7 avwstats pstat_3.0 -V displays the number of voxels per cluster avwmaths 3.0_tstat1 -mas pstat_3.0 tstat_3.0 masks voxels with pstat or tstat less than 3.0 avwstats++ tstat_3.0 -P 50 returns value of median tstat
Page 62 Baybars, Chang, Jiang, Onofrey, Ponnappa, Ponte, Ramgopal, Shen, Trevisan, Zheng
B. FA brain spatial normalization versus RD brain spatial normalization
Figure 8: Fractional anisotropy image of 13-year-old female Journal of the PGSS Adolescent Myelin Maturation Page 63
Figure 9: Radial diffusivity image of 13-year-old female Page 64 Baybars, Chang, Jiang, Onofrey, Ponnappa, Ponte, Ramgopal, Shen, Trevisan, Zheng
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BIOPHYSICS TEAM PROJECTS
The Isotope Effects of Deuterium on the Enzyme Kinetics of Alcohol Dehydrogenase
Sai Alla, Tanvi Bharathan, Colin Comerci, Anuj Desai, Natania Field, Eva Gillis-Buck, Caroline Hsu, Isha Jain, Zebulin Kessler, Michele Kim, Molly Kozminsky, Tara Magge, Isabela Negrin, Mikael Owunna, Caroline Rogi, Silpa Tadavarthy, Harina Vin, Michael Wang, Edward Wu, Christopher Zimmerman
Abstract
The purpose of this research was to investigate the effects of deuterium replacement on the enzyme activity of alcohol dehydrogenase, the primary enzyme associated with alcohol metabolism. Enzyme activity was measured by continuous spectroscopic analysis of the production of NADH. The concentrations of enzyme ADH and coenzyme NAD+ were varied to determine their effect on the initial rate of reaction. The concentrations of deuterated and non-deuterated alcohols were varied to determine the disparity in their respective initial rates of reaction. ADH concentrations were varied with a constant concentration of deuterated as well as non-deuterated alcohols to investigate the rate-limiting step of the reaction. Varying NAD+ and alcohol concentrations showed hyperbolic correlations with initial rate. Varying enzyme concentration showed that the rate-limiting step for reactions with ethanol occurred after the formation of NADH while the rate-limiting step occurred before or during the formation of NADH for reactions with isopropanol. There was no experimental evidence of an isotope effect.
I. Introduction
Alcohol dehydrogenase (ADH) catalyzes the oxidation of alcohols in liver-alcohol metabolism to produce aldehydes and ketones. Specifically, ADH degrades ethanol into acetaldehyde and isopropyl alcohol into acetone. Liver ADH forms a complex consisting of the coenzyme nicotinamide adenine dinucleotide (NAD+), therefore making an active dimer. This dimer then binds to the alcohol substrate and leads to the formation of the products. Figure 1 is a schematic of the overall reaction:
Figure 1: ADH reaction1
A. Enzymatic Structure
The enzyme alcohol dehydrogenase specific to this study was taken from horse liver. The enzyme has two Zn2+ components and two binding sites: one for NAD+ and the other for the substrate. Shown in Figure 4 is a schematic of the ADH active sites.
Figure 2: 3-D ADH2 Page 70 Alla, Bharathan, Comerci, Desai, Field, Gillis-Buck, Hsu, Jain, Kessler, Kim, Kozminsky, Magge, Negrin, Owunna, Rogi, Tadavarthy, Vin, Wang, Wu, Zimmerman In order to catalyze the oxidation of alcohols, ADH, whose tertiary structure can be seen in Figure 2, reduces integral coenzyme NAD+ with a hydride and uses a zinc ion to electrostatically stabilize the alcohol’s oxygen. The conversion process is completed by a proton transfer in which zinc removes two hydrogen atoms from the ethanol, forming acetaldehyde. Since ADH requires the cofactor NAD+ to bind to the substrate before forming the enzyme-substrate complex, this reaction is order-specific (as shown in Figure 3).
.
Figure 3: Formation of the Enzyme-Substrate Complex
3 For this reaction to occur, NAD+ must be reduced to NADH. Finally, Figure 4: ADH Structure the aldehyde or ketone product and the NADH are released from the enzyme. A specific activation energy and reaction rate is associated with each step of this reaction.2
B. Deuterium
Deuterium is a stable, naturally-occurring isotope of hydrogen and contains one proton and one neutron in its nucleus. The structure of both hydrogen and deuterium can be seen in Figure 5. Because of the additional neutron, deuterium has a greater mass: 2.01355 g/mol versus hydrogen’s 1.00794 g/mol. The increased mass in the nucleus causes a greater attraction for the electron and lowers the radius of the electron cloud. The higher attraction of deuterium’s nucleus also causes stronger bonding between deuterium and other atoms; therefore, a greater amount of energy is required for the breaking of these bonds (Nelson and Trager 1481-1498).
Figure 5: Hydrogen (left) and Deuterium (right)4
Journal of the PGSS Deuterium Isotope Effects on ADH Kinetics Page 71
C. Quantum Mechanics
1. Deuterium Isotope Effect
The effects of deuterium isotope substitution on reaction rate (k) can be determined theoretically through quantum mechanical analysis. The basis for these calculations relies heavily upon the transition state, 7 illustrated in Figure 6, and the activation energy (Ea). An unstable complex is formed during this brief period of time (10-15s), resulting in an unstable equilibrium. In a series of reactions, the reaction with the transition state of the highest energy level is the rate- limiting step. Biological catalysts, such as ADH, lower the energy of the transition state as well as the activation energy of a reaction. Activation energy is defined by collision theory as the amount of energy required for a reaction to progress. If two molecules lacking the kinetic energy to react happen to collide, Figure 6: Transition State5 they will not have enough activation energy to overcome the transition state.
Hydrogen and deuterium have different bond strengths due to their nuclear masses; therefore, their activation energies are different. This is called the isotope effect and is due to the quantum mechanical differences in hydrogen and deuterium’s zero point energies (E0). The zero point energy is the lowest possible amount of energy that a system may contain otherwise known as the system’s ground state. In the most notable method of calculating zero point energy,
, [1]
it is defined in terms of Planck’s constant and vibrational frequency. See Appendix A for nominal values. Figure 7 demonstrates the isotope effect using the harmonic oscillator quantum model.
By dividing the zero point energy of hydrogen by the zero point energy of deuterium, it is determined that
Figure 7: Isotopic Energy States6
. [2]
Vibrational frequency (v), which is necessary to completely derive Equation [2], involves the frequency of bond oscillation between two atoms; it is given by the equation Page 72 Alla, Bharathan, Comerci, Desai, Field, Gillis-Buck, Hsu, Jain, Kessler, Kim, Kozminsky, Magge, Negrin, Owunna, Rogi, Tadavarthy, Vin, Wang, Wu, Zimmerman
, [3] where is the reduced mass of the atom. The reduced mass is found with the atomic masses of carbon, hydrogen, and deuterium using the single mass equation,
. [4]
Equation 3 describes the interaction between molecules as though there is a spring connecting them. The molecules vibrate with a frequency that is affected by k, which is analogous to the spring constant, and a
reduced mass that accounts for any bonds formed by the atom. By inserting this equation into the ratio H/ D, it is determined that
. [5]
For detailed derivations, reference Appendix B.
2. The Arrhenius Equation
The Arrhenius equation can be used to calculate the expected isotope effect by relating the k values for the dissociation of the carbon-hydrogen and carbon-deuterium bonds. The rate constant is generally affected by the temperature, pressure, and atomic bond strength. The Arrhenius equation,
, [6]
provides a remarkably accurate calculation of k, where A is the pre-exponential (or frequency) factor found through experimentation and R is the gas constant. The ratio of the rate constants for hydrogen and deuterium is found to be
. [7]
This value allows for a comparison between the velocities of the reactions with hydrogen and deuterium bonds. The isotope effect is derived from the comparison of these reaction velocities.
This relationship can also be given using bond frequency in place of activation energy as Journal of the PGSS Deuterium Isotope Effects on ADH Kinetics Page 73
, [8]
where H and D are nominal values.
D. Enzyme Kinetics
1. General Kinetics
Enzyme kinetics is the study of the rates at which chemical reactions are catalyzed by enzymes. The reaction rate, k, is referred to as the reaction velocity, v, in enzyme kinetics. Enzyme kinetic theory is crucial to the analysis of the enzyme, both to identify methods for enzyme analysis and to understand basic mechanisms. In general, enzyme reactions fit the model
. [R1]
In this reaction, Z is the enzyme, S is the substrate, ZS is the enzyme-substrate complex, and ka, kd, and kcat are the rate constants for their respective reactions.
2. Michaelis-Menten Model
The velocity of the reaction is calculated using
. [9]
In this equation, the Michaelis constant, KM, is the substrate concentration when the velocity is half of the maximum velocity. It is also represented by the rate constants as
7 Figure 8: Michaelis-Menten Model . [10]
Low Km values indicate that the ZS is very tightly bound and rarely dissociates without first progressing to
Z+P, and vice versa, for greater Km values. In a broader sense, Km describes an enzyme’s affinity for the substrate and how efficiently products will be formed.
The maximum velocity can also be given by
[11]8 Page 74 Alla, Bharathan, Comerci, Desai, Field, Gillis-Buck, Hsu, Jain, Kessler, Kim, Kozminsky, Magge, Negrin, Owunna, Rogi, Tadavarthy, Vin, Wang, Wu, Zimmerman
where ZT is the total amount of enzyme in the reaction. If v is analyzed for different concentrations of the
reactants, seen in Figure 8, the graph will level off at a maximum velocity, Vmax. The maximum velocity occurs once the substrate is in excess compared to the enzyme.
Equation 9 shows the direct proportionality between V and [S] when [S] approaches Km. On the contrary, when [S] is much greater than Km, the enzyme becomes saturated with the substrate and the rate reaches
Vmax, the enzyme’s maximum rate of catalysis.
3. Burst Kinetics
As the enzyme is initially added to the substrate solution, there is an initial curve that deviates from the Vi line, as seen in Figure 15. This enzymatic kinetic burst, called the burst phase, is measured by extending the Vi line to the y-axis. This burst is equal to the enzyme concentration, so any given point on the linear portion of the graph is directly proportional to the enzyme concentration. It is possible to use this information to develop another method for determining the presence of a burst. If the ratio of enzyme concentration to the product concentration at a selected time on the linear graph is the same for all concentrations of the enzyme, a burst has occurred. If there is no such correlation between enzyme concentration and product concentration, no burst has occurred.
Amount of product vs. Time
Amount of enzyme Amount of product Burst phase
Selected time
Figure 9: Extrapolation for Burst Kinetics9
Journal of the PGSS Deuterium Isotope Effects on ADH Kinetics Page 75
Because the amount of product is measured by the formation of NADH, burst kinetics in the mechanism for ADH catalysis can only be observed if the greatest transition state exists following the ZNADHP step in Figure 4. Since the reaction rate will be slower to surpass this transition state, the amount of substrate and NADH can be measured. If activation energy is increased at this step, a larger kinetic burst will appear; this is mainly because the reaction will take longer to reach the final step of the mechanism, where the final products are released.
Experiments studying burst kinetics can be used to determine which step in the reaction is the rate-limiting step. In ADH catalysis, there are three distinct steps with an unstable intermediate in the middle, as shown in Figure 4. If the rate-limiting step occurs before or during the formation of NADH, there will be no rapid NADH production at the start of the reaction, meaning there will be no burst. If the rate-limiting step occurs after the formation of NADH, NADH is formed quickly until no more free enzyme is available. The rapid NADH production is the burst. However, free enzyme is released more slowly, limiting the reaction cycle and slowing later NADH production.10
II. Materials and Methods
A. Reagents
Seven reagents were used in all of the procedures. Shown in Table 1 is a list of the reagents, their concentrations, and additional information about their preparation.
Table 1: Reagents11
Reagent Concentration Name Additional Info
A 50 mM Sodium Pyrophosphate Buffer pH 8.8 at 25°C
B 95% (v/v) Ethanol n/a
C 15mM -Nicotinamide Adenine Dinucleotide Prepared fresh. Solution (NAD+) D 10mM Sodium Phosphate Solution Monobasic
E 10mM Sodium Phosphate Buffer Dibasic pH 7.5 at 25°C F 10mM Sodium Phosphate Buffer with 0.1% Prepared in Reagent Bovine Serum Albumin (Enzyme E with Albumin Diluent) G 0.75 units/ml Alcohol Dehydrogenase (ADH) Prepared immediately Enzyme Solution before use
Page 76 Alla, Bharathan, Comerci, Desai, Field, Gillis-Buck, Hsu, Jain, Kessler, Kim, Kozminsky, Magge, Negrin, Owunna, Rogi, Tadavarthy, Vin, Wang, Wu, Zimmerman B. Quality Control of the Enzymatic Assay of Alcohol Dehydrogenase
The first assay was a quality control test which analyzed the effectiveness of ADH. A solution was prepared containing 1 mg/ml of ADH in cold dibasic sodium phosphate buffer. This solution was then diluted to 0.75 units/ml with cold bovine serum albumin. To perform the assay, three reagents were combined in a 1mL cuvette: 0.433 mL of 50mM sodium pyrophosphate buffer, 0.033 mL of 95% ethanol, and 0.500 mL of NAD+. To this solution, 0.033 mL of the 15mM ADH solution was added. The cuvette was immediately mixed by inversion three times. It was then placed in the spectrophotometer in order to record the increase in absorbance at 340 nm for approximately six minutes.11
C. Varying ADH Concentration
The purpose of varying the concentration of ADH was to determine if a higher concentration resulted in better enzyme activity. Three 1 mL cuvettes were prepared with the first three reagents listed above. Several volumes of 15mM ADH were then added: 0.033 mL, 0.066 mL, and 0.099 mL.
D. Varying NAD+ Concentration
The purpose of varying the concentration of NAD+ was to determine if NAD+ had an effect on the activity of ADH. The volumes of NAD+ were varied while following the procedure for quality control. The concentrations used were 0.0 mM, 0.15 mM, 0.75 mM, 1.5 mM, 3 mM, 4.5 mM, 6 mM, and 7.5 mM. Deionized water was added respectively to retain a constant amount of solution, 1 mL. The only variation from the quality control procedure was that the cuvettes were placed in the spectrophotometer for two minutes each instead of six minutes each.
E. Matrix of Deuterated and Non-Deuterated Forms of Ethanol and Isopropanol
The purpose of varying the concentrations of the non-deuterated and deuterated alcohols is to find the
maximum velocity, Vmax, and the substrate concentration at half the maximum velocity, Km, for these specific substrates.
1. Non-Deuterated Forms of Ethanol and Isopropanol
Following the quality control procedure described above, sodium pyrophosphate buffer, varying concentrations of 95% non-deuterated ethanol, and 15mM NAD+ were combined in 1 mL cuvettes. The specific concentrations of 95% ethanol used were 0.0865 mM, 0.137 mM, 0.274 mM, 0.3425 mM, 0.549 mM, 1.1 mM, 2.2 mM, 4.4 mM, 8.78 mM and 17.56 mM.
Journal of the PGSS Deuterium Isotope Effects on ADH Kinetics Page 77
The same procedure was followed using 95% non-deuterated isopropanol instead of non-deuterated ethanol. The concentrations of isopropanol used were 205 mM, 102.5 mM, 51.25 mM, 25.63 mM, 12.81 mM, 6.406 mM, 3.203 mM, 1.602 mM and 0.4004 mM. A twofold dilution series was produced in order to minimize error between samples. The only other deviance from the procedure was that the cuvettes were placed in the spectrophotometer for two minutes each. Experiments were generally conducted in duplicates or triplicates to lower standard deviations.
2. Deuterated Forms of Ethanol and Isopropanol
Deuterated ethanol and isopropanol were then substituted for non-deuterated ethanol and isopropanol. The concentrations of ethanol used were 18.03 mM, 9.016 mM, 4.508 mM, 2.254 mM, 1.127 mM, 0.5635 mM, 0.2817 mM, 0.1409 mM, 0.07043 mM and 0.03522 mM. The cuvettes were placed in the spectrophotometer for 2.5 minutes each.
The concentrations for isopropanol used were 424 mM, 212 mM, 106 mM, 53 mM, 26.5 mM, 13.25 mM, 6.625 mM, 3.313 mM, 1.656 mM, 0.8281 mM and 0.41405 mM. The cuvettes were placed in the spectrophotometer for four minutes each.
G. Burst Kinetics
Following the quality control protocol, the first three reagents (sodium pyrophosphate buffer, non-deuterated ethanol, and NAD+) were combined in 1 mL cuvettes. 15mM ADH was added to each cuvette in various concentrations: 0.015mL, 0.021mL, 0.027mL, 0.033mL, 0.039mL, 0.045mL. Three samples of each concentration were prepared. The cuvettes were then placed in the spectrophotometer for one minute each. This procedure was repeated three times using non-deuterated isopropanol, deuterated ethanol, and deuterated isopropanol in place of non-deuterated ethanol.
III. Results and Discussion
A. Quality Control Testing
Quality control is a process that helps to normalize results. The process of quality control confirms that the enzyme has the same activity as that reported by the manufacturers. In this experiment, the quality control test involved determining the units per milliliter value for the stock solution of ADH. A single unit of enzyme converts 1.0 µmole of ethanol to acetaldehyde per minute at pH 8.8 and 25 °C.
In the presence of ADH and NAD+, ethanol is oxidized to acetaldehyde and NADH. This oxidation was monitored with a spectrophotometer set at 340 nm because NADH, a product of this reaction, absorbs light Page 78 Alla, Bharathan, Comerci, Desai, Field, Gillis-Buck, Hsu, Jain, Kessler, Kim, Kozminsky, Magge, Negrin, Owunna, Rogi, Tadavarthy, Vin, Wang, Wu, Zimmerman at this wavelength. The slope of the spectra reading (change in absorbance over time) was measured. This quantity should be proportional to the amount of NADH produced.
By determining the slope of the spectrophotometer readings, the units/mL of enzyme in solution were determined for the diluted sample: ∆A( )(1)(df) / enzymemLUnits = 340nm/min [12]11 )033.0)(22.6(
where ∆A340nm/min is the slope of the graph produced by the spectra, 1 is the total volume in milliliters of assay, df is the dilution factor (0.033), 6.22 is the extinction coefficient of NAD+, and 0.033 is the volume in milliliters of enzyme used. Table 2 shows the results of these experiments.
Table 2: Quality Control Data Set 1 Sample Number Units/mL ADH 1 0.272818 2 0.2927 3 0.31212 Average 0.292546
The average units/mL for ADH was then determined to be 0.00965. Finally, because only 0.033 mL ADH were used to produce 0.00965 units, it was possible to calculate units/1mL by using a simple proportion: unitsx = 00965.0 units [12]10 mL 033.01 ADHmL
x = 1/293.0 enzymemLunits solution . The original stock sample therefore had 0.293 units/1mL solution.
The second quality control data set, as shown in Table 3, resulted in a similar average of 2.99 units/mL.
Table 3: Quality Control Data Set 2 Sample Number units/mL ADH 1 0.189394 2 0.274286 3 0.337576 4 0.424242 5 0.214848 6 0.414109 7 0.365391 8 0.188542 9 0.388288 10 0.194388 Average 0.299106 Journal of the PGSS Deuterium Isotope Effects on ADH Kinetics Page 79
B. Varying ADH Concentration
By calculating the slope of the spectra readings, the absorbance of the ADH solution at varying concentrations over 120 seconds was determined. Initial velocity was calculated using the following equation: ∆A
V = ∆t , [13]11 i εB where A/ t is the change in absorbency over time (slope of generated graph), is the millimolar extinction coefficient of NADH at 340 nm (6.22 mM-1cm-1), and B is the path length (1 cm).
The collected data is shown in Figure10.
Varying ADH Concentration Effect on Initial Rate
1.E-03 y = 6E-06x - 1E-05 9.E-04 R2 = 0.9986 8.E-04 y = 5E-06x + 8E-05 2 7.E-04 R = 0.9909
6.E-04
5.E-04
4.E-04 Experiment 1 Experiment 2
Initial Rate (mM/sec) Rate Initial 3.E-04 Linear (Experiment 2) 2.E-04 Linear (Experiment 1)
1.E-04
0.E+00 0 50 100 150 200 Concentration of ADH (mM): Hold [ethanol] constant
Figure 10: Varying ADH Effect on Initial Rate
According to this graph, there is a linear relationship between the amount of ADH in units and initial rate. This is confirmed by the R2 values of the lines since values close to one suggest linearity. A first-order rate Page 80 Alla, Bharathan, Comerci, Desai, Field, Gillis-Buck, Hsu, Jain, Kessler, Kim, Kozminsky, Magge, Negrin, Owunna, Rogi, Tadavarthy, Vin, Wang, Wu, Zimmerman reaction depends on the concentration of only one reagent. Therefore, the reaction that produced the results shown in the graph qualifies as a first-order rate reaction. The rate of this reaction is directly proportional to the concentration of ADH. As the reactant is consumed, both the concentration and the rate of reaction drop.
C. Varying NAD+ Concentration
In order to analyze the results of varying NAD+ concentration, a graph of initial rate vs. concentration of NAD+ was made. The results suggest that at small concentrations, ranging from 0 mM to approximately 0.75 mM, the concentration of NAD+ does have an effect on the enzyme activity of ADH. This is represented by the initial upward trend of the graph. This part of the graph also shows that the reaction rate is increasing as NAD+ becomes more concentrated. It is assumed that ethanol is found in excess up until this point because all of the NAD+ is binding to the ethanol. At higher concentrations, ranging from approximately 1.5 mM to infinity, ADH is saturated with NAD+, and increasing concentrations of NAD+ have no measurable effect on the reaction rate. These trends are illustrated by the hyperbolic curve shown in Figure 11. Analysis of the curve suggests a maximum enzyme activity, Vmax, of 0.00577 mM/sec. The Km was found to be 0.0710 mM.
vi HmM minute L NAD vs Initial Rate 0.007 @D
0.006
Vmax=0.00577 mM/sec 0.005
0.004
0.003 Km= 0.0710 mM
0.002
0.001
@NAD D,mM 2 4 6 8 Figure 11: Initial Rate vs. Concentration of NAD+
D. Quantum Mechanics
1. Arrhenius Frequency Equation Journal of the PGSS Deuterium Isotope Effects on ADH Kinetics Page 81
The reduced mass equation, equation 4, was used to calculate the H and D. These values were inserted H D into the E0 ratio, equation 5, to find E0 / E0 . Afterwards, Equation 2 was used to determine D from the
nominal value of H. Equation 8 was then used to calculate the ratio of kH/kD to be 6.81.
2. Standard Arrhenius Equation
D H To verify the previous answer, the nominal value of Ea -Ea was inserted into equation 7. This calculation yielded a value for kH/kD of 6.88. For detailed calculations, reference Appendix C.
E. Deuterated and Non-Deuterated Forms of Ethanol and Isopropanol
The primary goal of this experiment was to determine changes in ADH enzyme kinetics due to the substitution of hydrogen with deuterium isotope. These effects were observed in the alcohol substrates isopropanol and ethanol. The initial experiment with non-deuterated substrates was conducted to replicate the hyperbolic Michaelis-Menten curve of ADH converting ethanol to acetate. A decrease in substrate concentration with a constant amount of enzyme should theoretically result in a declining velocity of NADH formation (a byproduct of alcohol substrate catalysis). This results in a hyperbolic relationship which reflects the gradual leveling off of initial velocity. At high concentrations the enzyme-substrate complex is maximized due to enzyme saturation and the reaction rate cannot increase.
The results for the non-deuterated Velocity vs. Concentration of Ethanol samples were originally inconclusive as the scatter plot of the data did not accurately reflect the upward hyperbolic trend of the Michaelis-Menten curve. The inconsistencies of the data were originally attributed to mechanical error of the spectrophotometer. A second trial run on a different spectrophotometer produced similar erroneous results. Due to the enzyme’s relatively short shelf-life, it was hypothesized that the enzyme had — non-deuterated isopropanol --- deuterated isopropanol
been denatured during storage. Despite Vmax = 0.000256519 mM/sec Vmax = 0.000223663 mM/sec Km = 0.581367mM Km = 0.609925mM further repetition of the experiment with Figure 12: Velocity vs. Concentration of Ethanol freshly prepared enzyme, the results still appeared counterintuitive. It was then proposed that the sodium pyrophosphate reagent may not have been prepared at the correct pH-level. These conditions would lead to an environment that is not conducive Page 82 Alla, Bharathan, Comerci, Desai, Field, Gillis-Buck, Hsu, Jain, Kessler, Kim, Kozminsky, Magge, Negrin, Owunna, Rogi, Tadavarthy, Vin, Wang, Wu, Zimmerman towards enzyme activity. To eliminate such potential errors, in addition to other possible contaminations, a new batch of reagents was made for further experimentation. In spite of such measures, atypical results were still collected. It was thus concluded that the source of error originated in the choice of dilution series. Upon closer analysis of the data and further research, it was discovered that the ethanol samples were much too concentrated, leading to complete ADH saturation at all tested dilutions. Since only the plateau of the Michaelis-Menten curve was represented originally by the data, the dilution series had to be extended to lower concentrations in order to encompass the entire curve. The ranges of the dilution series were extended appropriately, yielding data to support the Velocity vs. Concentration of Isopropanol
theoretical model. Isopropanol does not bind as effectively as ethanol; thus, further dilutions of isopropanol were not necessary to complete the Michaelis-Menten curve. Errors in isopropanol data were resolved after all reagents were remade.
Once consistent data for the non-deuterated
compoun ds was obtained, the same — non-deuterated isopropanol --- deuterated isopropanol effective procedure was executed with Vmax = 0.0000545585 mM/sec Vmax = 0.0000433781 mM/sec deuterated ethanol and isopropanol. The Km = 3.67913 mM Km = 0.506591 mM resulting data was consistent with the Figure 13: Velocity vs. Concentration of Isopropanol theoretical predictions, producing scatter plots that fit the Michaelis-Menten curve . Com parisons between the non-deuterated and deuterated samples of isopropanol and ethanol can be made by referencing the above figures in which both are plotted on the same graph. Mathematica 5.1 was employed to statistically determine whether a significant isotope effect resulted from the substitution of deuterium for hydrogen.
Vmax Km Non-deuterated 0.000256519 0.581367 Ethanol Deuterated 0.0002236 63 0.6099 25 Ethanol Non-deuterated 0.0000545585 3.67913 Isopropanol Deuterated 0.0000433781 0.506591 Isopropanol Journal of the PGSS Deuterium Isotope Effects on ADH Kinetics Page 83
Analysis of the non-linear regression provided values for the Vmax and KM. When varying deuterated and non- deuterated ethanol concentration, the ratio of the Vmax of the non-deuterated ethanol over the Vmax of the deuterated ethanol was close to 1, indicating that deuterium did not alter the reactivity of the ADH with ethanol. When varying deuterated and non-deuterated isopropanol concentration, the ratio of the Vmax of the
non-deuterated isopropanol over the Vmax of the deuterated isopropanol was close to 1 as well, indicating
that the deuterium did not affect the reactivity of ADH with isopropanol. The Vmax ratio of approximately 1 is much smaller than 6.88, the value calculated using the Arrhenius equation. Brooks and Shore report that a EG1 ratio of greater than 2 indicates a partial isotope effect . The observed Vmax ratio was less than 2. Therefore, no isotope effect was observed in the 12 ethanol or isopropanol reaction. Figure 14: Vmax and Km Comparisons
F. Burst Kinetics VmaxH KcatH = VmaxD KcatD To determine whether or not a reaction exhibits burst kinetics, the direct relationship between the Ethanol 1.15 absorbance of light at 340nm and concentration of Isopropanol 1.26
ADH was observed. The pre-steady-state, the initial Figure 15: Ethanol and Isopropanol Ratios burst that occurs in burst kinetics, usually lasts no longer than a few milliseconds and cannot be detected directly. However, one can still determine whether a burst occurred by extrapolating the theoretically linear curve of the steady-state to the moment at which the enzyme was added and catalysis began as shown by the dotted line in Figure 8. Amount of product vs. Time
Amount of enzyme Amount of product Burst phase
Time (seconds) Start time
Figure 8: Extrapolation for Burst Kinetics9
Page 84 Alla, Bharathan, Comerci, Desai, Field, Gillis-Buck, Hsu, Jain, Kessler, Kim, Kozminsky, Magge, Negrin, Owunna, Rogi, Tadavarthy, Vin, Wang, Wu, Zimmerman Since the reaction did not begin at the exact time the enzyme was added, it was impossible to extrapolate to the y-intercept of the steady-state reaction at the start of the reaction; therefore, a common point after adding the enzyme was chosen.
Results were taken for both non-deuterated and deuterated ethanol and isopropanol.
Ethanol: Average Absorbance vs. Concentration
0.09
0.08
0.07
0.06
0.05 Non-Deuterated Ethanol 0.04 Deuterated Ethanol
Absorbance 0.03
0.02
0.01
0 0 0.01 0.02 0.03 0.04 Concentration (units/mL)
Figure 16: Average Absorbance vs. Concentration
The average absorbance was plotted against the appropriate concentration for both non-deuterated and deuterated ethanol. Both data sets appear to be linear.
Journal of the PGSS Deuterium Isotope Effects on ADH Kinetics Page 85
Isopropanol: Average Absorbance vs. Concentration
0.06
0.05
0.04
Non-Deuterated Isopropanol 0.03 Deuterated Isopropanol
Absorbance 0.02
0.01
0 0 0.01 0.02 0.03 0.04 Concentration (units/mL)
Figure 17: Isopropanol Average Absorbance vs. Concentration The average absorbance of isopropanol was plotted against the appropriate concentration for both non- deuterated and deuterated isopropanol.
In order to establish the presence of a burst, the data was used to calculate a ratio between each average absorbance and the one preceding it. The same was done for each concentration of enzyme. The difference was taken between these ratios. If the differences were close to zero, a burst occurred. Figures 16 and 17 will appear to be linear, indicating that the change in enzyme was directly proportional to the change in rate. If the differences were not close to zero, there was no burst, and the graph will not be linear. The same values were calculated for the deuterated substrate.
Figures 16 and 17 show that ethanol has a burst and isopropanol does not. Both ethanol graphs are linear or near-linear as proved by the comparison of the ratios of the concentrations and the absorbance values, however a downward vertical shift was displayed by the deuterated substrate. While the non-deuterated isopropanol graph appeared to be linear, the ratio comparisons performed showed otherwise; also the absorbance values for both the non-deuterated and deuterated substrates were in the same range.
Since there is a burst when the ethanol is used, the rate-limiting step is after the production of NADH. In the case of isopropanol, there is no burst, so the rate-limiting step is before or during the production of NADH: the binding step. This occurrence is due to the structure of the substrates. Ethanol is better suited to fit the active site of ADH and thus requires less energy to bind. Compared to ethanol, isopropanol has a methyl Page 86 Alla, Bharathan, Comerci, Desai, Field, Gillis-Buck, Hsu, Jain, Kessler, Kim, Kozminsky, Magge, Negrin, Owunna, Rogi, Tadavarthy, Vin, Wang, Wu, Zimmerman group instead of a hydrogen. Therefore, isopropanol is less appropriate for the binding site of ADH. It requires more energy to bind, and the binding becomes the rate-limiting step.
IV. Conclusion
The data obtained from the variance of ADH concentration indicates that the rate of reaction is directly proportional to the concentration of ADH. That is, as the concentration of ADH increases, the initial rate of reaction increases.
When testing the effects of NAD+ concentration on initial reaction rate, a hyperbolic curve was observed. It was determined that the reaction rate increases as concentration increases at low concentrations of NAD+. However, at high concentrations of NAD+, the enzyme becomes saturated and the reaction rate is unaffected. This is demonstrated by the initial upward direction of the curve, which then flattens out as the enzyme becomes saturated at higher concentrations. Thus, NAD+ has an effect on reaction rate only when its concentration is low.
The data collected from varying the concentrations of deuterated and non-deuterated ethanol and isopropanol indicates that there is no deuterium isotope effect for either substrate. Calculated ratios for the maximum velocity of the non-deuterated to that of the deuterated for both substances were much less than values which would characterize a partial or full isotope effect.
The data collected from varying the concentration of the enzyme suggest that the rate limiting step in the catalysis of ethanol occurs after the formation of NADH. The converse is true in the catalysis of isopropanol. This conclusion was reached due to the burst present in the reaction with ethanol while it was absent in the presence of isopropanol. Since the ethanol bound more readily to the ADH than the isopropanol, it was concluded that ethanol was a better substrate. The data also suggests that the deuterated ethanol has a slower reaction rate than the non-deuterated ethanol as visible in graph (blah); deuterium substitution had no effect on the reaction rate of isopropanol. Observations from the experiments which varied ethanol and isopropanol concentrations show that there is no isotope effect with deuterated ethanol nor with deuterated isopropanol, conflicting with varied enzyme inferences.
For isopropanol, the burst kinetics study supported the conclusion of the substrate variation study. Isopropanol exhibited no isotope effect, suggesting that the rate-limiting step of the reaction is before or during the formation of NADH. Isopropanol also exhibited no burst, supporting this conclusion. There is, however, a discrepancy between the data from the enzyme variation study and the substrate variation studies. No isotope effect was observed for ethanol, yet the ethanol did exhibit a burst. This discrepancy opens possibilities for further research.
Journal of the PGSS Deuterium Isotope Effects on ADH Kinetics Page 87
Sources of error in the study include rounding due to pipetting accuracy and the inconsistency of reagents used. Additional sources of error include the denaturing of enzyme because of excess concentration of ethanol and the use of reagents with variable pH. Also, the spectrophotometers used had an inherent error.
V. Acknowledgements
The team would like to thank:
Our teaching assistants: Adnan Bashir, Liz Dicocco, and Ryan Melnyk,
Dr. Christopher Borysenko for his help and support,
Carnegie Mellon University for the use of laboratory facilities,
Mike for his diligence and persistence,
Isabela and Silpa for compiling the slideshow and paper,
Charlotte Barton for laboratory assistance when needed,
Dr. Barry Luokkala for his support, and the Pennsylvania Governor’s School for the Sciences for the amazing experience.
VI. References
1 “Chemical Compound.” Britanica Concise Encyclopedia. 2007. Encyclopedia Britannica Online. 26 July 2007
2 H. Ekund, S. Ramaswamy, and B. Plapp. "Structure Explorer." RCSB Protein Database. 24 July 2007. 23 July 2007
3 Maas, Stefan. "Blackboard Academic Suite." Blackboard 7.1 @ Lehigh. 2002. Lehigh University. 24 July 2007
4 Image:Hydrogen Deuterium Tritium Nuclei Schmatic-ja.png." Wikipedia. 24 Jul 2007
5 M. Dewar et al. "Computational Procedures." Semiempirical Methods and Parameters. 2004. Semichem, Inc. 22 July 2007
6 K. Lim and W. Coleman. "The Effect of Anharmonicity on Diatomic Vibration." J of Chem Edu. Aug. 2005. American Chemical Society. 24 July 2007 Page 88 Alla, Bharathan, Comerci, Desai, Field, Gillis-Buck, Hsu, Jain, Kessler, Kim, Kozminsky, Magge, Negrin, Owunna, Rogi, Tadavarthy, Vin, Wang, Wu, Zimmerman
7 Manske, Magnus. ”Image:Michaelis-Menten.png.” Biology Daily. 9 May 2005. 25 July 2007
8 “Michaelis-Menten Equation.” 24 August 2004. Wellesley College. 23 July 2007.
9 Pre-steady-state kinetics. 25 July 2007.
10 M. Bender and L. Brubacher. Catalysis and Enzyme Action. New York: McGraw, 1973.
11 “Enzymatic Assay of Alcohol Dehydrogenase (EC1.1.1.1).” Sigma, Sigma-Aldrich. 09/04/98.
12 R. Brooks and J. Shore "Effect of Substrate Structure on the Rate of the Catalytic Step in the Liver Alcohol Dehydrogenase Mechanism." Biochemistry 10 (1971): 3855+.
Appendix A: Nominal Values
A. Arrhenius Frequency Equation
B. Standard Arrhenius Equation
Journal of the PGSS Deuterium Isotope Effects on ADH Kinetics Page 89
C. Enzyme Kinetics
Appendix B: Derivations
A. Zero Point Energy Ratio
B. Standard Arrhenius Equation
C. Frequency Arrhenius Equation Page 90 Alla, Bharathan, Comerci, Desai, Field, Gillis-Buck, Hsu, Jain, Kessler, Kim, Kozminsky, Magge, Negrin, Owunna, Rogi, Tadavarthy, Vin, Wang, Wu, Zimmerman
Appendix C: Calculations
A. Reduced Masses
B. Zero Point Energy Ratio
C. Vibrational Frequency (D) Journal of the PGSS Deuterium Isotope Effects on ADH Kinetics Page 91
D. Frequency Arrhenius Equation
E. Standard Arrhenius Equation
CHEMISTRY TEAM PROJECTS
Journal of the PGSS Computational Predictions of Chromatographic Selectivity Page 95
Computational Predictions of Chromatographic Selectivity of Stationary Phases
Tyler Badgley, William Carpenter, Sumit Rastogi, Ross Widenor, and John Zhang
Abstract
Gas chromatography is a technique that is used to separate and identify components of a sample according to particle size and interactions between the stationary phase and the components. Using molecular modeling for gas chromatography means research is less expensive in terms of time and resources compared to experimental gas chromatography research. Because high selectivity is one characteristic of good stationary phases, the use of a polysiloxane stationary phase, which is known to have superior selectivity, is employed for molecular dynamics simulations. The selectivity was found to be comparable to experimental data, proving the effectiveness of the model.
I. Introduction
A. Gas Chromatography
Gas chromatography is a method of separating and identifying compounds in a sample. One example of a gas chromatograph is called an FID, or a flame ionization detector. A liquid sample, or analyte, is injected into the column via a syringe in order to separate the compounds. Upon exiting the column, the compounds are ignited by a flame to produce ions and electrons, which are detected. If the original sample is a solid, then it is necessary to make a solution by combining the solid sample with a general solvent. The sample is then moved through a column by an inert carrier gas, which is called a mobile phase. The gas should not react with the analyte, so a typical inert carrier gas would be helium or nitrogen. The column is a long, looped, thin tube which is coated with a stationary phase. A good stationary phase achieves three criteria: thermal stability, negligible bleed, and high selectivity for compounds of interest. A high selectivity is necessary for a good stationary phase because the selectivity is the separation of compounds.
Specifically, selectivity refers to the separation of two compounds in the sample. This separation is represented by a numerical factor. If the selectivity is low, several compounds of the sample will elute together, resulting in poor separation. With respect to thermal stability, if the experimental temperature is too low, then the retention time for the compounds will not be efficient. If the temperature is too high, the system is no longer stable and the stationary phase can be burnt or the sample may fail to separate properly. Negligible bleed refers to a lack of separation of the compound due to breakdown of the stationary phase.
Page 96 Badgley, Carpenter, Rastogi, Widenor, Zhang
The stationary phase is a microscopic layer of liquid, or polymer, inside the column, that will interact with the various components as they pass through the column. The inert carrier gas will move the sample through the column, but the sizes of the particles, as well as their interactions with the stationary phase, will cause them to elute at different rates, which is the goal. The time it takes the components of the sample to exit the column is referred to as the retention time. As the chemicals exit the column, they are detected and identified electronically, typically with a mass spectrometer.
Synthesizing stationary phases can be a time-consuming and expensive task and using molecular modeling for prediction of selectivity would make stationary phase development simpler and more efficient. We have chosen polysiloxane stationary phase as the primary focus of our research because of its ability to separate many types of compounds. Experimentally, polysiloxane is a general use stationary phase, good for numerous applications, such as forensic science, environmental science, food chemistry, and pharmaceuticals. We have also chosen to model a derivative of the solvent xylene in order to represent the behavior of a variety of analytes.
B. Molecular Modeling
Molecular modeling is a technique using computational methods to mimic, or model, the behavior of atoms and molecules. The modeling program performs mathematical computations to solve the complex intra- and intermolecular properties that exist between atoms and molecules. Computer software, such as HyperChem®, allows the user to build molecules, determine stable comformations, and model the dynamics of systems.
Two types of simulations were utilized during this molecular modeling research: geometry optimization (GO) and molecular dynamics (MD). Geometry optimization determines the confirmation in which the molecule is most stable and also calculates the minimum energy. Molecular dynamics is a form of simulation that allows molecules to interact for a period of time under specific conditions that obey the laws of physics. This is carried out using a chosen force field, or functional set of parameters that describes the potential energy of a system of particles.
Molecular modeling is being used to streamline the process of stationary phase design by developing a technique for predicting chromatographic selectivities of stationary phases. These computational predictions of the interactions between stationary phases and analytes should be proven with experimental data in order for the model to be considered accurate. Stationary phases and analytes 1 modeled for this research were based on the work from ShenTP PT et al because the theoretical results should
reflect the high selectivity determined experimentally.P
Journal of the PGSS Computational Predictions of Chromatographic Selectivity Page 97
The polysiloxane liquid crystal stationary phase and xylene derivative analytes were initially modeled with 2 the AMBERTP PT force field to determine the free energy of the molecular complex. The free energy (calculated in kcal/mol) was then used to calculate selectivity:
∆∆ = ∆∆ − ∆∆ ⋅TSHG (1)
Entropy is assumed to be zero for the computations, so free energy can be assumed to equal enthalpy (∆∆H). The enthalpy is calculated relatively between two stationary phase/analyte systems with the following equation:
=∆∆=∆∆ − ()(( + AHHHG − H − ()(()) + AH )) (2) + ASP 1 SP 1separate + ASP 2 SP 2separate
H SP in the equation represents the energy of the stationary phase while An represents the energy of each respective xylene derivative analyte, with n indicating the specific isomer. Separate refers to the analyte and/or stationary phase energy individually. All simulations were run at a constant temperature
(T) of 85K. These results were entered into the following equation to solve for the selectivity ( 1/2B ),B where ½ refers to the degree of separation between two different compounds in a column:
⎛ ∆∆G ⎞ −⎜ ⎟ ⎝ RT ⎠ α 2/1 = e (3)
II. Methods
1 3 In extension of papers by both Shen et al.P P and Suzanne D. GardnerTP ,PT nine isomers of 4’-cyanobiphenyl- 4-propoxy-methylsiloxane (Figure 1) and four xylene derivatives were modeled in HyperChem® (Student Edition 7.5). The xylene derivatives were aligned in the “pocket” created between two biphenyl groups attached to the polysiloxane backbone of the stationary phase, such that the biphenyl groups and xylene derivatives would stack together along the backbone. Each of the nine polysiloxane isomers were identified by first, the orientation (ortho, meta, or para) of the biphenyl group with respect to the polysiloxane backbone and second, the orientation of their cyano- group with respect to biphenyl. For example, the m,p-polysiloxane isomer (Figure 2) is characterized by the biphenyl group being orientated meta to the rest of the polysiloxane stationary phase and the cyano- group being orientated para to the biphenyl chain. The four xylene derivatives were 1,4-dimethyldodecylbenzene (Figure 3), 1,5- dimethyldodecylbenzene, 2,3-dimethyldodecylbenzene, and 2,4-dimethyldodecylbenzene.
Page 98 Badgley, Carpenter, Rastogi, Widenor, Zhang
The simulations were run using the AMBER force field. The simulations were first geometry-optimized to find a stable energy conformation. Then the molecular dynamics simulations were performed using the geometry-optimized structure of each stationary phase/analyte system. The MD simulations were run for 1 10 ps at a temperature of 85K. This temperature was selected based on experimental data.P P
® Figure 1: p,p-polysiloxane isomer in HyperChemP P
Figure 2: m,p-polysiloxane isomer
Journal of the PGSS Computational Predictions of Chromatographic Selectivity Page 99
® Figure 3: 1,4-dimethyldodecylbenzene analyte in HyperChemP P
These methods were the basis for the main group project. Five individual projects were conducted by modifying the above methods. The results and discussion for these individual projects are found in section V.
III. Results
The selectivity of 1,5-dimethyldodecylbenzene and 1,4-dimethyldodecylbenzene was calculated for each isomer of the polysiloxane stationary phase. For simplicity, the pairs of analytes used to calculated selectivity will be referred to as 1,5/1,4-dimethyldodecylbenzene. Selectivity for 2,4/2,3- dimethyldodecylbenzene was also calculated with each isomer of the polysiloxane stationary phase. These results are shown in Tables 1 and 2, respectively. Overall, the selectivities for the theoretically- modeled stationary phase interactions were comparable to the experimental data, which resulted in high selectivity factors of approximately one.
Page 100 Badgley, Carpenter, Rastogi, Widenor, Zhang
In the case of 1,5/1,4-dimethyldodecylbenzene, almost all of the selectivities of the stationary phase isomers were approximately a factor of one. Some of the most apparent anomalies occurred in the o,p- and m,p-polysiloxane isomers. With the calculations of 2,4/2,3-dimethyldodecybenzene with the stationary phase isomers, nearly all selectivities were once again close to a factor of one. The only notable outlier in this set was the p,m-polysiloxane isomer.
Table 1: Selectivity of 1,5/1,4-dimethyldodecylbenzene Polysiloxane α 5,1 Isomer 4,1 o,m 0.981 o,p 0.885 m,p 1.119 p,p 1.009 p,o 0.947 m,m 0.998 p,m 1.013 m,o 1.006 o,o 1.029
Table 2: Selectivity of 2,4/2,3-dimethyldodecylbenzene
Polysiloxane α 4,2 Isomer 3,2 o,m 1.096 o,p 1.004 m,p 1.005 p,p 0.910 p,o 1.064 m,m 1.105 p,m 0.859 m,o 1.004 o,o 1.025
Journal of the PGSS Computational Predictions of Chromatographic Selectivity Page 101
IV. Discussion
A. Selectivity Values Based on Stationary Phase Orientation
Although the selectivity was calculated to be approximately the experimental value of one, some anomalies, based on polysiloxane orientation, were present. The anomalous selectivity values for the 1,5/1,4-dimethyldodecylbenzene and 2,4/2,3-dimethyldodecylbenzene with the m,p-polysiloxane isomer and the 1,5/1,4-dimethyldodecylbenzene with the o,p-polysiloxane isomer can be attributed to the orientation of the biphenyl and cyano- groups on the polysiloxane side chains (Figure 4).
Figure 4: (a) and (b) o,p- and m,p-polysiloxane isomers are steric inhibitors, while (c) p,p- is not because the analyte can easily align in the pocket formed by two p,p-side chains.
In each of these anomalous polysiloxane stationary phase isomers, it is apparent that the cyanobiphenyl bends at a severe angle, due to the initial orientation of the side chains. This shape prevents the desired stacking of the analyte in the polysiloxane isomer pockets (the space formed between the polysiloxane backbone and side chains). The dimethyldodecylbenzene is sterically inhibited because the bond between the oxygen and the cyanobiphenyl liquid crystal keeps the cyanobiphenyl at a near right angle to the chain (Figure 4).
Page 102 Badgley, Carpenter, Rastogi, Widenor, Zhang
B. Selectivity Values Based on Analytes
Comparisons of the selectivity values of 1,5/1,4-dimethyldodecylbenzene and 2,4/2,3- dimethyldodecybenzene indicate that differences in their orientation have an effect on the selectivity. In several cases, stationary phase orientations that yielded high selectivity with the 2,4/2,3- dimethyldodecybenzene did not exhibit the same high selectivity with the 1,5/1,4- dimethyldodecybenzene. For instance, o,p-polysiloxane isomer demonstrated a poor selectivity of 0.885 for 1,5/1,4-dimethyldodecylbenzene, but the same polysiloxane orientation resulted in a selectivity of 1.004 when simulated with 2,4/2,3-dimethyldodecylbenzene.
C. Overall Selectivity
Selectivity values calculated with the theoretical model were comparable to experimental data. These values were approximately one, which model stationary phases for gas chromatography applications. Because this model simulates and reproduces experimental data, we can assume that the model could be used to simulate additional stationary phases. In addition, the model could be further researched to predict exactly which orientations are preferable during a simulation and furthermore, why certain orientations are preferable.
III. Mini-Projects
A. The Effect of Temperature on Selectivity
1. Introduction
For the group research, all the simulations were run at the same temperature of 85 K. In this project, the simulations were rerun at a significantly higher temperature to study the effects on selectivity. During a GC run, a higher temperature results in an increase in elution rate of the components of a sample. However, if the temperature is too high, the polysiloxane stationary phase can either burn inside the column or the compounds can elute at the same time.
2. Methods
For this project, the p,o-polysiloxane isomer and m,m-polysiloxane isomer and the xylene derivative pairs ® were simlulated using HyperChemP .P The temperature was increased to 563 K and both geometry optimization and molecular dynamics simulations were performed on the systems using the AMBER force field. Selectivity was calculated based on the resulting energies according to Equation 3. Journal of the PGSS Computational Predictions of Chromatographic Selectivity Page 103
3. Results
Selectivity of 1,5/1,4-dimethyldodecylbenzene was calculated for the p,o- and m,m-polysiloxane isomers of the stationary phase. These results are shown in Table 3. The selectivities of 2,4/2,3- dimethyldodecylbenzene of those same isomers are shown in Table 4. The higher temperatures resulted in drastically higher energies from the simulation the energies were nearly ten times higher than those seen at the temperature of 85 K. Also upon examination of the simulation, the analytes were forced out of their polysiloxane pockets. However, the selectivities did not change nearly as much as the energy values would suggest.
Table 3: Selectivity of 1,5/1,4-dimethyldodecylbenzene
Polysiloxane 5,1 α , 85 K α 5,1 , 563 K Isomer 4,1 4,1 p,o 0.946 1.124 m,m 0.998 1.067
Table 4: Selectivity of 2,4/2,3-dimethyldodecylbenzene
Polysiloxane 4,2 α , 85 K α 4,2 , 563 K Isomer 3,2 3,2 p,o 1.064 0.998 m,m 1.105 0.901
Figure 5: p,o-polysiloxane isomer with 3,4-dimethyldodecylbenzene
Page 104 Badgley, Carpenter, Rastogi, Widenor, Zhang
4. Discussion
As shown in Tables 3 and 4, selectivities for the increased temperature simulations were comparable to the lower temperature results. The selectivity factors results indicate that this particular stationary phase and temperature combination would not cause all of the components of the sample to elute at the same time. However, while viewing the simulation, the analytes move out of the pockets of the polysiloxane, which will cause them to elute with the rest of the compounds. Hence, this early elution will result in difficulty detecting the solvent at the end of the run, which is typically where we expect to see it. So while theoretically, selectivity encourages good separation at higher temperatures for a polysiloxane stationary phase, the graphics employed by HyperChem® illustrate that the final conformation of the system is not desirable (Figure 5). It can be concluded that attempting to separate a sample using gas chromatography at a temperature of 563 K will not make it more effective.
B. The Effect of Force Fields on Selectivity
1. Introduction
Molecular modeling force fields are used to calculate the interactions of atoms and molecules. AMBER, for instance, models substances by computing the energy between covalent bonds, the energy due to electron configuration, the energy due to the torsion of bonds, and the non-bonded energy between all 4 atoms. BIO+CHARMMTP PT operates on much the same principle, but in addition to the standard quantum interaction computations, BIO+CHARMM also has explicit solvent parameters and the ability to group 5 atoms together so that they interact as larger structures, which speeds the modeling simulationTP .PT This ability to group atoms becomes particularly useful when simulating very large biological molecules, and it is one reason to test the effects of BIO+CHARMM on the polysiloxane models.
2. Methods
The methods of this study were the same as those of the group experiment, but the molecular dynamics was changed from the original AMBER force field to BIO+CHARMM force field. The study also differed in that only the o,m- and o,p-polysiloxane isomers were modeled.
3. Results
With respect to this study, the o,m-polysiloxane isomer selectivities are significantly lower than the o,p- polysiloxane isomer when modeled with the BIO+CHARMM force field compared to those obtained with the AMBER force fields. Despite this discrepancy, the selectivity values of the BIO+CHARMM force fields are not significantly different from the group results (Tables 1 and 2). Journal of the PGSS Computational Predictions of Chromatographic Selectivity Page 105
Table 5: Comparison of the selectivity factors of 1,5/1,4-dimethyldodecylbenzene and 2,4/2,3- dimethyldodecylbenzene in the AMBER and BIO+CHARMM force fields.
AMBER BIO+CHARMM 5,1 Polysiloxane Isomer α 4,1 o,m 1.096 0.121 o,p 1.004 1.004 4,2 Polysiloxane Isomer α 3,2 o,m 0.981 0.124 o,p 0.885 1.001
4. Discussion
The difference between the selectivities of the AMBER force field and the CHARMM force field is statistically insignificant (p<0.07, n=26) when comparing the BIO+CHARMM values in Table 5 of this sub- experiment to the data in Tables 1 and 2 of the group experiment. Unfortunately, this means it cannot be statistically proven that the change in force fields has a significant effect on the outcomes of a particular selectivity modeling experiment. This is discouraging because data obtained from experiments 6 conducted by researchers at the University of HoustonTP PT indicates that there are significant effects on molecular simulations depending on the force field employed. However, the Houston researchers were performing simulations of biological molecules (specifically, DNA) when comparing the AMBER and CHARMM force fields. These experiments deal with polysiloxane stationary phases, which are not biological molecules.
Additionally, CHARMM is designed to simulate biological molecules (hence it’s ability to group atoms) while AMBER is designed as a universal molecular modeling application (all atoms are explicit). 7 Moreover, the methods and calculations employed by CHARMM and AMBERTP PT are subtly different, affording greater speed to CHARMM and greater versatility to AMBER. CHARMM, for instance, defines 8 its solvent explicitlyTP ,PT which saves calculation time. These differences become apparent in a variety of experimental simulations.
For instance, a comparison of the effects of different force fields on trialanine conformations (a molecule much smaller than DNA, and more similar to polysiloxane in composition and complexity) conducted by researchers at the Frankfurt Institute of Physical and Theoretical Chemistry concluded that changes in force fields (e.g. AMBER to CHARMM) caused results to “differ considerably” in all areas, including free 9 energy valuesTP .PT Noting these discrepancies, it makes sense that a cursory comparison of average selectivities still suggests that a change in force fields has some effect.
Page 106 Badgley, Carpenter, Rastogi, Widenor, Zhang
For example, the group experimental average for the selectivity of the all the polysiloxane isomers calculated in AMBER was 1.003, and the experimental average for the two polysiloxane isomers in this sub-experiment was 0.992 (also calculated in AMBER). In contrast, the experimental selectivity average for the CHARMM polysiloxane isomers was a much lower 0.5625.
Considering that AMBER’s selectivity results suggest that polysiloxanes are a viable chromatography stationary phase ( is approximately a value of one) and that experimental evidence agrees (polysiloxane stationary phases are now ubiquitous in gas chromatographs), it is apparent that AMBER is the better molecular modeling force field for the simulation of polysiloxanes and similar compounds.
C. The Effect of Dicyanobiphenyl Liquid Crystals on Selectivity
1. Introduction
The structure of the polysiloxane molecule can be altered at the liquid crystal. Due to the interactions between the analyte and liquid crystal, the effect of different liquid crystals on the selectivity of the polysiloxane stationary phase needs to be examined. The addition of a cyano- group to the polysiloxane will change the complexity and size of the molecule (Figure 6). This could have an influence on the selectivity of the stationary phase.
Figure 6: m,p-polysiloxane isomer with dicyanobiphenyl liquid crystal
The cyanobiphenyl liquid crystal is used on a polysiloxane backbone because its interactions yield good 10 selectivity for a wide variety of analytesTP .PT Other polysiloxane stationary phases do not offer high selectivity for the same range of compounds as the cyanobiphenyl liquid crystal. Cyano- groups have polar interactions with the analytes; therefore, relative to the cyanobiphenyl liquid crystal, the dicyanobiphenyl liquid crystal would offer additional desirable interactions with the dimethyldodecylbenzene isomers. This study was designed to examine the effects of the dicyanobiphenyl as a liquid crystal on the polysiloxane stationary phase. Journal of the PGSS Computational Predictions of Chromatographic Selectivity Page 107
2. Methods
The methods of this study were identical to the group study conducted on the cyanobiphenyl liquid crystal polysiloxane stationary phase with the exception of the cyanobiphenyl liquid crystal being replaced with a dicyanobiphenyl group. Selectivity was calculated for the m,p- and p,p-polysiloxane isomers with the
(CN)2B liquidB crystal attached to the cyanobiphenyl side chain (Figure 6).
3. Results
The results for the two liquid crystals modeled in this study are shown in Table 6. While the selectivities of the dicyanobiphenyl liquid crystal isomers differ slightly from the cyanobiphenyl liquid crystal isomers, a noticeable difference is not apparent. Specifically, the selectivity for the dicyanobiphenyl was closer to the desired value of one for the m,p-polysiloxane isomer. However, in the case of the 2,4/2,3- dimethyldodecylbenzene, the value was below one, which is not desirable.
Table 6: Effects of cyanobiphenyl and dicyanobiphenyl liquid crystals on selectivity
Cyanobiphenyl Dicyanobiphenyl 5,1 Polysiloxane Isomer α 4,1 p,p 1.009 1.030 m,p 1.119 1.000 4,2 Polysiloxane Isomer α 3,2 p,p 0.910 1.020 m,p 1.005 0.992
4. Discussion
Because there is not a trend for improvement in selectivity values when modeling a dicyanobiphenyl liquid crystal, no solid conclusion can be made as to the effectiveness of this change. More research would need to be done in order to make a credible comparison between these two liquid crystals. This future research would include modeling all nine stationary phase isomers with both liquid crystals.
D. The Effect of Analyte Orientation on Selectivity
1. Introduction
In the group research, all simulations were performed by modeling the stationary phase with the analyte placed in the liquid crystal pocket tail-first. For this investigation, the analytes were inserted head-first. It Page 108 Badgley, Carpenter, Rastogi, Widenor, Zhang is important to study the effect of analyte orientation because in an actual gas chromatography run, it is difficult to determine which orientation the analyte will prefer in the pocket: head-first or tail-first.
2. Methods
In order to run a tail-first simulation, the p,m- and p,p-polysiloxane isomers were modified in the ® HyperChemP P student program by flipping the analytes so that the heads, or benzene rings, faced toward the inside of the pocket. These systems were first geometry-optimized using the AMBER force field, and then simulated using molecular dynamics. Using the resulting molecular dynamics energies, selectivity was calculated using Equation 3.
3. Results
The selectivity of 1,5/1,4-dimethyldodecylbenzene was calculated for the p,p- and p,m-polysiloxane isomers and these results are shown in Table 7. The selectivity of 2,4/2,3-dimethyldodecylbenzene was also calculated for the same stationary phase isomers, and the results is shown in Table 7 as well.
The p,p-polysiloxane isomer showed selectivities of approximately one in both comparisons. The p,m- polysiloxane isomer, on the other hand, had selectivities of far less than 1 when both pairs of analytes were compared.
Table 7: Selectivity of 1,5/1,4-dimethyldocecylbenzene and 2,4/2,3-dimethyldodecylbenzene
Polysiloxane 5,1 4,2 α α 3,2 Isomer 4,1 p,o 0.946 1.064 m,m 0.998 1.105
4. Discussion
While both the p,p-polysiloxane isomer and the p,m-polysiloxane isomer had acceptable selectivity when the analytes were placed in the pockets tail-first, the p,m-polysiloxane isomer failed to maintain its good selectivity when the analytes were inserted head-first instead. Therefore, in an actual GC run, analytes that interact with a p,m-polysiloxane isomer in a head-first manner would have a lower selectivity. This would make a p,m-polysiloxane isomer less effective because the peaks would be unidentifiable. Future research may give insight into why this difference in selectivity is present, as well as which of the nine stationary phase orientations are preferable. Journal of the PGSS Computational Predictions of Chromatographic Selectivity Page 109
E. The Effect of Functional Groups on Selectivity
1. Introduction
When researching gas chromatography, the size and shape of the analytes, solid phase, and liquid crystal determine the retention rate of the process. As a result, modifying the liquid crystal attached to the 11 stationary phase can have a significant effect on the selectivity of the modelTP .PT The liquid crystal cyano- group of the group project was changed to biphenylcarboxylate ester for this study, which is significantly larger than the cyano- group. Compared to the cyano- liquid crystal, the biphenylcarboxylate ester does not simply have a linear relationship to the biphenyl molecule on the stationary phase side chain. Furthermore, if the new model achieves great selectivity, experiments can also be done to test the 11 efficiency and stability, as both are affected by the functional groupP .P Increases in efficiency and stability would reduce the cost of running experiments and allow effective modeling at different temperatures, respectively.
2. Methods
The cyano- group was replaced with biphenylcarboxylate ester. Every other variable, such as the polysiloxane backbone, the position of the benzene bonds, and the orientation of the analytes, remained the same. Each polysiloxane isomer was simulated with the same analytes as the group project. The molecular dynamics simulation was run using the AMBER force field at 85K.
3. Results
The selectivity values of the m,o- and o,o-polysiloxane isomer with biphenylcarboxylate were calculated for both 1,5/1,4-dimethyldodecylbenzene and 2,4/2,3-dimethyldodceylbenzene and the results are shown in Table 8 below. All selectivity values are close to one, which demonstrates good selectivity in every combination of stationary phase and analyte. Page 110 Badgley, Carpenter, Rastogi, Widenor, Zhang
Table 8: Comparison of the selectivity of 1,5/1,4-dimethyldodecylbenzene and 2,4/2,3- dimethyldodecylbenzene with cyano- and biphenylcarboxylate ester liquid crystals
Cyano- Biphenylcarboxylate Side Compound Ester 5,1 Polysiloxane Isomer α 4,1 m,o 1.006 1.013 o,o 1.029 1.011 4,2 Polysiloxane Isomer α 3,2 m,o 1.004 1.034 o,o 1.025 1.030
4. Discussion
The selectivity values of the two modified polysiloxanes indicated negligible differences between the cyano- and the biphenylcarboxylate ester- polysiloxane systems. The largest difference in selectivity between the two samples is 0.03, which is negligible. This experiment showed that either liquid crystal could be used for gas chromatography, depending on availability and cost. Nevertheless, it is worth noting that the biphenylcarboxylate ester liquid crystal and polysiloxane stationary phase produced worse selectivity than the cyano- group liquid crystal and polysiloxane stationary phase in the m,o-polysiloxane isomer while yielding better selectivity in the o,o-polysiloxane isomer. Furthermore, the m,o-polysiloxane isomer showed a greater difference than the biphenylcarboxylate ester liquid crystal and o,o-polysiloxane isomer. These differences demonstrate that it would be meaningful to run the molecular dynamics simulations using other isomers of polysiloxane with biphenylcarboxylate ester as the liquid crystal.
IV. Acknowledgements Dr. Barry Luokkala Suzanne Gardner Laura Anzaldi Pennsylvania Department of Education Pennsylvania Governor’s School for the Sciences Mellon College of Science Carnegie Mellon University Journal of the PGSS Computational Predictions of Chromatographic Selectivity Page 111
V. References
1 TP PT Y. Shen, W. Li, A. Malik, S.Reese, B. Rossiter, and M. Lee, "Cyanobiphenyl-substituted polymethylsiloxane encapsulated particles for packed capillary column supercritical fluid chromatography," J. Microcolumn Separations 7, (4), 411-419, (1995). 2 TP D.PT Pearlman, D. Case, J. Caldwell, W. Ross, T. Cheatham, D. Fergusen, G. Siebel, U. Singh, P. Weiner, and P. Killman. AMBER 4.1; University of San Fransisco: San Francisco, CA, 1995. 3 TP PT S. Gardner and P. Schettler. "Computational predictions of chromatographic selectivity of polysiloxane liquid crystal stationary phases," Juniata College, Department of Chemistry, Huntingdon, PA 16652, May 2004. 4 TP PT B. R. Brooks, R. E. Bruccoleri, B. D. Olafson, D. J. States, S. Swaminathan, and M. Karplus, “CHARMM: A program for macromolecular energy, minimization, and dynamics calculations,” J. Comp. Chem. 4, 187-217 (1983). 5 TP C.PT Brooks, conference handout from "Overview of Molecular Modeling and Dynamics" on Jun. 5, 2000 in Workshop on Methods and Applications of Molecular Dynamics to Biopolymers, Jun. 4-7 2000, Pittsburgh Supercomputing Center. 21 July 2007. http://www.psc.edu/general/software/packages/charmm/tutorial/brooks/mmd.pdf. 6 TP PT M. Feig and B. Pettitt, “Experiment vs Force Fields: DNA Conformation from Molecular Dynamics Simulations.” J. Phys. Chem. B, 101, (38), 7361 - 7363 (1997). 7 TP PT Y. Duan, C. Wu, S. Chowdhury, M. Lee, G. Xiong, W. Zhang, R. Yang, P. Cieplak, R. Luo, T. Lee, J. Caldwell, J. Wang, and P. Kollman, “A point-charge force field for molecular mechanics
simulations of proteins based on condensed-phase quantum mechanical calculations.” JT Comp
Chem T 24, (16) 1999-2012 (2003). 8 TP PT C. Brooks, J. Chen, and W. Im, “Balancing solvation and intramolecular interactions: toward a consistent generalized born force field (CMAP opt. for GBSW),” J Am Chem Soc 128, (11), 3728-3736, (2006). 9 TP PT Y. Mu, D. Kosov, and G. Stock, “Conformational Dynamics of Trialanine in Water. 2. Comparison of AMBER, CHARMM, GROMOS, and OPLS Force Fields to NMR and Infrared Experiments.” J. Phys. Chem. B, 107, (21), 5064 - 5073 (2003). 10 TP PT A. Malik, S. Reese, S. Morgan, J. Bradshaw, and M. Lee. "Dicyanobiphenyl polysioloxane stationary phases for capillary column gas chromatography," Chromatographia 46, (1/2), 79-84, (July 1997). 11 TP PT K. Markides, M. Nishioka, B. Tarbet, J. Bradshaw, and M. Lee, "Smetic biphenylcarboxylate ester liquid crystalline Polysiloxane Stationary Phase for Capillary Gas Chromatography." Analytical Chemistry. 57, 1296-1299, (1985).
Page 112 Badgley, Carpenter, Rastogi, Widenor, Zhang
Appendix A: Compounds Simulated with Molecular Modeling
Figure 1: p,p-4’-cyanobiphenyl-4-propoxy-methylsiloxane (referred to as p,p-polysiloxane isomer)
Journal of the PGSS Computational Predictions of Chromatographic Selectivity Page 113
Figure 2: The four xylene derivatives: 1,4 dimethyldodecylbenzene, 1,5-dimethyldodecylbenzene, 2,3-dimethyldodecylbenzene, and 2,4-dimethyldodecylbenzene
Page 114 Badgley, Carpenter, Rastogi, Widenor, Zhang
Figure 3: Biphenylcarboxylate ester liquid crystal
Figure 4: Cyanobiphenyl liquid crystal
Figure 5: Dicyanobiphenyl liquid crystal Journal of the PGSS Computational Predictions of Chromatographic Selectivity Page 115
Appendix B: The nine isomers of 4’-cyanobiphenyl-4-propoxy- methylsiloxane (referred to as polysiloxane)
Page 116 Badgley, Carpenter, Rastogi, Widenor, Zhang
Appendix C: Steric inhibition of xylene derivative by the cyanobiphenyl side chain orientation in p,p- and o,p-polysiloxane isomers
Journal of the PGSS Effects on Resistance and Lattice Structure Page 117
Effects on Resistance and Lattice Structure Using Substitution-Based Superconductors
Kevin Cockerham, Jessica Frey, Milton Herrold, Isaac Johnson, Healy Ko, Jonathan Lucas, Susan Phan, Ryan Rosario, and Jesse Ru
Abstract
The purpose of this experiment was to create compounds that exhibit superconductivity at higher temperatures and have a higher current carrying capacity than the YBa2Cu3O7-x compound. To achieve this result, various compounds were substituted into the YBa2Cu3O7-x ceramic compound at the Yttrium and Barium positions based on similar atomic radii and charge. Overall, the full substitution of europium for yttrium was most effective, because it was able to superconduct at a higher temperature and had a higher current carrying capacity than the YBa2Cu3O7-x.
I. Introduction A. History of Superconductors
Superconductors were first discovered in 1911 when Dutch physicist Heike Kammerlingh Onnes passed a current through a pure mercury wire and measured its resistance while lowering the temperature.1 Until that point, scientists had hypothesized that the resistance of a sample would decrease as temperature was lowered, creating a better conductor. However, when Onnes reached the temperature of 4.2 K, the resistance of the mercury wire completely vanished. This absence of resistance at extremely low temperatures soon came to be known as “superconductivity”.
Since then, research into this phenomenon has expanded due to increasing interest in the subject of superconductivity. In 1933, German researchers Walter Meissner and Robert Ochsenfeld reported evidence of the Meissner effect and perfect diamagnetism.2 This discovery was very important in showing that superconductors had many more unique properties than regular conductors. The Meissner effect is responsible for levitation due to the repulsion of magnetic fields.
By 1957, the first widely accepted theory on superconductivity was formed by American physicists John Bardeen, Leon Cooper, and John Schrieffer. Their theory later came to be known as BCS Theory. BCS Theory outlined the principle of Cooper pairs, in which two electrons interact due to an attractive force.
Many experiments have been conducted in order to find a material that would exhibit superconductivity at higher temperatures. The critical temperature (Tc) of a material is the highest temperature at which it behaves as a superconductor. By 1953, a vanadium-silicon compound was shown to superconduct at 17.5 K. By 1962, Westinghouse created the first commercial Page 118 Cockerham, Frey, Herrold, Johnson, Ko, Lucas, Phan, Rosario, Ru superconducting wire using an alloy of niobium and titanium.3 In 1986, Alex Müller and Georg Bednorz discovered the first ceramic compounds to act as superconductors. Ceramics are normally insulators, so the concept of using ceramics to superconduct was originally overlooked. Müller’s and Bednorz’s experimentation showed that the lanthanum-barium- copper-oxygen ceramic compound they tested actually superconducted at 30 K. Only a year later, a team at the University of Alabama-Huntsville discovered a ceramic compound, known as the 1-2-3 ratio, or 1 YBa2Cu3O7-x, that superconducted at a remarkable 92 K. Certain elements with ionic radii similar to those of yttrium and barium and with the same charge as yttrium or barium can be substituted into this 1-2-3 compound at varying ratios to produce new ceramics with unique Tc values. As the first material to superconduct at temperatures above that of liquid nitrogen, YBa2Cu3O7-x opened up new possibilities for commercial applications of superconductors.
Currently, most superconducting materials are classified in one of two categories: Type I or Type II. Type I superconductors are very pure samples of lead, mercury, or tin, that typically superconduct at extremely low temperatures, from .0003 K to only 7 K. Type II superconductors tend to be able to conduct at much higher temperatures, though still far below room temperature.
For the YBa2Cu3O7-x compound and its derivatives, current research shows that the Tc can range from 60 K to over 107 K, depending on the type and ratio of substitutions made for yttrium and barium. Over the past few years, many surprising new discoveries have been made using these ceramic compounds. To date, the ceramic compound of (Hg0.8Tl0.2)Ba2Ca2Cu3O8.33 superconducts at the highest temperatures of 138 K.1
To date, all superconductors still only work at extremely low temperatures, so they need some coolant, such as liquid nitrogen, in order to function. However, if a material could be found that superconducts at warmer temperatures, preferably room temperature, then the applications for superconductors would be more universal.
B. The Theory Behind Superconductors 1. BCS Theory
The motion of an electron can be represented by a wave. In a perfect conductor, the electron wave moves freely through a crystal lattice without any scattering from lattice vibrations if the wave is initially in phase with the lattice. However, with current technology, it is impossible to create a structure in which there is zero vibrational energy. The electrons interact with the lattice, and the wave motion is Figure 1: Cooper Pair Moving Through Crystal Lattice 5 Journal of the PGSS Effects on Resistance and Lattice Structure Page 119 disrupted by the vibrational energy of the lattice. This causes the electrons to be scattered. When the electrons are scattered, momentum and energy must be conserved. As a result, electron-interaction creates a phonon, a particle that possesses a quantifiable momentum, which gains some of the momentum of the electron. When this phonon makes contact with another electron, a weak attractive force is created. At normal temperatures, the attractive force created by the electron-lattice interaction is too weak to overcome vibrational energies, but at temperatures, less than 40 K, the attractive force created by the electron-lattice interaction is enough to form Cooper pairs.4 Due to the quantum mechanical properties of Cooper pairs, when momentum is applied to these Cooper pairs, the net momentum of all the pairs in the system is conserved, i.e. there is no scattering. Because there is no scattering of the Cooper pairs, there is no resistance in the system. 4
The discovery of the isotope effect by Maxwell in 1950 further supported BCS Theory. In his experiments, Maxwell found that the isotopic mass directly affected the critical temperature of a superconductor. By using different isotopes of an ion in the compound, Maxwell was able to change the vibrational frequency of the ion without affecting its lattice structure or significantly altering its chemical properties. This discovery proved that the lattice itself affects the superconducting electrons.4
Previous experiments show that good superconductors make poor room-temperature conductors, and good room-temperature conductors make poor superconductors due to this electron-lattice interaction. Because superconductors require higher-energy phonons in order to create the attractive forces necessary to form Cooper pairs, a large amount of an electron’s initial energy must be used. On the other hand, good room-temperature conductors conserve the greatest amount of the electron’s initial energy as possible, thus an electron transfers a minimal amount of energy to the creation of a phonon.
BCS Theory attempts to explain superconductive properties in Type I and Type II superconductors, but loses credibility in higher temperature superconductors.
2. Modern Theories
The discovery of high-Tc superconductors created from copper oxides formed a new class of superconductors that BCS Theory was unable to adequately explain. There are many different models attempting to explain the phenomenon of high-Tc superconductivity, with some resembling BCS Theory and others contradicting BCS Theory. It has been generally accepted that the interaction between electrons and phonons is involved in superconductivity, but it is also argued that there is another predominant causality creating superconducting materials. There is no widely accepted theory that explains why electrons can pair at high temperatures creating high-Tc superconductors, but there are three main branches of theory under which all other theories follow. These include: pairing through electron-phonon interaction (BCS-like theories), pairing through magnetic correlation of the electrons, and pairing through the exchange of electronic polarization resonances.6 Page 120 Cockerham, Frey, Herrold, Johnson, Ko, Lucas, Phan, Rosario, Ru a. Polaron and Bipolaron Model
Polaron and bipolaron superconductivity is somewhat similar to BCS Theory, in that it attempts to explain the phenomenon of superconductivity by addressing the interaction between electrons and phonons. If an electron travels slowly enough through a lattice, the interaction between the electric fields of the lattice and electron will cause phonons to be emitted. This interaction causes the electron to become surrounded by a “cloud of phonons,” also known as distortion, which possesses a strong attraction for the 7 Figure 2: Small polaron electron. This process creates a quasi-particle known as a polaron. When two polarons are in close proximity, they may share distortions in order to lower their energy and create an attractive force. If this attractive force is large enough, a bipolaron may form. Polarons with large radii and bipolarons may form a Bose-Einstein condensate below the temperature for Bose condensation.8
A Bose-Einstein condensate is a substance made up of bosonic particles, which are particles that can share quantum states in the lowest quantum state of energy. Thus it cannot lose energy, but only gain it. Due to this property, Bose-Einstein condensates have unique abilities. Most notably, they can instantly transport heat and energy, so the Bose-Einstein condensates formed by the polarons or bipolarons in superconductors can instantly transport current. According to quantum mechanical equations associated with this theory, superconductors can potentially have critical temperatures close to room temperature.9 b. Excitonic Model
In order for a substance to conduct according to the excitonic model, there must be alternating layers of insulators with highly polarizable molecules and conductors.10 The electron-lattice interaction still produces an attractive force via the exchange of phonons, but this coupling does not explain why the copper-oxide superconductors have such high Tc values. The excitonic model mainly attributes the higher Tc value to Coulombic attraction that occurs between an electron and an exciton. An exciton is a quasi-particle made up of a pair containing an electron and a hole, a region of positive charge created by the absence of an electron. This exciton can couple with an electron, thus creating pairing which causes superconductivity.11
Journal of the PGSS Effects on Resistance and Lattice Structure Page 121
C. Lattice Structure
1. Y1Ba2Cu3O7-x
The YBa2Cu3O7-x compound has a Perovskite crystal structure, which is a typical orthorhombic crystal structure
that resembles CaTiO3 crystals. In each lattice of
YBa2Cu3O7-x, there is a central yttrium ion, one barium ion above and below the yttrium, and copper and oxygen ions surrounding the yttrium and barium ions. In the lattice, there are yttrium planes, copper and oxygen planes, and barium and copper planes.
The Cu-O bonds, which allow for the transfer of electrons, must be coplanar for the Y-123 superconductor to function properly. These bonds are the key to superconductivity in
any known high-Tc superconductor. Experimentation has
shown that misaligned Cu-O bonds cause the Tc to reduce or superconductivity to disappear completely, because the electrons must change momentum in order to travel across the Cu-O bonds. A change in momentum denotes scattering, which denotes resistance. This was found to Figure 3: Y-12312 be true in the tetragonal crystal structures created by variations in the Y-123 crystal lattice created through substitution or different oxygen stoichiometries.13
D. Resistance Theory 1. Resistivity Resistance measures a material’s opposition to the flow of an electric current and can be defined as the ratio of voltage over current (R = V/I) with units of ohms. It is caused by the scattering of electrons as they move through a metal lattice, resulting in the loss of energy in the form of heat.14
In addition to the mathematical calculation, a general trend for resistance can also be found based on the temperature and the type of material being tested, including insulators, metals, semiconductors, and superconductors. Insulators have a consistent and extremely high resistance, regardless of temperature. Metals are considered good conductors because they readily conduct electricity at room temperature. Semiconductors will only conduct electricity at certain temperatures. The higher the temperature, the more likely the semiconductor will conduct electricity. Inversely, at very low temperatures, semiconductors work as insulators. Superconductors have resistance at most temperatures; however, when the temperature has been lowered below the critical temperature, resistance falls abruptly to zero.15
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Resistance can change for superconductors depending upon which elements are mixed together. If the superconducting compound is not completely pure, properties of the other conducting sources may be displayed in the graphs. For example, if a compound is formed from both superconducting materials and semiconducting materials, the graph will appear to superconduct at first; however, instead of resistance increasing linearly and almost vertically at the critical temperature, that line will contain bulges that indicate the presence of semiconducting materials.
Properties of Superconductors 1. Zero Resistance In a perfect crystal lattice, electrons in phase with the lattice can propagate through the lattice experiencing no scattering, and therefore no resistance. Vibrational energies cause a pure crystal lattice to deviate from a perfect periodic pattern. These deviations interfere with the conduction of electrons causing scattering, which creates resistance. As temperature decreases, the overall vibrational energy decreases, which, in metals, causes a decrease in resistance. Superconducting substances behave somewhat differently. They behave like metals until they are cooled to a specific temperature, called the “transition temperature” or the “critical temperature,” which varies from substance to substance. At the critical temperature, these special substances enter the superconducting state, where they exhibit no resistance.
In order to scientifically prove that superconductors truly possess zero resistance, an experiment that does not simply measure the current with an ammeter had to be performed. This had to be done because an ammeter is not precise enough. Rather, the magnetic field density surrounding the superconductor could be Figure 4: Perfect cubic lattice16 measured. If the magnetic field decays over time, then there was some resistance. Two physicists, Quinn and Ittner, measured the magnetic field decay. From this experiment, it was deduced that the resistivity of a superconducting metal to be 4×10-25 ohm-meters, which is a negligible amount.17
Journal of the PGSS Effects on Resistance and Lattice Structure Page 123
2. Meissner Effect and Diamagnetism
The zero resistance property of The zero resistance property of superconductors necessitates frictionless electron movement, such that the energy in the system of the superconductor is conserved within the superconductor. Metal conductors lose energy, in the Figure 5: Magnetic Field Lines for form of light and heat, at room temperature, due to resistance. Conversely, superconductors have frictionless movement of energy, because of the lack of resistance. Superconductors also exhibit other properties involving another form of energy, known as diamagnetism. Diamagnetism is a form of magnetism which repels any form of magnetic field near it. This property is 19 observed when another Figure 6: Meissner Effect magnetic field is present near a diamagnet. Diamagnetism allows magnetic fields to exist around the superconductor, but not inside it. This is because the magnetic field flows around the atoms inside the superconductor. Diamagnetism changes the flow of electrons within the superconductor to degrade and cancel out any magnetic flux, or field, from an applied magnetic field outside it. The opposing magnetic field makes the orbital spin of the electrons opposite that of the applied magnetic field, resulting in mutual repulsion. Since there is no resistance this can continue indefinitely. The electrons create a magnetic field outside the superconductor which pushes back with a repulsive force. This property is observed as the Meissner Effect. Figure 5 shows how the magnetic field lines behave when the temperature is greater than the Tc, and when the temperature is less than the Tc. If a magnet is placed above a superconductor while the superconductor is below the critical temperature, then the field produced from the magnet creates a magnetic field around the superconductor. This repels the magnet causing it to levitate. Figure 6 displays the phenomenon. Since there is no resistance, levitation can continue indefinitely, without the need for current being passed through the superconductor.17
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E. Applications of Superconductors
Superconductors have diverse, practical uses in today’s world. With their special properties, they have revolutionized technology.20
Their property of magnetic levitation assists in attempts of constructing new technologies, including nearly-frictionless trains. By effectively eliminating friction, the amount of wasted energy released as heat is reduced to essentially zero. This allows levitating trains to travel at much faster speeds than non-levitating trains.20
Although superconductors are used to improve transportation, they also have an impact in neurology and military affairs. The efficiency of Magnetic Resonance Imaging (MRI) has been significantly enhanced by the discovery and application of superconductors. Before superconductors were used, it would take up to five hours to generate a single MRI image. With the use of superconductors, MRI machines require only about 30 to 90 minutes to compile the data. A section of the Korean Research Institute of Standards and Science (KRISS), The Korean Superconductivity Group, has invented the Superconducting Quantum Interference Device (SQUID). This contrivance is capable of detecting miniscule magnetic fields. It is most commonly used for locating metallic instruments such as land mines.21
Superconductors do not currently have many other characteristic usages. This is due to operation exclusively at cryogenic temperatures. The ultimate goal is to synthesize a superconductor that operates at room temperature. If such a superconductor was created, it would provide many more benefits to society.20
Journal of the PGSS Effects on Resistance and Lattice Structure Page 125
II. Purpose
For this experiment, different elements were selected to substitute into the YBa2Cu3O7-x superconducting compound with two distinct goals in mind. First, the substituted compounds should superconduct at a temperature higher than the critical temperature of the parent compound. The second goal was to make a superconductor with a higher current carrying capacity. The purpose of enhancing these two superconducting properties was to improve the range of possible applications. A high Tc compound could lead to superconduction without cryogenic equipment; and a high current carrying capacity compound would allow for more complex applications, which demand more current.
ErBa(2)Cu(3)O(7-x) Ho(0.6)Dy(0.1)Y(0.1)Eu(0.1)Er(0.1)Ba(2)Cu(3)O(7-x) Current vs. Critical Temperature Current vs. Critical Temperature 80 90 70 80
70 60
60 50
50 40 I (m A)
I (mA) 40 30 30 20 20 10 10
0 0 91.5 92 92.5 93 93.5 94 94.5 91.5 92 92.5 93 93.5 94 94.5 Temperature (K) Temperature (K)
Figure 7: High Current Carrying Capacity Figure 8: Low Current Carrying Capacity
III. Procedure
A. Synthesis of Superconductors 1. Stoichiometry
This is the process used to synthesize superconductors based on YBa2Cu3O7-x. The first step was to find suitable elements to substitute, based on ionic radii and charge, for Y and Ba in the
YBa2Cu3O7-x superconductor. After determining what substitutions to make, the first step was to use stoichiometry to find the precise amount of reactants required to make 3.75 grams of the new superconductor. Figure 9 displays an example of the calculations.
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Base material Materials Used
YBa2Cu3O7-x Er2O3 Substituted as: BaCO3 ErBa2Cu3O7-x CuO
Molar Mass= MassEr + 2MassBa + 3MassCu = 167.26 + 2× 137.33+ 3× 63.55 = 632.57 g
MolesSuperconductor= 3.75/Molar Mass =3.75/632.57 = .00592819767 moles
Moles of Er = MolesSuperconductor = 0.00592819767 moles Moles of Ba= MolesSuperconductor × 2 = .00592819767× 2 = 0.011856395 moles Moles of Cu = MolesSuperconductor × 3 = .00592819767 × 3 = 0.017784593 moles
Recipe Grams of Er2O3 = Moles of Er ×(2MassEr + 3MassO) = 0.00592819767× (2× 167.26 + 3× 15.9994) = 2.268 grams
Grams of BaCO3 = Moles of Ba × (MassBa + MassC + 3MassO) = .011856395× (137.33 + 12.011+ 3 ×15.9994) = 2.340 grams
Grams of CuO = Moles of Cu ×(MassCu + MassO) = .017784593 ×(63.55+15.9994) = 1.415 grams
Figure 9: Sample Calculations for Stoichiometry
2. Measuring/Mixing
The masses of the reactants were measured to four significant figures to insure all the reactants completely reacted. The measuring was done in a weighing boat, based on the calculations from the stoichiometry. Once the measuring was completed, the mixture was ground together in order to thoroughly mix the reactants. This process required five to ten minutes of hard grinding with the mortar and pestle. Afterwards, the mixed powder was uniform in color and texture.
3. Firing/Pressing
The uniform powder was then loaded into pure aluminum boats and carefully set into the furnaces. Furnace 1 was heated to 975 degrees Celsius while Furnace 2 was heated to 950 degrees Celsius. Journal of the PGSS Effects on Resistance and Lattice Structure Page 127
Firing Sequence
1200
1000
800
Furnace 1 600 Furnace 2
Temperature (K) Temperature 400
200
0 0 1020304050 Time (hr)
Figure 10: Firing Sequence
Oxygen was piped through the furnaces to guarantee there was enough oxygen available to produce the proper oxygen ratio (07-x) in the superconductor. After one day in the furnace, the mixture was a uniform powder, which was pressed into thin disks.
In order to press the powder into disks, a die set and hydraulic press were used. The powder was loaded into the die set up to the two millimeter level. After loading the powder, the die set was placed into the hydraulic press. First, the pressure was increased to 10,000 psi and then allowed to drop. This process was repeated, increasing the pressure until it remained at 20,000 psi. This process formed a brittle but solid disk of superconducting material. The disks were then fired again with the same firing sequence as before. After the Figure 11: Hydraulic Press second firing sequence, the final product was produced. This product was then sent to the second portion of the lab: testing resistance and x-ray crystallography.
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B. X-Ray Diffraction After the samples’ firing for the second time, x-ray diffraction was conducted to determine the lattice structures and lattice parameters of the pellets. Initially, each pellet was cut into two pieces – one larger than the other. The smaller sampling was broken up and ground into a fine powder using a mortar and pestle. A thin layer of the ground sample was loaded onto a plate that was lightly coated in vacuum grease, a silicone lubricant. Then, the x-ray machine was activated and set up in order to load the sample and the plate into the machine. The machine was set to 40 kV and 30 mA. The machine was also set at 2 / , so that the sample moved at half the rate of the detector while the x-ray Figure 12: X-Ray Machine tube remained stationary. Figure 12 shows the X-ray machine.
While the machine was running the sample, a program called “dsSiemac” presented the data from the x-ray diffraction on a 2 / graph. To standardize the procedure, the goniometer, a device that allows each sample to be rotated about certain angles, was always set to 20.0°. After the spectra from the x-ray diffraction was obtained, it was transferred into “X-ray Fit,” a program that fits the orthorhombic lattice parameters of a, b, and c onto the spectra. When the a, b, and c parameters were inputted, the program displayed red vertical lines that represented the positions of the major peaks that have crystal lattice structures with the given parameters.
The purpose of this analysis was to match the red lines to the major peaks present on the graph in order to identify or verify the crystalline structure. Through this analysis, it became evident how the crystal lattice structures of the samples differed from that of the standard YBa2Cu3O7-x commercial sample. The placements of the lines, which corresponded to the lattice parameters, confirmed the structure to be orthorhombic. The graphs also revealed the relative purity of the pellets. Using the “Scaled Multiplot Program,” which allows graphs to be plotted together for comparison, samples with similar substitutions were plotted against a standard base YBa2Cu3O7-x graph in order to compare their peaks.
C. Resistance Testing An electrical circuit was used to determine the superconducting capabilities of the samples. This circuit began with a power supply of 18 volts, using 2 9-volt batteries set in series, next to an on- off switch that controlled the entire circuit. To the right of the switch were five different resistors positioned parallel to each other. Since their function is to restrict the current, each could be chosen separately by a switch apparatus. The resistor values were as follows: R1 = 18 kΩ, R2 = 9 kΩ, R3 = kΩ, R4 = 1.8 kΩ, and R5 = .3 kΩ. The amount of current that flows through the sample depended on which resistor is chosen for the test: the higher the resistor value, the lower the current. The current then traveled through the standard resistor, set at 10 Ω, parallel to an volt- ohm-current meter that displayed its value for each test run. Figure 13 shows the circuit schematic.
Journal of the PGSS Effects on Resistance and Lattice Structure Page 129
Figure 13: Circuit Schematic
The next part of the circuit included its connecting units and the sample. Four metal prongs held the sample in place. The two inner prongs were connected to an amplifier, a device used to control a larger amount of energy, while the two outer prongs were connected to the circuit, allowing the current to flow through one part of the sample and test for electrical conduction. A sensitive temperature wire positioned adjacent to the sample enabled the cool-down temperature and critical temperature to be monitored.
From the sample, the current passed through a wire connected to another switch. This switch controlled the transfer of the current from the circuit itself to the computer program. This program read the samples’ current, voltage, and temperature, and created graphs that plotted temperature versus resistance.
The resistance data collection began with a cool-down process. The tube containing the sample was first flushed with helium to facilitate cooling and prevent condensation on the inside of the tube. Then, the tube was inserted into the Dewar (see Figure 14), a long cylinder containing liquid nitrogen. This process allowed the sample to be cooled very quickly from room temperature to a range of 77-80 K.
Page 130 Cockerham, Frey, Herrold, Johnson, Ko, Lucas, Phan, Rosario, Ru
After cool-down, the R1 resistor was used to pass a current of about 1 mA through the sample. The tube containing the sample was raised slightly out of the liquid nitrogen so that the tube itself remained submerged in the liquid nitrogen, but the part of the tube containing the sample was positioned above the liquid surface. This caused the temperature to rise very slowly (generally in increments of approximately 0.1 K) from around 80 K to 110 K. As the temperature increased, the computer plotted a graph of temperature versus resistance. If this graph resembled a curve characteristic of a metal or a semiconductor, or if the sample did not display zero resistance even at the lowest temperature range of the liquid nitrogen, testing was stopped after the first run. However, if the graph did reach zero resistance, the tests were repeated at approximately 2 mA, 6 mA, 10 mA, and 65 mA. During these tests, the computer recorded the samples’ temperature, voltage, current, and resistance, and this data was later analyzed in Excel.
Dewar Helium gas
Glass Vacuum tube
Baseboard with sample
Liquid nitrogen
Figure 14: Dewar Device
Journal of the PGSS Effects on Resistance and Lattice Structure Page 131
IV. Data/Analysis A. Suitable Elements for Substitution
Element Ionic radii (Å) Valence charge Yttrium (Y) 0.90 +3 Holmium (Ho) 0.90 +3 Dysprosium (Dy) 0.91 +3 Europium (Eu) 0.95 +3 Erbium (Er) 0.89 +3 Lanthanum (La) 1.061 +3
Barium (Ba) 1.35 +2 Strontium (Sr) 1.12 +2 Lead (Pb) 1.19 +2
Table 1: Suitable Substitutions
B. Synthesized Compounds Synthesized Compounds YBa2Cu3O7-x EuSr2Cu3O7-x ErBa2Cu3O7-x Ysr2Cu3O7-x EuBa2Cu3O7-x EuBaSrCu3O7-x LaBa2Cu3O7-x Ho.5Er.5Sr2Cu3O7-x EuBa2/3Pb2/3Sr2/3Cu3O7-x Ho.5Er.5SrBaCu3O7-x Y.5Ho.5Ba2Cu3O7-x Y.75Ho.25Ba2Cu3O7-x
Eu.2Y.4Ho.4Ba2Cu3O7-x Y.6By.3Ho.1Ba2Cu3O7-x
Ho.25Dy.15Y.3Eu.1Er.2Ba2Cu3O7-x Ho.6Dy.1Y.1Eu.1Er.1Ba2Cu3O7-x Ho.2Y.2Dy.2Eu.2Er.2Ba2Cu3O7-x Y.3Ho.25Dy.15Eu.1Er.2Ba2Cu3O7-x
Table 2: Synthesized Compounds
Page 132 Cockerham, Frey, Herrold, Johnson, Ko, Lucas, Phan, Rosario, Ru
D. X-Ray Data/Analysis
Lattice Parameters Compound a b c Superconductivity
Yba2Cu3O7-x (commercial powder 1) 3.822 3.888 11.68 Yes
Yba2Cu3O7-x (sample 2) 3.812 3.867 11.6 Yes
LaBa2Cu3O7-x 3.832 3.898 11.85 Potential Superconductor
EuBa2Cu3O7-x 3.83 3.888 11.71 Yes
ErBa2Cu3O7-x 3.822 3.888 11.58 Yes
Y.5Ho.5Ba2Cu3O7-x 3.822 3.888 11.65 Semiconductor
Ho.25Dy.15Y.3Eu.1Er.2Ba2Cu3O7-x 3.822 3.875 11.58 Yes
Ho.2Dy.2Y.2Eu.2Er.2Ba2Cu3O7-x 3.822 3.888 11.68 Yes
Ho.6Dy.1Y.1Eu.1Er.1Ba2Cu3O7-x 3.822 3.888 11.68 Yes
Y.6Dy.3Ho.1Ba2Cu3O7-x 3.815 3.888 11.68 Yes
Eu.2Y.4Ho.4Ba2Cu3O7-x 3.822 3.888 11.58 Yes
EuBaSrCu3O7-x 3.822 3.888 11.58 High Resistance
EuBa.66Pb.66Sr.66Cu3O7-x 3.835 3.73 11.9 High Resistance
Ho.5Er.5BaSrCu3O7-x 3.822 3.78 11.65 Potential Superconductor
EuSr2Cu3O7-x n/a n/a n/a High Resistance
Ysr2Cu3O7-x n/a n/a n/a High Resistance
Ho.5Er.5Sr2Cu3O7-x n/a n/a n/a High Resistance
Table 3: Lattice Parameters
Yba2Cu3O7-x Samples (Graph 1)
Yba2Cu3O7-x (sample 1)*
Yba2Cu3O7-x (sample 2)
Parent
Yba2Cu3O7-x
Yba2Cu3O7-x
Graph 1
The samples of the non-commercial parent compounds have very similar structure to the commercial compounds. The only noticeable difference in structure for each sample is that a secondary peak is protruding forth from the largest peak in each graph: while in the commercial Journal of the PGSS Effects on Resistance and Lattice Structure Page 133 powder, the largest peak has no secondary peaks. As a consequence, the lattice parameters changed slightly as seen in Table 3.
Single 123 Substitutions (Graph 2)
LaBa2Cu3O7-x
EuBa2Cu3O7-x
ErBa2Cu3O7-x
Parent
LaBa2Cu3O7-x
EuBa2Cu3O7-x
ErBa2Cu3O7-x
Graph 2
In the single 123 substitutions, lanthanum, europium, and erbium each replaced yttrium in three separate samples. The major peak and the four double peaks were present in each sample. However, in the lanthanum sample, several extraneous peaks were present to the left of the major peak. Also, two of the double peaks in the same sample became single peaks. These different peaks led to a significant difference in all three lattice parameters. The europium and erbium substitutions were essentially identical to the commercial parent sample with only slight shifts in the 2 theta value at which the peaks occurred.
Partial Yttrium Substitution (Graph 3)
Y.5Ho.5Ba2Cu3O7-x
Parent
Y.5Ho.5Ba2Cu3O7-x
Graph 3
Page 134 Cockerham, Frey, Herrold, Johnson, Ko, Lucas, Phan, Rosario, Ru
In the partial holmium substitution, the graph was extremely similar to the standard base graph. The necessary peaks were present with no significant deviations found on the graph. The lattice parameters were similar to those of the commercial sample. However, the secondary peak found on the major peak mirrored that found on the non-commercial parent sample rather than the commercial sample.
“Fruit Salad” Substitutions (Graph 4 and 5)
Ho.25Dy.15Y.3Eu.1Er.2Ba2Cu3O7-x
Ho.2Dy.2Y.2Eu.2Er.2Ba2Cu3O7-x
Ho.6Dy.1Y.1Eu.1Er.1Ba2Cu3O7-x
Y.6Dy.3Ho.1Ba2Cu3O7-x
Eu.2Y.4Ho.4Ba2Cu3O7-x
Parent
Ho.25Dy.15Y.3Eu.1Er.2Ba2Cu3O7-x
Ho.2Dy.2Y.2Eu.2Er.2Ba2Cu3O7-x
Ho.6Dy.1Y.1Eu.1Er.1Ba2Cu3O7-x
Graph 4
Parent
Y.6Dy.3Ho.1Ba2Cu3O7-x
Eu.2Y.4Ho.4Ba2Cu3O7-x
Graph 5 The graphs of the “fruit salad” substitutions, in which combinations of holmium, dysprosium, europium, and erbium partly replaced yttrium, greatly resembled the graphs of the parent compounds. However, in each of these graphs, the separation between the doublets was more Journal of the PGSS Effects on Resistance and Lattice Structure Page 135 distinct. Regardless, the lattice parameters of the “fruit salad” substitutions were closer to the actual commercial sample than those of the single substitution compounds.
Partial Barium Substitutions (Graph 6)
EuBaSrCu3O7-x
EuBa.66Pb.66Sr.66Cu3O7-x
Ho.5Er.5BaSrCu3O7-x
Parent
EuBaSrCu3O7-x
EuBa.66Pb.66Sr.66Cu3O7-x
Ho.5Er.5BaSrCu3O7-x
Graph 6
In the partial barium substitutions, the aberrations from the standard yttrium, barium, copper graph were varied. In the compound that contained lead, the major peak was a double peak as opposed to a single one; it contained numerous extraneous peaks; and it was missing the last doublet. As a result, the lattice parameters of the lead compound were significantly different. However, the other two partial barium substitutions did not deviate much from their parent compounds.
Page 136 Cockerham, Frey, Herrold, Johnson, Ko, Lucas, Phan, Rosario, Ru
Full Barium Substitutions (Graph 7)
EuSr2Cu3O7-x
YSr2Cu3O7-x
Ho.5Er.5Sr2Cu3O7-x
Parent
EuSr2Cu3O7-x
YSr2Cu3O7-x
Ho.5Er.5Sr2Cu3O7-x
Graph 7
The full strontium for barium substitutions exhibited complete deviations from the commercial samples. In each of the graphs, there were three or more major, large peaks. In addition, there were superfluous smaller peaks throughout the graph. Interestingly, these full strontium substitutions were unable to be fit using the “X-ray Fit” program because these graphs were neither orthorhombic nor tetragonal in structure.
Ionic Radii Ratio vs. Cell Volume 1.6
1.4
1.2 1
0.8
0.6
Ionic Radii Ratio 0.4
0.2
0 167.0 168.0 169.0 170.0 171.0 172.0 173.0 174.0 175.0 176.0 177.0 178.0 3 Cell Volume (Å ) Graph 8: Ionic Radii Ratio vs. Cell Volume
A general observation is the compounds substituting yttrium or barium had larger ionic radii, then their cell volumes, which were obtained by multiplying the a, b, and c lattice parameters, were larger as well. Thus, there was a linear correlation between ionic radii and cell volume (see graph 8). In the full barium substitutions, the graphs of those compounds had extraneous secondary peaks that did not match those of the standard graph. Journal of the PGSS Effects on Resistance and Lattice Structure Page 137
E. Resistance Data
Table 4 shows the list of superconductors, potential superconductors (compounds with similar lattice structure with the Y-123 but did not show superconductivity above 77 K.), High Resistance Samples (compounds that did not superconduct at all) and semi/superconductor (compound that displays both the characteristics of a semiconductor and a superconductor). Table 5 lists all the superconductors in order of increasing Tc.
Tested Superconductors Potential High Resistance Semi/Super Superconductors Samples Conductor Yba2Cu3O7-x (commercial) LaBa2Cu3O7-x EuSr2Cu3O7-x Y.5Ho.5Ba2 Cu3O7-x Yba2Cu3O7-x (lab synthesized) Ho.5Er.5Sr2Cu3O7-x Ysr2Cu3O7-x Y.6Dy.3Ho.1Ba2Cu3O7-x EuBaSrCu3O7-x EuBa2Cu3O7-x Ho.5Er.5Sr2Cu3O7-x ErBa2Cu3O7-x EuBa.66Pb.66Sr.66Cu3O7-x Ho.25Dy.15Y.3Eu.1Er.2Ba2Cu3O7-x Ho.6Dy.1Y.1Eu.1Er.1Ba2Cu3O7-x Ho.2Y.2Dy.2Eu.2Er.2Ba2Cu3O7-x Eu.2Y.4Ho.4Ba2Cu3O7-x
Table 4: Results of Resistance Testing
Sample Tc (K) – from lowest to highest Yba2Cu3O7-x (commercial powder) 90.5 Yba2Cu3O7-x (lab synthesized) 92.0 Ho.2Y.2Dy.2Eu.2Er.2Ba2Cu3O7-x 92.3 Y.6Dy.3Ho.1Ba2Cu3O7-x 92.6 Ho.25Dy.15Y.3Eu.1Er.2Ba2Cu3O7-x 92.8 Ho.6Dy.1Y.1Eu.1Er.1Ba2Cu3O7-x 92.8 Eu.2Y.4Ho.4Ba2Cu3O7-x 92.9 EuBa2Cu3O7-x 94.1 EuBa2Cu3O7-x 96.1 Table 5: Tc Values at 1 mA for all Superconducting Samples
Page 138 Cockerham, Frey, Herrold, Johnson, Ko, Lucas, Phan, Rosario, Ru
F. Resistance Analysis
The raw data needed to be converted from resistance to resistivity. Resistance is a measurement that is directly related to the size of the material. Changing the size of a material changes its resistance. Resistance can be converted into resistivity, which is a measurement that does not depend on the size of the sample. The resistivity of materials can be directly compared even if the materials are of different sizes. The formula for converting the resistance of the samples to resistivity is:
= resistivity R = resistance A = cross-sectional area of the sample l = length between two innermost silver paste strips on sample length
Figure 15: Diagram of Sample
The resistance data was converted to resistivity, and a graph was created that plotted the resistivity of the sample against the temperature at each current for every sample that superconducted. This graph was used to determine the critical temperature of that sample for each current. The critical temperature was determined by expanding the area where the resistance became zero and approximating where the data points reached zero. Graph 9 shows that the critical temperature for Ho0.6Dy0.1Y0.1Eu0.1Er0.1Ba2Cu3O7-x is approximately 92.8 K at 1 mA. The critical temperature was approximated for each current, and then this data was used to create another graph. This new graph plotted the critical temperature at each current for
Journal of the PGSS Effects on Resistance and Lattice Structure Page 139
Ho 0.6Dy0.1Y0.1Eu 0.1Er 0.1Ba2Cu3O7-x Resistivity vs. Temperature 1.00E-04
8.00E-05
/cm) 6.00E-05 1 mA Ω
, ( 2 mA ρ 4.00E-05 6 mA 11 mA 2.00E-05 68 mA
Resistivity, 0.00E+00 92 92.5 93 93.5 -2.00E-05 Zero resistance at 92.8 K Temperature (K) for 1 mA
Graph 9: Resistivity vs. Temperature Graph expanded to determine Critical Temperature
the sample. A trend line was formed in the graph and that trend line showed the current carrying capacity of the material. Graph 10 shows an example of this graph and its trend line for
Ho0.6Dy0.1Y0.1Eu0.1Er0.1Ba2Cu3O7-x.
Ho0.6Dy0.1Y0.1Eu0.1Er0.1Ba2Cu3O7-x Current Carrying Capacity: Current vs. Critical Temperature
80
70
60
50 Non-Superconducting
40
30 Current (mA)
20
10 Superconducting
0 92 92.1 92.2 92.3 92.4 92.5 92.6 92.7 92.8 92.9 Temperature (K)
Graph 10: Determine Current Carrying Capacity
Page 140 Cockerham, Frey, Herrold, Johnson, Ko, Lucas, Phan, Rosario, Ru
The trend line divides the graph into two parts. Any pair of current and temperature that falls below the trend line would allow the sample to superconduct. If the pair is located above the trend line, then the sample is no longer superconducting. Graph 11 shows that
Ho0.6Dy0.1Y0.1Eu0.1Er0.1Ba2Cu3O7-x superconducts with 20 mA of current at 92.3 K. However, the sample would not superconduct at 20 mA if the temperature was at 92.5 K.
A graph was created that plotted all of the trend lines against each other. This graph is shown in Graph 11. This graph can be used to compare the current carrying capacities and critical temperatures of all the samples. The parent compound, Yba2Cu3O7-x, is depicted by the large blue squares on the far left. Each of the new superconductors is also exhibited on the graph. If the current carrying capacity trend line of a sample is greater, then it can conduct more current while still retaining zero resistance.
The Yba2Cu3O7-x can conduct relatively little current when it first reaches zero resistance and still stay superconducting. The EuBa2Cu3O7-x compound is represented by the trend line farthest to the right. Its current carrying capacity trend line is much steeper and the critical temperature at each current is higher than the Yba2Cu3O7-x. All the other new superconductors also improved upon the current carrying capacity and critical temperature of the Yba2Cu3O7-x.
The data analysis involved the creation of three graphs for each new superconductor. The first graph plotted resistivity against temperature. The second graph showed the critical temperatures for each of the currents ran on the sample. The final graph plotted the critical temperature versus current and was used to determine the current carrying capacity of the sample. The first graph of each new superconductor was very similar and only the scale changed. The first graph always resembled Graph 11. While the element is superconducting, the resistance is at zero. The sample then reaches a point where it is no longer superconducting, and the resistance immediately begins to increase very rapidly. The rapid change in resistivity then lessens into a relatively slowly rising linear rate of change. These three stages are characteristic of any superconductor. The only graphs that were important for analysis were the second and third graphs.
Er Ba 2Cu3O7-x Resistivity vs. Temperature
3.00E-03
2.50E-03
2.00E-03 1 mA /cm)
Ω 2 mA , ( 1.50E-03 ρ 6 mA 1.00E-03 11 mA 5.00E-04 67 mA
Resistivity, 0.00E+00 92 93 94 95 96 97 98 -5.00E-04 Temperature (K)
Graph 11: Graph of ErBa2Cu3O7-x that shows resistivity versus temperature
Journal of the PGSS Effects on Resistance and Lattice Structure Page 141
Yba2Cu3O7-x (Noncommercial Sample):
This is the noncommercial sample of the parent compound. It was compared to the commercial sample to prove that the procedure for creating superconductors was producing good samples. This noncommercial sample matched closely with the commercial sample.
YBa 2Cu3O7-x Resistivity vs. Temperature
1.00E-04
8.00E-05
1 mA /cm) 6.00E-05 Ω 2 mA , ( ρ 4.00E-05 6 mA 11 mA 2.00E-05 67 mA
Resistivity, 0.00E+00 91 91.5 92 92.5 93 -2.00E-05 Temperature (K)
Graph 12
YBa 2Cu3O7-x Current Carrying Capacity: Current vs. Critical Temperature
0.07
0.06
0.05
0.04
0.03
Current (mA) Current 0.02
0.01
0 91.2 91.3 91.4 91.5 91.6 91.7 91.8 91.9 92 92.1 Temperature (K)
Graph 13
Page 142 Cockerham, Frey, Herrold, Johnson, Ko, Lucas, Phan, Rosario, Ru
EuBa2Cu3O7-x:
The EuBa2Cu3O7-x was a single substitution for yttrium. Its current carrying capacity was much higher than the parent compound’s capacity and the substitution raised the critical temperature from around 90 K to 96 K. The resistance was tested twice at 11 mA. The data for both tests was poor, so error bars were added to the graph to show the possible range of data.
EuBa 2Cu3O7-x Resistivity vs. Temperature
2.00E-04
1.50E-04 1 mA /cm)
Ω 2 mA
, ( 1.00E-04 ρ 6 mA 5.00E-05 11 mA 67 mA 0.00E+00 Resistivity, 95.4 95.6 95.8 96 96.2 96.4 96.6 -5.00E-05 Temperature (K)
Graph 14
Eu Ba 2Cu3O7-x Current Carrying Capacity: Current vs. Critical Temperature
80 70 60 50 40 30
Current (mA)Current 20 10 0 95.7 95.75 95.8 95.85 95.9 95.95 96 96.05 96.1 96.15 Temperature (K)
Graph 15
Ho0.25Dy0.15Y0.3Eu0.1Er0.2Ba2Cu3O7-x:
This five-substitution for yttrium led to a rise in critical temperature of about 2.5 K.
Journal of the PGSS Effects on Resistance and Lattice Structure Page 143
Ho0.25Dy0.15Y0.3Eu0.1Er 0.2Ba2Cu3O7-x Resistivity vs. Temperature
1.40E-04 1.20E-04 1 mA /cm) 1.00E-04 Ω 2 mA , (
ρ 8.00E-05 6 mA 6.00E-05 11 mA 4.00E-05 67 mA
Resistivity, 2.00E-05
0.00E+00 92 92.5 93 93.5 94 Temperature (K)
Graph 16
Ho0.25Dy0.15Y0.3Eu0.1Er0.2Ba2Cu3O7-x Current Carrying Capacity: Current vs. Critical Temperature
70
60
50
40
30
Current (mA)Current 20
10
0 92.35 92.4 92.45 92.5 92.55 92.6 92.65 92.7 92.75 92.8 92.85 Temperature (K)
Graph 17
Ho0.2Y0.2Dy0.2Eu0.2Er0.2Ba2Cu3O7-x:
None of the runs for the 67 mA current test yielded good data. This led to a bad data point on the current carrying capacity graph. Numerous things could have caused the poor data for the 67 mA Page 144 Cockerham, Frey, Herrold, Johnson, Ko, Lucas, Phan, Rosario, Ru runs. It is obvious that the data is bad because the data shows that the superconductor has a higher critical temperature at 67 mA then it does at 11 mA. This conclusion is impossible. The compound showed a slight improvement in critical temperature.
Ho0.2Y0.2Dy0.2Eu0.2Er0.2Ba2Cu3O7-x Resistivity vs. Temperature
5.00E-04
4.00E-04
3.00E-04 /cm) /cm) 1 mA Ω
, ( 2 mA ρ 2.00E-04 6 mA 11 mA
1.00E-04 67 mA Resistivity,
0.00E+00 91.5 91.7 91.9 92.1 92.3 92.5 92.7 92.9
-1.00E-04 Temperature (K)
Graph 18
Ho0.2Y0.2Dy0.2Eu0.2Er0.2Ba2Cu3O7-x Current Carrying Capacity: Current vs. Critical Temperature
80
70
60
50
40
30 Current (mA)
20
10
0 90.8 91 91.2 91.4 91.6 91.8 92 92.2 92.4 92.6 Temperature (K)
Graph 19
Journal of the PGSS Effects on Resistance and Lattice Structure Page 145
ErBa2Cu3O7-x:
The effects of this single substitution for yttrium were very similar to the EuBa2Cu3O7-x, but to a lesser degree.
Er Ba 2Cu3O7-x Resistivity vs. Temperature
1.00E-03
8.00E-04 1 mA /cm) 6.00E-04 Ω 2 mA , ( ρ 4.00E-04 6 mA 11 mA 2.00E-04 67 mA
Resistivity, 0.00E+00 92 92.5 93 93.5 94 94.5 95 -2.00E-04 Temperature (K)
Graph 20
Er Ba 2Cu3O7-x Current Carrying Capacity: Current vs. Critical Temperature
90
80
70
60
50
40
Current (mA) Current 30
20
10
0 91.5 92 92.5 93 93.5 94 94.5 Temperature (K)
Graph 21 Page 146 Cockerham, Frey, Herrold, Johnson, Ko, Lucas, Phan, Rosario, Ru
Eu0.2Y0.4Ho0.4Ba2Cu3O7-x:
This substitution led to slight improvements in current carrying capacity and critical temperature.
Eu 0.2Y0.4Ho 0.4Ba2Cu3O7-x Resistivity vs. Temperature
5.00E-05
4.00E-05
1 mA /cm) 3.00E-05 Ω 2 mA , ( ρ 2.00E-05 6 mA 11 mA 1.00E-05 67 mA
Resistivity, 0.00E+00 90.5 91 91.5 92 92.5 93 93.5 -1.00E-05 Temperature (K)
Graph 22
Eu 0.2Y0.4Ho 0.4Ba2Cu3O7-x Current Carrying Capacity: Current vs. Critical Temperature
90
80
70
60 50 40
Current (mA) Current 30 20 10 0 90.5 91 91.5 92 92.5 93 93.5 Temperature (K)
Graph 23 Journal of the PGSS Effects on Resistance and Lattice Structure Page 147
Ho0.6Dy0.1Y0.1Eu0.1Er0.1Ba2Cu3O7-x:
This mostly holmium substitution for yttrium led to marginal improvements the current carrying capacity and critical temperature.
Ho 0.6Dy0.1Y0.1Eu 0.1Er 0.1Ba2Cu3O7-x Resistivity vs. Temperature 1.00E-04
8.00E-05
/cm) 6.00E-05 1 mA Ω
, ( 2 mA ρ 4.00E-05 6 mA 11 mA 2.00E-05 68 mA
Resistivity, 0.00E+00 92 92.5 93 93.5 -2.00E-05 Temperature (K)
Graph 24
Ho 0.6Dy0.1Y0.1Eu 0.1Er 0.1Ba2Cu3O7-x Current Carrying Capacity: Current vs. Critical Temperature
80 70 60 50 40 30 Current (mA) Current 20 10 0 92 92.2 92.4 92.6 92.8 93 Temperature (K)
Graph 25 Page 148 Cockerham, Frey, Herrold, Johnson, Ko, Lucas, Phan, Rosario, Ru
Y0.6Dy0.3Ho0.1Ba2Cu3O7-x:
This compound resembled the other multiple yttrium substitutions in that there was a marginal increase in current carrying capacity and critical temperature.
Y0.6Dy0.3Ho 0.1Ba2Cu3O7-x Resistivity vs. Temperature
6.00E-05
5.00E-05
4.00E-05 1 mA /cm)
Ω 2 mA , ( 3.00E-05 6 mA 2.00E-05 10 mA 1.00E-05 65 mA Resistivity, ρ 0.00E+00 91 91.5 92 92.5 93 -1.00E-05 Temperature (K)
Graph 26
Y0.6Dy0.3Ho 0.1Ba2Cu3O7-x Current Carrying Capacity: Current vs. Critical Temperature
70
60
50
40
30 Current (mA) 20
10
0 91.4 91.6 91.8 92 92.2 92.4 92.6 92.8 Temperature (K)
Graph 27 Journal of the PGSS Effects on Resistance and Lattice Structure Page 149
Ho0.5Er0.5SrBaCu3O7-x: The critical temperature of this sample is below 77 K. The graph is characteristic of a superconductor, which suggests that this compound would superconduct if lowered to a low enough temperature. The only current run on the sample was 1 mA because the other currents would not have changed any conclusion that could be drawn from the data.
Ho0.5Er0.5SrBaCu3O7-x Resistivity vs. Temperature
0.001 0.0009 0.0008
0.0007 /cm) /cm)
Ω 0.0006 , ( ρ 0.0005 1 mA 0.0004
0.0003 Resistivity, 0.0002 0.0001
0 77 97 117 137 157 Temperature (K)
Graph 28
Page 150 Cockerham, Frey, Herrold, Johnson, Ko, Lucas, Phan, Rosario, Ru
Y0.5Ho0.5Ba2Cu3O7-x:
The graph for Y0.5Ho0.5Ba2Cu3O7-x (Graph 30) is not singly characteristic of semiconductors or superconductors. The left part of the curve resembles that of a superconductor, while the right side resembles the curve of a semiconductor. It is reasonable to conclude that
Y0.5Ho0.5Ba2Cu3O7-x would superconduct if the temperature was lowered far enough, because the x-ray diffraction data showed a lattice very similar to the Y-123 parent compound, and the resistivity is trending toward zero.
Y0.5Ho0.5Ba2Cu3O7-x Resistivity vs. Temperature
0.0103
0.0102
0.0101 /cm) /cm)
Ω 0.01 , ( ρ 0.0099 1 mA
0.0098
Resistivity, Resistivity, 0.0097
0.0096
0.0095 0 50 100 150 200 250 300 Temperature (K)
Graph 29 Journal of the PGSS Effects on Resistance and Lattice Structure Page 151
G. X-Ray/Resistance Analysis
Compound Cell Volume Ionic Ratio Tc (1 mA)
Yba2Cu3O7-x (commercial powder 1) 173.6 1 90.5
Yba2Cu3O7-x (sample 2) 171.0 1 92
LaBa2Cu3O7-x 177.0 1.339 40s
EuBa2Cu3O7-x 174.4 1.056 96
ErBa2Cu3O7-x 172.1 0.989 94.1
Y.5Ho.5Ba2Cu3O7-x 173.1 1 n/a
Ho.25Dy.15Y.3Eu.1Er.2Ba2Cu3O7-x 171.5 1.005 92.8
Ho.2Dy.2Y.2Eu.2Er.2Ba2Cu3O7-x 173.6 1.011 92.4
Ho.6Dy.1Y.1Eu.1Er.1Ba2Cu3O7-x 173.6 1.006 92.8
Y.6Dy.3Ho.1Ba2Cu3O7-x 173.2 1.003 92.55
Eu.2Y.4Ho.4Ba2Cu3O7-x 172.1 1.011 92.9
EuBaSrCu3O7-x 172.1 0.95 n/a
EuBa.66Pb.66Sr.66Cu3O7-x 170.2 0.935 n/a
Ho.5Er.5BaSrCu3O7-x 168.3 0.935 n/a
EuSr2Cu3O7-x n/a 0.886 n/a
Ysr2Cu3O7-x n/a 0.83 n/a
Ho.5Er.5Sr2Cu3O7-x n/a 0.871 n/a Table 6: Compiled Data
V. Conclusion A. Non-superconducting Samples
Eight of the samples tested in this experiment never exhibited zero resistance within the temperature range of liquid nitrogen. Two of these compounds, LaBa2Cu3O7-x and
Ho.5Er.5Sr2Cu3O7-x, yielded sharply decreasing resistance vs. temperature curves that resembled portions of superconducting graphs before they plummeted to zero resistance due to the absence of a drop-off points on the graphs. Furthermore, the great resemblance of the X-ray graphs of
LaBa2Cu3O7-x and Ho.5Er.5Sr2Cu3O7-x to that of the parent compound suggested that their crystals were structurally similar (See Graphs 2 and 6). This similarity signified the possibility of these two compounds reaching superconductivity at temperatures below that of liquid nitrogen and hence out of the measuring capabilities of this experiment.
For five of the samples, quick tests using an ohm-meter at room temperature revealed such high resistance that they were not cooled down and tested for superconductivity at all. These five samples were EuSr2Cu3O7-x, YSr2Cu3O7-x, EuBaSrCu3O7-x, Ho.5Er.5Sr2Cu3O7-x, and
EuBa.66Pb.66Sr.66Cu3O7-x. All of them contained Strontium as either a full or partial replacement for barium, indicating the failure of strontium to substitute for barium. Though the charges of their Page 152 Cockerham, Frey, Herrold, Johnson, Ko, Lucas, Phan, Rosario, Ru ions were the same, strontium’s ionic radius was notably smaller than that of barium’s, which could explain why it did not fit similarly into the lattice structure of the crystal (See Table 1). Not surprisingly, major differences existed between the X-ray graphs of the parent compound,
YBa2Cu3O7-x, and those of the Sr-containing compounds. Where the diffraction graph of the former displayed one major peak, those of the latter depicted three or more smaller peaks (See Graph 7).
One of the samples in this experiment, Y.5Ho.5Ba2 Cu3O7-x, yielded a peculiar graph that seemed to combine features of both a semiconductor and a superconductor (See Graph 29). From 90 – 140 K, the resistance steeply decreased as temperature decreased in a curve characteristic of a superconductor. However, from 150 – 300 K, resistance decreased with increased temperature in the manner of a semiconductor. This odd amalgamation could have resulted from the sample partially retaining the separate characteristics of its components. Mixing multiple substitutions in for a single element, yields in random crystals. The distribution of the substituted elements is random throughout the compound. It is possible that the .5Ho.5Ba2 Cu3O7-x, crystal had both semiconducting components and superconducting components.
B. Successful Superconductors
Current Carrying Capacity and Critical Temperatures of Superconductors
90 YBa(2)Cu(3)O(7-x)
80 Eu(0.2)Y(0.4)Ho(0.4)Ba(2)Cu(3)O(7-x) 70 Y(0.6)Dy(0.3)Ho(0.1)Ba(2)Cu(3)O(7-x) 60
50 Ho(0.6)Dy(0.1)Y(0.1)Eu(0.1)Er(0.1)Ba(2)C u(3)O(7-x) 40 ErBa(2)Cu(3)O(7-x) Current (mA) 30 EuBa(2)Cu(3)O(7-x) 20
10 Ho(0.25)Dy(0.15)Y(0.3)Eu(0.1)Er(0.2)Ba(2 )Cu(3)O(7-x)
0 Ho(0.2)Y(0.2)Dy(0.2)Eu(0.2)Er(0.2)Ba(2)C 87 89 91 93 95 97 u(3)O(7-x) Temperature (K)
Graph 30: Compare all the samples
In this experiment, nine compounds successfully superconducted within the temperature range of liquid nitrogen. They included two YBa2Cu3O7-x parent compounds, two compounds with complete substitution for yttrium, and five “fruit salad” compounds in which a mixture of elements replaced Journal of the PGSS Effects on Resistance and Lattice Structure Page 153
yttrium. The noncommercial sample of YBa2Cu3O7-x began to superconduct at exactly 92.0, a value that matched previously published values for this compound, thus proving the viability of the techniques for synthesis and testing used in this experiment.
Two of the three samples in which yttrium was completely substituted by another element superconducted at even higher temperatures than the parent compound, while one did not superconduct within the temperature range of liquid nitrogen at all. ErBa2Cu3O7-x and
EuBa2Cu3O7-x had the highest Tc values of all the compounds tested (94.1 K and 96.1 K, respectively) and greater current carrying capacity than the parent compound YBa2Cu3O7-x (See graph 11). In contrast, LaBa2Cu3O7-x never reached zero resistance in this experiment, though the decreasing shape of its graph opened the possibility of it superconducting at temperatures below that of liquid nitrogen. Variations in ionic radii could provide a possible explanation for the unexpected behavior of the lanthanum-substituted compound: the radii of Er+3 and Eu+3 were very close to that of Y+3, while La+3‘s radius was notably larger (See Table 1). The larger size of the La+3 ions could have prevented it from fitting into the crystal lattice structure in the same way that Y+3 would, thus altering the compound’s superconducting abilities.
Five mixed “fruit salad” compounds superconducted successfully, and their transition temperatures fell in the 92 K – 94 K range, higher than the parent compound but lower than the full substitution compounds. Despite the wide variety of elements substituted, the X-ray graphs of these “fruit salad” compounds depicted a great resemblance to that of the parent compound, indicating that the crystal structures of these compounds and YBa2Cu3O7-x were very similar (See Graphs 4 and 5). The resistivity data seemed to indicate that these multiple-substitution samples combined characteristics of their complete substitution counterparts. For example, the Tc value of
Eu.2Y.4Ho.4Ba2Cu3O7-x fell between those of YBa2Cu3O7-x and EuBa2Cu3O7-x (See Table 5 and Graph 30). Interestingly, “fruit salad” compounds containing holmium and at least two other elements in place of yttrium functioned as superconductors, while those with holmium and only one other element did not.
The compounds ErBa2Cu3O7-x, EuBa2Cu3O7-x, Y.6Dy.3Ho.1Ba2Cu3O7-x, Eu.2Y.4Ho.4Ba2Cu3O7-x,
Ho.2Y.2Dy.2Eu.2Er.2Ba2Cu3O7-x, Ho.25Dy.15Y.3Eu.1Er.2Ba2Cu3O7-x, and Ho.6Dy.1Y.1Eu.1Er.1Ba2Cu3O7-x all began superconducting at temperatures above 92.0 K, the maximum Tc value for the parent compound (See Table 5). Furthermore, the steeper slopes of their current capacities curves demonstrated their ability to carry more current at a given temperature (See Graph 30). Therefore, the creation of these seven superconducting compounds based on the structure of
YBa2Cu3O7-x fulfilled the twin goals of this experiment: to increase critical temperature and current carrying capacity.
Page 154 Cockerham, Frey, Herrold, Johnson, Ko, Lucas, Phan, Rosario, Ru
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19 Magnetic Forces. 1987. Energy Production and Conservation, Materials Science. Pacific Northwest National Laboratory. 25 July 2007
20 "Uses for Superconductors." Superconductors.Org. June 2007. 23 July 2007
21 "SQUID, Superconducting Quantum Interference Device." ESA Technology. 5 Oct. 2000. 23 July 2007
Acknowlegements
Dr. Butera for his time, dedication, advice, and patience for our mistakes
Dr. Barry Luokkala
The Pennsylvania State Government
The Governor of Pennsylvania, Edward Rendell
The Department of Education
Faculty and Staff of PGSS
Catherine Vinci for keeping us entertained in the lab (Dancing Queen, anyone?)
John Paul for being amazing
Mrs. Butera for brownies…YUM!
Liz for staying awake all night so we could finish our paper
Walther Meissner for confusing us….
…and Europium for being such a great substitute
COMPUTER SCIENCE TEAM PROJECTS
Journal of PGSS LexerEvolution Page 159
LexerEvolution
Christian Bruggeman, Jonathan Chu, Tony Jiang, Ian McGuier, Christopher Shotter, Rebecca Triano
Abstract
This paper discusses the creation and implementation of the newly designed LexerEvolution, a lexical analyzer generator for the programming language ML. Information is provided concerning the features and functionality of ML. The problems with previous lexical analyzer generators are discussed, along with how the newly created LexerEvolution addresses these problems.
I. Introduction
A. Background
1. ML (Programming Language)
ML, metalanguage, was originally designed by Robin Milner as a functional language to solve proofs in the Formal Logical System. As the programming language grew in popularity, it was equipped with imperative power and an exception mechanism.1
ML has many unique features incorporated into its design:
Call-by-value evaluation strategy will evaluate an expression before passing it as an argument. This strategy is known as “eager” evaluation, as opposed to “lazy” evaluation, which only evaluates expressions immediately before they are required.
First class functions allow the programmer to define a function and recall it.
User-defined types permit the programmer to create new types for a specific task. This gives the programmer much more flexibility, allowing him to create his own structure and hide types from others who might modify them.
Static typing requires functions and variables to have a well-defined type at compile time. A component of the ML compiler called the type checker enforces this requirement before compilation. 2
Type inference lets a programmer omit type specification for most variables and functions. The type checker will infer the type from the function context, which makes programs more compact and easier to write.
Polymorphism enables the programmer to create a function of abstract type and apply it to any number of concrete types. An example of polymorphism is a list reversal function, which reverses the elements in a list no matter what types the elements are. Without polymorphism, a programmer would have to define a unique list reversal function for every possible type. Because types can be user-defined, there are an unlimited number of types, and the implementation of any nontrivial program would be prohibitively tedious. 3
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Pattern matching describes the ability of a programmer to define a pattern and requires the compiler to force the data into said pattern, which allows the programmer to access the data in a format of his choice. 4
Exception handling recognizes an error that occurred during the execution of a program. ML then checks the code before the error and tries to discover possible sources of or a solution to the problem. The error will be reported to the user immediately; this distinguishes ML from some programming languages, such as C, that are not able to gracefully recover from such errors.
Unlike other programming languages, ML uses expressions rather than states and largely avoids mutable data, information that can be changed after it is originally computed. ML, a primarily functional language, does have imperative features, sequences of commands for the computer to perform.
ML has been applied in bioinformatics and financial systems. These applications include genealogical databases and peer-to-peer client/server programs. 5
2. Terminology
A finite state machine is an abstract representation of a set of states and rules, known as transitions, that describe how to move a input stream among the states. There are two major types of finite state automata in computational theory: nondeterministic finite state machines and deterministic finite state machines.
A nondeterministic finite state machine (NFA) is similar to a DFA, but an NFA can transition to any number of a set of “next” states. Rather than selecting one transition, an NFA selects all possible transitions from a given state simultaneously. It will then determine the transition path that enables it to reach an accepting state if one exists.
A
State ε State B B 2 3 A
State A 1 Input B A
State ε State A A 4 5
B
Figure 2: Nondeterministic Finite State Machine
Journal of PGSS LexerEvolution Page 161
The input state begins in State One, and from there moves to State Two and State Four, because an NFA selects all possible next states. The stream then follows both paths until it either reaches a final accepting state or has exhausted all transitions.
In a deterministic finite state machine (a DFA), for any given state, there is a unique “next” state for any possible input. DFAs are efficient and practical, but they have limited power. A DFA, for example, cannot be used for the regular expression anbn, which would be used to ensure that parentheses are balanced.
A Input
State State B 1 2 B
A
Figure 1: Deterministic Finite State Machine (DFA)
The input stream begins in State One. If the input is of type B, it will return to State One; if it is of type A, it will then move to State Two, the accepting state. If there are remaining characters in the input stream, the stream can then continue back to State One or stay in State Two, depending on the characters’ type. The input stream will only be accepted if it ends in an accepting state. Any character other than A or B causes the DFA to fail.
A Regular Expression (RegExp) is a sequence of symbols that describes a set of strings. Using regular expressions, one can describe a set of strings, such as the set of all real numbers, without listing all its possible options. For example, the regular expression (ab)* would describe the empty set, ab, abab, ababab, etc.
Tokens are literal elements within a string.
Lexemes are the basic units in lexical analysis. 6
Tokenizer is what classifies the section of strings that are input into tokens.
3. Description of a Lexer
A lexer is a tool used in an early stage of compilation that converts a source file into a stream of lexemes, the minimal units of a programming language. A lexer encompasses several processes. First, the scanner reads the source file and converts it into a stream of characters. The tokenizer then converts the stream of characters into a stream of tokens, which represent single units of information. When tokenizing, the tokenizer will ignore all white space, line breaks, and comments. Finally, the tokens are converted into lexemes through comparison to a set of regular expressions, which represent various classes of lexemes recognized by the language. 7
II. Improvements made
A. Evolution of Lexer
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Lexers have been developed for different programming languages. Each lexer contains different features, but performs the same basic functions. Below are the three most common lexical analysis generators.
1. Lex - Lex is a programming tool for the UNIX system designed only for the C language. It uses regular expressions to describe sets of strings. From this it produces a program for lexical analysis based on the sets of strings. Unfortunately, it places limitations on string length.
2. Flex (Fast Lexical Analyzer Generator) – Flex is a free alternative to Lex, written for C and C++, that provides greater conciseness and efficiency than handwritten code. It reads a description of the desired lexer as a source file and generates code for the lexer. According to the flex manual, flex is “a tool for generating scanners: programs which recognized lexical patterns in text. Flex reads the given input files, or its standard input if no file names are given, for a description of a scanner to generate.” 8
3. ML-Lex – ML-Lex is a variation of Lex for the ML language. It’s similar to Lex but produces an efficient ML program. It improves on Lex by not limiting string size, and it is more efficient than a hand-coded lexer. 9
B. Addressing problems
Current lexers share one major flaw: they fail to implement type safety. Type safety is the ability of the lexer to recognize type errors. LexerEvolution is one the first type-safe lexers, a lexer that will not blindly insert code without verifying that it will successfully compile. It uses a construct known as a functor to dynamically ensure type safety based on the defined specifications.
III. Components of LexerEvolution
A. LexerEvolution Map
SourceLexer File B Regexps C NFA Description
D
E Source File F Lexer DFA
Figure 3: Step by step map of LexerEvolution
The map above shows the step by step process of how LexerEvolution operates.
B. Source File to Characters
Journal of PGSS LexerEvolution Page 163
This stage of LexerEvolution uses a team-created function in order to peel characters one by one from a source file. This is implemented using a wrapper function, which allows LexerEvolution to pass the function as a first-class parameter and which presents a defined type without requiring the user to have knowledge of the underlying implementation.
C. Regexps to NFA
There are eight regular expressions that have been defined in our code:
Epsilon is the empty string.
State ε State
1 2 Input
Figure 3: Epsilon Regular Expression
In the regular expression “ε ”, the input stream begins in state one. If the stream is empty, it transitions to the final accepting state.
Character matches a chosen character to the input. If the character matches, the NFA proceeds to the next state; anything other than the chosen character will cause the NFA to fail.
State a State 1 2 Input
Figure 4: Character Regular Expression
The regular expression “Character” begins in State One and transitions to State Two, the accepting state, if the input stream matches the specified character.
Concatenation is the fusion of two elements into a single entity. The LexerEvolution uses concatenation to join regular expressions, for example, the fusion of “a” and “b” into “ab.”
State a State b State 1 2 3 Input
Figure 5: Concatenation Regular Expression
The regular expression that represents Concatenation begins with the input stream entering State One. If the first specified regexp matches, it transitions to State Two; otherwise, the stream fails. Once it transitions to State Two, the second specified character is compared to the stream. If it matches, it then transitions to State Three, which accepts it. If any other rexexp is entered, the stream fails. In the example above, the characters a and b would have to be adjacent for the string to be accepted.
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Union gives the NFA the ability to accept multiple inputs. In LexerEvolution, union allows a number of regular expressions to reach the same next state. This can be represented by a | b, which means “a or b.” A
State State 1 2 Input
B
Figure 6: Union Regular Expression
The regular expression Union begins with the input stream entering State One. The character is then tested against both regexp A or B. The character follows the appropriate transition and enters State Two, the accepting state. If the input character is not either type A or type B, the stream fails.
Null needs a lack of input; any input will cause the NFA to fail.
State State 1 Input 2
Figure 7: Null Regular Expression
The regular expression for Null is different than previous figures, because the stream will always fail.
Star requires any number of repetitions, from zero upwards, of the characters to which it is applied. ε Input
State A State State 1 2 3
ε
Figure 8: Star Regular Expression
Journal of PGSS LexerEvolution Page 165
In the regular expression Star, the input stream begins in State Two. The character will move into an accepting state if it is Epsilon or of type A. Star does not require the initial input to be repeated a certain number of times, but will fail if anything other than regexp A is entered.
Plus requires one or more repetitions of the set to which it is applied.
A
ε
State State State 1 2 3 Input
ε
Figure 9: Plus Regular Expression
The regular expression Plus begins with the input stream entering into State One. Unlike the regular expression Star, Plus requires that the entered character be repeated at least one time. After the desired character has been entered once, the NFA can transition between State Two and State Three.
Question Mark requires either zero or one inputs that satisfy the regular expression.
A
State State 1 2 Input
ε
Figure 10: Question Mark Regular Expression
In the regular expression Question Mark, the input begins in State One. If the input is not entered or entered only one time, then the regular expression will be satisfied and transition to State Two, the accepting state.
Each regular expression and finite state machine possesses a start state, a set of final states, and a list of rules that describes how to get from one state to the other.
In LexerEvolution, the user-defined regular expression is broken down into simpler elements and then stitched back together. After the non-deterministic finite automaton has been created, it will then be changed into a deterministic finite automaton.
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D. NFA to DFA
The conversion of an NFA to a DFA redraws the NFA, frequently increasing the number of states and transitions. This arrangement is ideal for a stream of characters to be tested against a pattern from a regular expression.
All NFAs can be written as DFAs, so the lexer does not change any information or the given value of the expression. It simply recreates the same specific function in a form that is more conducive to lexical analysis.
Additionally, after creating a DFA from the given NFA, LexerEvolution optimizes the DFA by combining similar state rules into a single list, utilizing MergeSort. MergeSort divides a list into single elements, compares them, and rearranges them into a newly ordered list.
If two different state rules apply to a distinct state, the two rules will be appended so that both rules may be exhausted. Each state of the DFA contains a list that has all possible transitions from the given state. This simplification is necessary for the DFA to generate a lexer.
E. DFA to Lexer
The lexer converts the string into a character list and peels each character from the character list. It feeds each character into a DFA to see if it reaches a final state. Every character list will eventually have its characters exhausted, at which point the DFA will terminate, return an error, and output the last sequence of characters that arrived at a final state.
IV. Conclusion
LexerEvolution is not a professional SML tool; it still requires a lot of effort before it can be implemented as a practical lexical analyzer generator. The user must input a single regular expression now; users should be able to input multiple regular expressions with identifiers. The Regexp to NFA stage is largely efficient, but the DFA still needs to be optimized to reduce the number of states and transitions. Errorchecking must be done throughout the code, and many more tests for error handling and exceptions must also be conducted. LexerEvolution expects a perfect input at this time.
LexerEvolution is, nevertheless, a working core for a full-featured lexical analysis generator, which has applications for programmers everywhere. It would replace ML-Lex as the standard lexical analysis tool for any user in the SML language.
V. References
Harper, Robert. Programming in Standard ML. Carnegie Mellon University. 2005. 1-192. 20 July 2007
Paulson, L. C. ML for the Working Programmer. Cambridge: Cambridge UP, 1991. 1-429.
Stansifer, Ryan. ML Primer. New Jersey: Prentice-Hall, Inc., 1992. 1-160.
1 Milner, Robin, Mads Tofte, and Robert Harper. The Definition of Standard ML. Cambridge: The MIT P, 1990. vii-viii
2 Cardelli, Luca, and Peter Wegner. On Understanding Types, Data Abstraction, and Polymorphism. Brown University. 1985. 3-4. 23 July 2007
3 Cardelli, Luca, and Peter Wegner. On Understanding Types, Data Abstraction, and Polymorphism. Brown University. 1985. 4-5. 23 July 2007
4 Ullman, Jeffrey D., Elements of ML Programming, Prentice Hall Englewood Cliffs, New Jersey 07632, 57
5 "ML Programming Language." Wikipedia. 26 July 2007
6 Hilfinger, P. N. Class Notes #2: Lexical. University of California. 2005. 3. 24 July 2007
7 Hilfinger, P. N. Class Notes #2: Lexical. University of California. 2005. 1-2. 24 July 2007
8 Paxson, Vern. Flex Version 2.5. 2001. 26 July 2007
Journal of the PGSS Project LPX Page 169
Project LPX: A Functional Web Design Language in SML
Melissa Elfont, Andrew Guenin, Haarika Kamani, Bill McDowell, and Jeff Ruberg
Abstract
Creating HTML tags for web page design is necessary to generate neat, professional-looking web pages. However, writing out enough HTML tags to make a full web page is not only tedious and repetitive, but also very error prone. Alternative languages, such as CSS and PHP, are more convenient, but still have some weaknesses that decrease their effectiveness. LPX is a simple web page markup language that novices can use to turn simple code into a manageable web page.
I. Introduction
The use of the World Wide Web has drastically increased. Our society’s need to share personal information over the Internet and to be constantly connected to others has led to a large increase in the number of web pages created and viewed. A creative, professional-looking web page is more likely to attract potential customers than a rarely updated one. LPX is a user-friendly language that one can use to design simple web pages easily and efficiently.
The web markup language, LPX, is a typed functional language written in SML. LPX allows the user to utilize any HTML tag and create his or her own functions that utilize any number of tags. Since the user can create new, personalized functions to use, LPX has the capability to generate more intricate web pages. LPX uses the functionality of SML to produce a series of functions and expressions to create a markup language. The syntax of LPX was written with efficiency, simplicity, and logicality in mind. In order to create LPX, we defined types, inference rules, base cases, and all of the functions necessary to make LPX work.
II. SML
Since SML is a strongly typed systemi, it was the most useful language to use for coding LPX. A typed system is a system that classifies every variable, function, and expression into a specific type. These types can then be manipulated into whatever the user wants the types to be. However, if a type is manipulated in a way that it should not be, an error is thrown. For example:
var foo = 7 //foo is of type int var bar = “ab” //bar is of type string print foo+bar
Once code is written in SML, it is implemented using a type checker, which verifies that each expression yields its expected type, and an evaluator, which computes the given expressions. A parser reads in the typed code for the type checker and the evaluator. An expression is only passed through the evaluator if it passes through the type checker without any problems. If types Page 170 Elfont, Guenin, Kamani, McDowell, Ruberg cannot be correctly evaluated, the compiler raises an error, and the user has to try to find the source of the error within his or her code. The type checker enforces the limits of each type, and should not allow the different types to mix.
III. Inference Rules
Inference Rulesii are psuedocode steps that show the basic trend of arguments within a code. The basic blueprint for an inference code is that the top layer indicates the necessary conditions for a code to work. The bottom layer describes the conclusion that the code producesiii.
Figure 1: A General Inference Rule
For example:
Figure 2: The ‘If’ Inference Rule
In an inference ruleiv, the top layer indicates that within the context of Γ, exp1 is of type Boolean, exp2 is of any type τ, and exp3 is also of any type τ. While the bottom indicates that if exp1 is true, then follow exp2 of a type τ, otherwise, follow exp3 of type τ.
In the first evaluating inference rule, the top layer sets the condition of exp1 being true within the environment of ∑. The bottom layer says that if the condition afore mentioned is true, then the if function will produce exp2.
In the second evaluating inference rule, the top layer sets the condition of exp1 being false within the environment of ∑. The bottom layer says that if the condition afore mentioned is true, then the if function will produce exp3.
IV.HTML
ML tags for web page design use is necessary to generate neat, professional-looking web pages. HTML, although applied in many languages, is not very useful when used by itself. Writing out enough HTML tags to make a full web page is tedious, unnecessarily complex, repetitive, and Journal of the PGSS Project LPX Page 171 error- prone. HTML tags can quickly become difficult to read when many styles are applied to a block of text. For example:
< b > < i > < u > Sample text < / u > < / i > < / b > would generate
Sample text
As web pages become more intricate, more and more tags must be used, and text files become crowded with hard-to-understand symbols. There are some languages designed to make writing HTML much easier. Cascading Style Sheets, or CSS, makes changing the attributes of HTML tags much easier by defining the attributes in one place instead of defining the attributes of the tags at each occurrence of the tags. Blocks of code can also have defined attributes, allowing for much easier formatting of the page and the ability to quickly change the formatting of the page. Because the attributes can be defined in a separate file, a whole website will use many less lines of code and will be able to be reformatted with more easily in one file instead of changing the attributes of every single tag in every single file. Unfortunately, different web browsers implement CSS differently, causing pages to be displayed differently from browser to browser.
Another language that makes writing HTML much easier is PHP. PHP is a server side language, which means that the script is run on a web server, and the output is sent to the client. The client computers never see the actual PHP code; just the HTML output that is generated by the PHP. The PHP language is also useful because it allows for database interaction, graphics creation, and complicated HTML generation. Although PHP is a very powerful language for server side evaluation, PHP does not have variable types, which often leads to unexpected bugs during runtime that may not be obvious until the webpage has been up and running for a period of time.
V. Type Theory
A typed system is a system that classifies every variable, function, and expression into a specific type. These types can be manipulated in a defined manor, and if a type is manipulated in a way that it is not supposed to be, an error is thrown. For example: var foo = 7 //foo is of type int var bar = “ab” //bar is of type string print foo+bar will raise an error in SML because an int cannot be added to a string. If this pseudo-code was implemented in a weakly typed language, this line might unexpected. In a weakly typed language, the output might be a string with the contents “7ab.” The language might assume that the programmer meant to convert 7 to a string and then append “ab” to that string. Most of the time, this is exactly what the programmer meant. However, in a more complicated example, Page 172 Elfont, Guenin, Kamani, McDowell, Ruberg this would not be what the programmer wanted. Say, for instance, that the program prompted the user for foo and bar, and the programmer was expecting an int and a string. If the user accidentally typed in an int and an int, the program would arithmetically add the two values, and that would produce an unexpected and undesirable result.
VI. A Typed Language
A typed language has, at its core, three basic processes. These processes, called the compilation process, are vital for any script of a typed language to run. The processes are the parser, the type checker, and the evaluator. Controlling these processes is accomplished by a control program that instructs the parser to parse the file, and then this control program stores the parsed file and sends it to the type checker and the evaluator. Typed languages that follow this compilation process include SML and Java. In LPX, the control process is a SML script called use3.sml. The control program in LPX reads the contents of the script and then sends them to the parser.
VII. The Parser
The parser is first step in the evaluation process of the script. The parser reads the contents of the script file and translates it into a standard format as defined by the datatype. The translated code does not resemble the original code in any way. For example, an if statement in LPX would look like:
if(condition) { exp1 } else { exp2 }
The parse would read that statement and translate it into a line with the following form:
EIf(condition, exp1, exp2).
The output is stored in a control program. If no parse errors rise during the parse stage, the control passes the output to the next step in the evaluation process. Errors that might arise from the parser include forgetting to close parenthesis, typing a keyword wrong, and using keywords as variable names, among others. If no errors occurred, the compiling process moves to the type checker.
VIII. The Type Checker
Typed languages, at their core, contain a type checker. The type checking stage at run-time occurs after parsing, but before evaluation. The type checker takes two arguments: a context of variables matched with their types, and the expression. The expression is recursively broken down into its sub expressions, and then the base case is the declaration. A declaration is an assignment of a variable. Once a declaration occurs, the type of the variable is added to the context and then associated with the variable name. Now, types can be evaluated based on the Journal of the PGSS Project LPX Page 173 types in the context. If a type rule is ever broken, then an error is raised and the type check fails. The output of LPX should be of type TText. Since the evaluator returns the type of expressions, the return type of type check should be TText. If this is not the case, then the control program raises an error. Since the type checker does not actually evaluate any code, a perfect piece of code should still run perfectly without passing through the type checker.
IX. The Evaluator
The evaluation in SML and LPX, done by the evaluator, is the last step in running a script. At the most fundamental level, the evaluator returns the value of an expression. In order to complete this process, the evaluator must consider each possible type of expression, and evaluate it to the correct type of value. The possible types of values each expression can evaluate to are defined in code as, Datatype value = VInt of int | VBool of bool | VText of string | VString of string | VCons of (value list) | VNil | VPair of value * value | VFun of Syntax.pattern * Syntax.exp * (Syntax.id -> value); The first four types of values are integers, booleans, texts, and strings. A text is the same as a string, except it represents the actual HTML output within LPX. The other four types of values are a list, an empty list, a pair of values, and a function. One of the simplest examples of an expression to evaluate is an integer.
EN (v1) => VInt v1
In this code, an integer expression consists of a single piece of information and evaluates to a single integer value. Since certain types of expressions contain simpler expressions within them, the evaluator functions recursively. An example of this phenomenon occurs in the evaluation of a list, where the evaluator must call itself to evaluate each expression in the list, and then return the value of the list itself. In addition to evaluating the usual expressions, the evaluator must also evaluate declarations. To accomplish this, it must take the expression and apply it to a pattern of identifiers. Before it does this, an environment of variables related to their values already exists. Through the evaluation of each declaration, the environment is expanded to include the new variables with their values.
(fn id => if id=id1 then v Page 174 Elfont, Guenin, Kamani, McDowell, Ruberg
else env id )
This function returns from the evaluation of declarations and represents a new environment with the new variables. To accomplish its purpose as a new environment, it returns the new value if given the new variable, but otherwise returns the variable’s value according to the previously existing environment.
X. Limitations
We made LPX operational, but there are still many features that had to be excluded from the language due to complexity and time restraints. In addition, there are some features, like negation and being able to perform recursion over a list, that we overlooked when laying the language’s foundations and did not realize as missing until after it was too late to change the parser to understand them. Although we built in all the features that we felt necessary for LPX and tried to make the syntax as simple and understandable as possible, once we tried to actually use the language, we found that the syntax worked very poorly with the language’s function and was very limiting. Because of this, the language is not at all user-friendly and gets almost as complex as plain HTML, if not more so, if trying to do anything actually useful for making a website.
XI. Potential Expansion
Given more time, we would like to expand upon a large number of things in LPX. First and foremost among these are the bugs, like not being able to handle recursion over lists, that we didn’t see coming and put a huge dent in the power of our language. However, even with those corrections, our language would not hold any particular edge over HTML, CSS, or PHP. Thus, there are several additions , which we would have liked to add on, given more time and knowledge, that would increase the user-friendliness of our language and make it generally more useable. Among these are an error handling system that would actually identify errors and point the user to their location and reason, as well as a series of built-in functions like those of CSS and PHP that would simplify things for the user.
Appendix A: Inference Rules
Journal of the PGSS Project LPX Page 175
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Journal of the PGSS Project LPX Page 177
References i Gordon S. Novack Jr. CS 307 Vocabulary. http://www.cs.utexas.edu/users/novack/cs307.index.html ii The Not So Short Introduction to L T E X2 ε. www.ctan.org/tex-archive/info/lshort/english/lshort.pdf iiiProf. Michael Erdmann. Computer Science 15-212 Spring 2007. Assignment 6. Carnegie Mellon University. ivShort Math Guide for Latex. www.ams.org/tex/short-math-guide.html
MATHEMATICS TEAM PROJECTS
Journal of the PGSS Three-Dimensional Lie Algebras Page 181
Isomorphisms of Three-Dimensional Lie Algebras
Frankie Chan, Josh Falk, Sam Radomy, David Wang, Sharon Warner
Abstract
In this paper, we classify Lie Algebras into two classes to find isomorphisms which help to simplify computations when working with the multiplication table. In order to find isomorphisms, we used Lie Algebra conditions and basis triples to reduce the number of parameters for each multiplication table and then analyzed these tables to find isomorphisms and non-isomorphisms. We found that Lie Algebras can be reduced by isomorphism to eight Algebras and a special family of Algebras.
I. Introduction
The Lie Algebras with which we are concerned are three-dimensional Algebras over the real numbers. We represent these Algebras with multiplication matrices for basis triples (),, wvu , representing all the rules of multiplication needed to define the Algebra.
⎛ wvu ⎞ ⎜ ⎟ ⎜ uwuvuuu ⎟
⎜ vwvvvuv ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ wwwvwuw ⎠
A multiplication table for a basis triple of a three-dimensional Algebra involves twenty seven parameters, each the coefficient of one component in every vector. Though this model is capable of representing all Lie Algebras, it does not provide great insight into the structure of Lie Algebras, the number of non-isomorphic Algebras, or the essence of each multiplication matrix. Indeed, many Algebras that do not obey the conditions imposed on a Lie Algebra also fall under this model. However, if we reduce the number of parameters, isomorphisms between Algebras appear. It is also much simpler to show that two Algebras are non- isomorphic when the number of parameters is reduced. Clearly, then, reducing the number of parameters provides theoretical insight into the structure of Lie Algebras.
There is, furthermore, a computational reason for developing isomorphisms and reducing the number of parameters. The greater the number of parameters, the more computationally complex it is to use the multiplication rules. However, if we eliminate parameters, computing products reduces to computing a simple change of basis, followed by multiplication with a greatly simplified set of rules. If we succeed in eliminating all parameters, a computationally difficult problem becomes rather easy. Page 182 Chan, Falk, Radomy, Wang, Warner
Therefore, our overall goal is to eliminate as many parameters as possible while developing isomorphisms (and non-isomorphisms) between Algebras. In the majority of cases, we are able to eliminate all parameters; in the worst-case scenario, only one parameter remains.
Algebras are vector spaces associated with vector multiplication; within the Algebra, vectors can be multiplied. The multiplication may fail to have either the commutative or associative properties. However, the multiplication is assumed to be bilinear. In our particular case, the multiplication has neither the commutative nor associative properties.
II. Definition of Lie Algebra
The features distinguishing Lie Algebras from other Algebras are three conditions known as the Lie Conditions. These are:
(L1) For all ∈ Ax , xx = 0 . (L2) For all , ∈ Ayx , + yzxy = 0 . (L3) For all ,, ∈ Azyx , ( ) + ( ) + ( )yzxxyzzxy = 0 The final condition is known as the Jacobi Condition. From now on, all Algebras considered are assumed to be three-dimensional Lie Algebras, with real numbers as coefficients. By manipulating Algebras’ basis triples through various methods, such as perturbation, scaling, and relabeling, the number of parameters and multiplication matrices can be reduced, revealing isomorphic Algebras.
III. Class I versus Class II
A Lie Algebra is described by a multiplication matrix with an underlying Vector Space. Initially the matrix for a basis triple (),, wvu involves twenty seven parameters. Condition (L1) eliminates the diagonal entries because any vector multiplied by itself is equal to zero. Condition (L2) eliminates nine parameters because the multiplication is anti-commutative.
⎛ 0 − wuuv ⎞ ⎜ ⎟ ⎜− uv 0 vw ⎟ ⎜ ⎟ ⎝ − vwwu 0 ⎠
Figure 3.1
There are still nine parameters remaining: ,,,,,,,, jhgfedcba in += + cwbvauuv , ++= fwevduvw , and = + + jwhvguwu . Furthermore, parameters can be reduced through the use of the Jacobi Condition. To use the Jacobi Condition, a classification has to be made. Journal of the PGSS Three-Dimensional Lie Algebras Page 183
A. Class I Algebras
A Class I Algebra is recognized as requiring that any two vectors multiplied form another vector in their span. More precisely, an Algebra is defined to be in Class I if it satisfies:
For all , yx ∈ A , xy = rx + sy for some real numbers r, s
Figure 3.2
We state this in its multiplication matrix form, for a basis triple ( ,, wvu ) , as
⎛ 0 +−+ fuewbvau )( ⎞ ⎜ ⎟ ⎜ +− bvau 0)( + dwcv ⎟ ⎜ ⎟ ⎝ +−+ dwcvfuew 0)( ⎠
Figure 3.3 B. Class II Algebras
A Class II Algebra is recognized as requiring that some two vectors multiplied form another vector not in their span. There thus exists a pair of vectors, u,v , such that uv is not a linear combination of u and v , so that ( ,, uvvu ) is a basis triple. We agree to consider only basis triples (),, wvu satisfying = wuv .
uv = w vw = au + bv + cw wu = du + ev + fw
Figure 3.4
Class I Algebras (Figure 3.3) are characteristically “flat” because the product vector is always coplanar with the two parent vectors. Conversely, Class II Algebras (Figure 3.4) have the possibility of producing a product that is not coplanar with the parent vectors.
IV. Exploration of Class I Lie Algebras
We begin with an unsimplified multiplication table.
⎛ 0 +−+ fuewbvau )( ⎞ ⎜ ⎟ ⎜ +− bvau 0)( + dwcv ⎟ ⎜ ⎟ ⎝ +−+ dwcvfuew 0)( ⎠ Page 184 Chan, Falk, Radomy, Wang, Warner
We choose a new basis triple ()wvu ',',' defined by
'= uu '= vv '= + wuw
Consider the product of v' and w' in this new basis:
'' = (− − ) − +dwbvudawv '''
Since in a Class I Algebra the product of v' and w' will be a linear combination of v' and w' , we have d =−a. By cyclically permuting the perturbation mentioned above, we find f =−c and −= eb . The new set of multiplication rules are:
uv = au − ev vw = cv − aw wu = ew − cu
It is clear that these rules satisfy the Jacobi Condition. To further reduce the number of parameters necessary for multiplication, consider another basis triple ( wvu ',',' ).
'= + pvruu (r ≠ 0) '= svv ()s ≠ 0 ' += qvtww ()t ≠ 0
By letting = at , −= cq , r =−a, and p = e, we arrive at the following multiplication rules:
u'v'= sau' v'w'=−saw' w'u'= 0
For simplification, we relabel (u',v',w') as (u,v,w). At this stage, there are two possible cases:
a = 0 (Case 1.1)
a ≠ 0 (Case 1.2)
If a = 0 , Case 1.1, the following rules result:
uv = 0 vw = 0 wu = 0 Journal of the PGSS Three-Dimensional Lie Algebras Page 185
While Case 1.1 satisfies Figure (3.2), it does not provide much insight because all products result in 0. We therefore go on to Case 1.2 where a ≠ 0 . After scaling vu '' and wv '' by 1 setting s = and relabeling the triple wvu ),,( , a
= uuv −= wvw wu = 0
To prove that these multiplication rules satisfy Figure (3.2), consider the multiplication of two arbitrary vectors, x and y .
= α + β + γwvux
++= ζεδ wvuy This yields:
xy = (αε − βδ )u + (γε − βζ )w
We note a linear combination of x and y :
ε − β = εα − βδ + γε − βζwwuuyx
Because this is the same as the product of x and y , the multiplication rules satisfy Figure (3.2).
V. Exploration of Class II Lie Algebras
We begin with the general multiplication matrix for any Class II Algebra:
= wuv ++= cwbvauvw ++= fwevduwu
Applying the Jacobi Condition gives us that: Page 186 Chan, Falk, Radomy, Wang, Warner
cb = fa ce = fb d = b
Figure 5.1
We observe that any Algebra of Class 2 with a multiplication having a = 0 and e ≠ 0 is isomorphic to an Algebra with a multiplication having a ≠ 0. This can be shown under a change of basis triple to (v,−u,w). Therefore, we limit our study to the two cases: a = e = 0, and a ≠ 0.
A. Case 1: = ea = 0
Let us start by assuming a = e = 0:
uv = w vw = bv + cw wu = du + fw
By the Jacobi Condition, cb = 0; there are three potential cases: b = c = 0; b = 0, c ≠ 0; b ≠ 0, c = 0.
First consider b = c = 0. Then by the Jacobi Condition, d = 0, and the table reduces to:
uv = w vw = 0 wu = fw
Assuming f ≠ 0, we can scale u to eliminate f , leaving:
uv = w vw = 0 wu = w
Figure 5.2
If, instead, f = 0, the multiplication table becomes:
uv = w vw = 0 wu = 0
Figure 5.3
Journal of the PGSS Three-Dimensional Lie Algebras Page 187
Now consider b = 0, c ≠ 0. Then by the Jacobi Condition, d = 0, and the table reduces to:
uv = w vw = cw wu = fw
We can scale v to eliminate c . Assuming f ≠ 0, we can scale u to eliminate f , leaving:
uv = w vw = w wu = w
Figure 5.4
If, instead, f = 0, the multiplication table becomes:
= wuv = wvw wu = 0
Figure 5.5
We now have four cases: (5.2), (5.3), (5.4), and (5.5). We claim that (5.2) and (5.4) are isomorphic to (5.5).
Theorem 5.1: There exist isomorphisms between (5.2), (5.4), and (5.5).
By using the basis triple (u',v',w'):= (u,v + u,w), the multiplication table for (5.4) becomes:
uv = w vw = 0 wu = w
Therefore, (5.4) is isomorphic to (5.2).
Furthermore, by using the basis triple (u',v',w'):= (v,−u,w), the multiplication table for (5.2) becomes: Page 188 Chan, Falk, Radomy, Wang, Warner
uv = w vw = w wu = 0
Therefore, (5.2) is isomorphic to (5.5). Since (5.4) is isomorphic to (5.2), all three Algebras are isomorphic.
We have demonstrated isomorphisms between three of the four cases in which a and b equal zero. However, we now wish to show that the four cases are not all isomorphic. Since (5.2), (5.4), and (5.5) are all isomorphic, showing that (5.5) is not isomorphic to (5.3) will suffice to show that there are two non-isomorphic Algebras.
Theorem 5.2: There exists no isomorphism between (5.5) and (5.3).
In (5.3), w has the property that, for all ∈ Ax , wx = 0. However, in (5.5), if for all ∈ Ax , zx = 0, then z = 0. Therefore, no isomorphism between these Algebras can exist.
Finally, consider b ≠ 0, c = 0. Then by the Jacobi Condition, f = 0.
uv = w vw = bv wu = bu
Since b ≠ 0, we can scale the basis vectors, leaving:
uv = w vw = v wu = u
B. Case 2: a ≠ 0
By invoking Jacobi Conditions, we draw upon the equalities established in (5.1). We recall the unsimplified rules:
uv = w vw = au + bv + cw wu = bu + ev + fw
We create a new basis triple (u',v',w') by letting Journal of the PGSS Three-Dimensional Lie Algebras Page 189
u'= u + sv ⎛ b⎞ v'= v ⎜ s = ⎟ ⎝ a⎠ w'= w
With the new basis triple, new multiplication rules arise:
u'v'= w' v'w'= au'+cw' ⎛ b2 ⎞ ⎛ cb⎞ w'u'= ⎜ e − ⎟v '+⎜ f − ⎟ w' ⎝ a ⎠ ⎝ a ⎠
cb By invoking the Jacobi Condition we realize that f = . Hence, a
⎛ b2 ⎞ w'v'= ⎜ e − ⎟v ' ⎝ a ⎠
⎛ b2 ⎞ ce⎜ − ⎟ = 0 ⎝ a ⎠
At this stage, two possible cases arise:
c = 0(Case 2.1)
c ≠ 0(Case 2.2)
b2 Assuming the first case, Case 2.1 (c = 0), while defining g = e − and g ∈ F , a uv = w vw = au wu = gv
Theorem 5.3: The Algebra obeying multiplication rules as in Case 1.2 is isomorphic to an Algebra of Type 2.1
Let A be an Algebra of Type 2.2.1 with basis triple (u',v',w') defined as ()u + v,u − v,−2w . Through multiplication rules, u'v'= w', v'w'= 2gv', and 1 w'u'= 2gu'. By letting g = a = , we find that v'w'= v' and w'u'= u', which is identical 2 to the multiplication rules of an Algebra of Type 1.2. QED.
If g = 0, we attempt to scale the basis triple by creating a new basis ()u',v',w' = ()u,sv,sw . We find the following multiplication rules: Page 190 Chan, Falk, Radomy, Wang, Warner
u'v'= w' v'w'= s2au' w'u'= 0
To simplify the multiplication, we set s =1 a . Depending on the sign of a , v'w' is either u' or − u' , creating two non-isomorphic Algebras.
If g ≠ 0, we again scale the basis vectors to eliminate the scalars , ∈ Rga . Given basis vectors (),, wvu and ()wvu ',',' , we set = = ',',' = wwsvvtuu . We find:
= ''' = stwvuw = 2auswv ''' = 2 gvtvw '''
Since we operate in the real numbers, we can set == 1,1 gtas . This gives us four cases depending on the sign of a and g ;
= wuv = uvw = vwu
Figure 5.6
= wuv = uvw −= vwu
Figure 5.7
= wuv −= uvw = vwu
Figure 5.8
= wuv −= uvw −= vwu
Figure 5.9
Lemma 5.4: The Algebra of Figure (5.8) is isomorphic to Figure (5.7). Journal of the PGSS Three-Dimensional Lie Algebras Page 191
We being by defining a new basis ( ) = ( ,,:',',' uwvwvu ). The new multiplication rules are as follows:
= wvu ''' −= uwv ''' = vuw '''
Relabeling wvu ',',' to ,, wvu , the new multiplication rules are identical to those of Figure (5.8). Therefore, the two Algebras are isomorphic.
Lemma 5.5: The Algebra of Figure (5.9) is isomorphic to Figure (5.7).
We define another basis ( ) = ( ,,:',',' −vuwwvu ). The new multiplication rules are as follows:
= wvu ''' −= uwv ''' −= vuw '''
Relabeling wvu ',',' to ,, wvu , the new multiplication rules are identical to those of Figure (5.9). Therefore, the two Algebras are isomorphic.
Theorem 5.6: Figure (5.7), Figure (5.8), and Figure (5.9) are isomorphic.
From Lemma 5.4 and Lemma 5.5, we know that (5.7) is isomorphic to (5.8), and (5.7) is isomorphic to (5.9). Because isomorphisms are transitive, (5.8) is isomorphic to (5.9).
Theorem 5.7: There does not exist an isomorphism between Figure (5.6) and Figure (5.7)
Suppose there does exist an isomorphism between the two cases. Let ()u',v',w' be a basis such that
'= + + cwbvauu ' ++= fwevduv ''' ()()()−+−+−== wbdaevafcducebfvuw
Assume that this new basis is the change of basis that exhibits the isomorphism. We first notice that according to the multiplication rules in Figure (5.7), we have ''' −=−= fwvuw . Also, by the property of being a change of basis, we have '' ( 2 2 ) ( 2 2 ) ( 2 +−−+−−−++−−= 2 )wfaacdbcefbudbabeacfdcvecbcfabdeauw
Page 192 Chan, Falk, Radomy, Wang, Warner
Because −= vuw ''' and (),, wvu is a basis, we find the following:
( 22 ) ( +−+=− becfacbdd ) ()22 ()+−+=− cfadbcaee ()22 ()+−+=− adbecbaff
Figure 5.10
Similarly, realizing that + + = = wvucwbvau ''' and also that '= ++ fwevduv and ''' ()()+−== − + ( − )wbdaevafcducebfvuw , we find that
( 22 ) ( +−+= cfbedfeaa ) ()22 ()+−+= cfabefdbb ()22 ()+−+= beadfedcc
Figure 5.11
It is possible to multiply all the equalities in Figure (5.10) and Figure (5.11) by the scalars on the left hand side and add the equalities, we find the following
fedcba 222222 =+++++ 0
Since ,,,,, ∈ Rfdecba , we know that = = = = = fdecba = 0 , which is impossible.
Corollary: (5.6) is not isomorphic to (5.8) or (5.9).
Suppose that an isomorphism between (5.6) and either (5.8) or (5.9) exists. Recalling that (5.8) and (5.9) are isomorphic to (5.7), and because isomorphisms are transitive, this would imply that (5.6) is isomorphic to (5.7). We know from Theorem 5.7 that this is impossible, and thus no isomorphism can exist between (5.6) and (5.8) or (5.9).
Theorem 5.8: There is no Algebra with basis triples (u,v,w) and ()u',v',w' such that ()()= uwwuvwuv 0,,,, and ( ) = ( −uwuwwvvu 0,',''','','' ) .
Let A be an Algebra with two basis triples, (u,v,w) and (u',v',w'), such that
= wuv = wvu ''' = uvw and −= uwv ''' wu = 0 uw = 0'' Journal of the PGSS Three-Dimensional Lie Algebras Page 193
Every product in this Algebra is a linear combination of u and w . Therefore ' += kwhuu for some , kh . Setting '= + + rwqvpuv , we have ''' −== kquhqwvuw . It then follows that ( ) ''' 2 +==−=+− 2 wkquhqwvukwhu . Since , kh are not both 0 , this requires q 2 −= 1, which is impossible
b2 Assuming the second case, Case 2.2 (c ≠ 0), which implies e − = 0, we have a uv = w vw = au + cw wu = 0
We set up another new basis triple (u',v',w') such that
'= puu '= qvv ''' == pqwvuw
1 By setting q =1 and p = while relabeling (u',v',w') to ()u,v,w , our new c multiplication rules are as follows:
uv = w ⎛ a⎞ vw = ru + w ⎜ r = ⎟ ⎝ c ⎠ wu = 0
Theorem 5.5: If A is an Algebra, and ( ,, wvu ) and (u',v',w') are two basis triples, with vw = ru + w and = +wurwv ''''' with ', ∈ Rrr as their multiplication rules, then r'= r.
Let A be an Algebra with basis triples (u,v,w) and (u',v',w') as indicated. Then the range of the multiplication is included in the span of w and + wru , which is the span of u and w ; it follows that w' and +wur ''' are included in this span, and hence so is u' . We then have, for suitable scalars ,,,, cbaqp (with b ≠ 0 ),
u'= pu + qw v'= au + bv + cw w'= u'v'= b()−qru + ()p − q w
Because u' is not collinear with w', we find
Page 194 Chan, Falk, Radomy, Wang, Warner
bq( 2r + pp( − q))≠ 0
Figure 5.6
Also, by the multiplication rules of this class,
v'w'= r'u'+w'= (pr'−bqr)u + (qr'+bp( − q))w .
This leads us to the following:
pr'= br() q + bp()− q (b ≠ 0) qr'= bbqr( ()+ p − q − ()p − q ) Figure 5.7
Multiplying the above equations by q and p respectively and equating, we find that
(b −1)(q2r + pp( − q))= 0
Because of (5.6) we infer that b =1. Putting this into (5.7), we get pr'= pr and qr'= qr. Furthermore, since p and q are not both 0 , we deduce that r'= r. Therefore v'w'= r'u'+w'= ru'+w'. QED.
From Theorem 5.4 we realize that for distinct values of r , there exist non-isomorphic Algebras.
VI. Conclusion
Through the classification of Lie Algebras, we managed to condense all Lie Algebras into nine multiplication tables: eight tables are distinct Algebras and one table is a family. Of the eight multiplication tables, two were part of Class I, and six were part of Class II. Within Class II, we further separated the tables by making certain assumptions for the scalars in each table. Also in this class, we found a family of Algebras which differ from the other tables in a scalar multiple of a vector.
The goal of this project was to find isomorphisms. We found isomorphisms by scaling and perturbing vectors and by redefining variables and basis triples. Also, we used the distributive property and the conditions of Lie Algebras until we could redefine the multiplication rules to match another class’s multiplication rules. Journal of the PGSS Three-Dimensional Lie Algebras Page 195
In further research of Lie Algebras, we could consider Fields other than R , such as the rational numbers and the complex numbers. As with the cross product and the Heisenberg Group, which are both known to be Lie Algebras, we could search for other applications. A final potential area of research would be to investigate higher-dimensioned Lie Algebras.
VII. Acknowledgements
We would like to thank Professor Juan Schäffer for his infinite knowledge and willingness to put time and effort into our project. Much was gained after our experiences with him. We would also like to acknowledge Leland Thorpe and Pulak Goswami for their help in comprehending the many aspects of Lie Algebras.
PHYSICS TEAM PROJECTS
Journal of the PGSS Gambling for the Truth Page 199
Gambling for the Truth: A Monte Carlo Simulation of Magnetic Phase Transitions
Andrew Helbers, Katie Jordan, Michael Petkun
Abstract
We investigated the phase transition of a magnetic material and its properties under varying temperatures. We wrote a program in Java to simulate the Ising model, a Monte Carlo simulation of the effects of temperature on magnetism. In this model, electron spins are arranged in a grid and flipped based on probability. Our main purpose was to find the critical temperature at which a ferromagnetic material loses its magnetic properties.
I. Introduction
A. Monte Carlo Method
The Monte Carlo method is a broad term for using random numbers to solve a problem. A model is created using probabilities to make calculations or to simulate events. Used in a wide variety of fields, from engineering to physics, the main purpose of a Monte Carlo simulation is to model a system which would be unrealistic to simulate deterministically.1 Therefore, most Monte Carlo simulations involve computers, which can perform experiments millions of times faster than humans. It is important to note, however, that it is impossible for computers to create truly random numbers. Instead, a computer’s “random number” generator actually produces pseudorandom numbers based on complex equations. Nevertheless, these numbers have the necessary statistical properties to be relied upon for Monte Carlo simulations.
1. Configuration Space and the Random Walk
In general, a Monte Carlo simulation can be visualized in terms of a configuration space which holds the states of the system. A configuration space is the collection of all of the possible positions of the system, while a single configuration is one possible position in which a system could be at any given time. To model how a system changes, a Monte Carlo simulation can be viewed as a random walk through the configuration space. In a random walk, the configuration at a given step is dependent on the previous configuration as well as on probability.2 To visualize this, one can picture a person walking along a path. For each step, there is a certain probability that he will take a step forward, and a certain probability that he will take a step backward. Page 200 Helbers, Jordan, Petkun
Therefore, his new position depends on where he was previously, but it also takes into account the probability of stepping in each direction.
2. Boltzmann vs. Non-Boltzmann Sampling
For a system, the probability for a given configuration can be calculated by using a Boltzmann distribution. A Boltzmann distribution shows that the probability of any given configuration is the probabilistic weight of this configuration over the sum of the weights of all possible configurations. While this would give an accurate model for a system, it is difficult to calculate for a system that has a large number of configurations, since the denominator would be almost impossible to calculate. Thus, we can use a non-Boltzmann sampling instead.3 One example of non- Boltzmann sampling, also known as importance sampling, serves to eliminate the configurations that have a negligible probability of happening under certain conditions. Instead of summing the probabilistic weights of all configurations, it only counts those that have a realistic chance of occurring, those that are of importance (hence the name “importance sampling”).
Figure 1: Non-Boltzmann Sampling
For a graph such as the one shown in Figure 1, only the shaded area would be counted in an importance sampling, since any of the rest of the area under the graph is negligible. In terms of a random walk, an importance sampling would not consider the probability of any position which has very little probability of occurring (such as the configuration resulting in the person taking a step forward ten meters). Therefore, a non-Boltzmann sampling is faster than a Boltzmann sampling, but still just as effective.
3. Using the Monte Carlo Method to Calculate π
The first Monte Carlo simulation that we performed was intended to calculate the value of π by throwing darts while blindfolded. The target is shown in Figure 2. Journal of the PGSS Gambling for the Truth Page 201
Figure 2: Original Dart Target
For a circle inscribed in a square, the ratio of the area of the circle to the area of the square is π/4 and can be derived as shown in Equation 1 (where r = radius of circle).4 __ circleofArea r 2 r 2 πππ === __ squareofArea r 2 4)2( r 2 4
Equation 1: Derivation of Ratio of Area of Circle to Area of Square
When we throw darts blindfolded, we are essentially choosing random points on the target. Given that each point on the target has the same probability of being hit by a dart, the ratio of darts thrown in a given region to darts thrown in a different region is the same as the ratio of the areas of the two regions. Thus, we can calculate π by multiplying the ratio of darts in the circle to darts in the square by four.
When we first performed this experiment, we found that throwing the darts blindfolded still produced a biased distribution of points. For example, one of us may have tended to throw most of the darts onto the right half, or the bottom, of the target. Therefore, we decided that the regions inside and outside of the circle should be interspersed in order to ensure that throwing darts consistently in one area did not have as much of a negative effect on the results. The modified target is shown in Figure 3. Page 202 Helbers, Jordan, Petkun
Figure 3: Modified Dart Target
Since the target was still comprised of circles inscribed in squares, the value of π could still be calculated in an identical manner, but now the darts are more likely to be spread out among the different regions. In other words, if the darts had the tendency to land on the bottom portion of the target, then this target still contains full circles in squares at the bottom. However, this system was not perfect, so we searched for a method that would be even more effective. Thus, we decided that shooting staples down five stories would yield better results for a couple of reasons. First, it is more efficient than throwing darts, since shooting staples can be done more rapidly, so more data points could be collected in a shorter period of time. Also, the scattering of staples would be more random, since gravity would not affect the outcome. When throwing darts, for example, gravity would tend to push the darts toward the bottom of the target, but when shooting staples downward, gravity does not affect the location of the staples on the target. After we shot staples and collected data, we found a value of π that was correct to one decimal place, based on calculations of error. For calculating an unknown value (in this case, π) with a distribution of σ uncorrelated events, the error is defined as ± , where N is the number of points taken (in N this case, darts thrown or staples shot), and σ is the standard deviation, or error, for one trial.5 Such a definition of error is necessary because of the fact that in most Monte Carlo simulations, the target value in unknown. Thus, error must be calculated using the simulated data. Because of this calculation of error, we could only keep one decimal place, since our value of N was approximately 100, giving an error on the order of 0.1. It then follows that it would be necessary to increase the number of data points in order to minimize this error. Journal of the PGSS Gambling for the Truth Page 203
Therefore, we determined that a computer-based Monte Carlo simulation might yield more effective results. We wrote a program in Java that simulated our experiment, virtually throwing darts at a circle inscribed in a square. The simple algorithm is as follows: First, pretend that we have a square with a side length of 1. Then, obtain two (pseudo)random numbers between 0 and 1 from the computer. These are the x- and y-coordinates of the dart thrown. Next, based on these coordinates, calculate the distance of the dart from the center of the circle. If the radius is less than or equal to 0.5, then the dart is inside the circle. To throw multiple darts, loop through this multiple times. The percentage of darts inside the circle multiplied by four should yield the approximation of π. Because of a computer’s fast runtime, millions of virtual darts can be thrown. As more data is collected, the value of π becomes more precise. Table 1 shows a sample of our data (20 trials each). Darts Mean Standard Thrown Value of Pi Deviation 10 3.18 0.129826 100 3.178 0.041055 1000 3.1396 0.012983 10000 3.14456 0.004105 50000 3.142228 0.001836 100000 3.140762 0.001298 500000 3.141969 0.000581 1000000 3.141495 0.000411 5000000 3.141778 0.000184 10000000 3.141727 0.00013 20000000 3.141545 9.18E-05 Table 1: Mean and Standard Deviation (Error Calculated) vs. Darts Thrown
It is evident that as more darts are thrown at the target, the calculated value of π becomes more accurate, as the mean approaches the true value of π and the deviation approaches zero.
B. The Ising model of Magnetism
1. Calculating Energy and Probability
Page 204 Helbers, Jordan, Petkun
The Ising model is a model of the way temperature affects magnetization. The Ising model uses a grid configuration where each element in the grid is either “up” or “down”. Its energy (E) is calculated by dot product using Equation 2.5
∑ •−= ssJE ji , ∈ngbji
Equation 2: Energy Equation
. Figure 4: Energy of Electron Spins
The energy of each element depends on the spins of the immediate neighboring spins. If the neighboring element is the same as the element in question (either “up” or “down”), +1 is added to the energy. If the neighboring element is opposite, -1 is added to the energy. For example, in Figure 4, E( ) = ( -J ) (-1 + 1 -1 -1) = 2J. The energy of each element in each configuration is calculated in this manner, and a total energy for each configuration is determined by taking the summation of the energies of each of its elements. The probability of each element in the configuration switching from “up” to “down” or “down” to “up” is determined by Equation 3.5
E − i e Tk EP i )( = −E j ∑e Tk j
Equation 3: Probability Function
The probability of the system having the energy Ei is equal to the weight of the Ei configuration
−Ei Tk ( e , where k is Boltzmann’s constant, and T is the temperature in units Kelvin.) divided by the sum of the weights of all of the possible configurations. As temperature decreases, the lowest energy configurations dominate and thus these configurations clearly become the most probable. As temperature tends toward infinity, the weight of all configurations approach 1, and all configurations become equally probable. The system will become random as each configuration Journal of the PGSS Gambling for the Truth Page 205 is equally accessible, and the element spins keep shifting due to the fact that the probability of being “up” is the same as the probability of being “down”.
2. Phase Transition and critical temperature
At a certain temperature, the material will change from being mostly aligned to being mostly random. This is called a phase transition and will take place at the critical temperature. Phase transitions exist in many different forms. Some are common, such as water changing phase from liquid to gas at a boiling point, or from ice to liquid at the melting point. Other types exist also, such as the transition of a material from being magnetized to being demagnetized at a certain temperature. Observing the effects of temperature on magnetism using the probability equation is an application of the Ising model because the probability of electrons becoming aligned in a magnet is dependent on the temperature of the material. When looking at magnetism as an Ising model, the “up” and “down” spins in the configuration grid can represent the spins of electrons in a magnetic material. The probability equation would be applied to determine the most probable configuration of electrons at a certain temperature, and random numbers would be used to compare to the probability and determine whether or not the electron spin would change to spin in the opposite way. In this way, the Ising model is solved with the Monte Carlo method of comparing random numbers to probabilities to simulate a system. Since magnetism is caused by the alignment of electron spins, and most of the element spins are aligned in an Ising model at low temperatures, the material would be magnetized at low temperatures. At high temperatures, however, the spins become random, thus decreasing the magnetization of the material because the spins would cancel each other out.
The critical temperature can be observed in the Ising model by plotting the Magnetization vs. Temperature for many sweeps of the probability equation. For convenience, this temperature would be measured in natural units of J/k where J is a constant dependent on the material that is being considered and k is Boltzmann’s constant, 1.38*10-23 m2 kg s-2 K-1. J is not known theoretically, but could be calculated experimentally by physically examining the magnetization of a material compared to temperature. The critical temperature would be the lowest temperature with a net fractional magnetism of 0, since at the critical point the graph drops off from a net fractional magnetization of about 1 at low temperatures when the material is magnetized, to 0 because the spins are random at high temperatures, canceling out magnetization. The critical temperature occurs where there is a discontinuity in the derivative of the graph with respect to temperature. The phase transition that occurs in the Ising model simulation is of the second order because there is not a discontinuity in the entropy of the system.6
Page 206 Helbers, Jordan, Petkun
3. Entropy, Energy, and the Ising model
As the system loops through the probability equation and approaches the most probable configuration at a particular temperature, the system maximizes entropy and minimizes energy. The system would have the least energy when the spins are aligned in the same direction, but would have the most entropy when they are shifting continuously and randomly from “up” to “down”.
4. Summary of the Ising model
The Ising model can use a grid configuration to simulate the effects of temperature on magnetism. After looping through the probability equation many times, the system will tend toward a specific magnetization where the entropy is maximized and the energy is minimized for that particular temperature. The system will also tend toward a net fractional magnetism of either 0 at high temperatures (because of the randomness of the spins) or 1 at low temperatures (because the spins are mostly aligned). If the spins are mostly aligned, which occurs at temperatures below the critical point, the material is considered magnetized. At temperatures above the critical point, the electrons are spinning in a random and unaligned way, thus canceling out the magnetism of the material.
II. Implementation of the Ising model
Using a Boltzmann sampling, it would be possible to evaluate the probability for all possible states for the entire array and calculate the average magnetization using an expectation value. However, this is impractical for large arrays, as the number of possible states to evaluate grows as an exponential function of the number of elements in the array. Thus, we use importance sampling. This means that only the most probable states are evaluated, saving time and computations. Consider a random walk. This simulation will start at a random configuration, and there will be a certain probability that the configuration will change to another. Eventually, the random walk will be drawn to the most likely states converging to a physically realistic system. This is effectively how the simulation will find the most probable states. A group of these states is then evaluated for their magnetic moments.
Being that the model describes an array of up and down configurations, it is natural to store it as such, an array. A function was developed to use a pseudorandom number and the probability function to flip or not flip a given element in the array. This function was then directed to act upon Journal of the PGSS Gambling for the Truth Page 207 all of the elements in the array, one-by-one. Adding together the elements and finding the magnetization and energy is a trivial matter of adding together each element. The energy is calculated by evaluating Equation 2. As the elements become further apart, the energy between them becomes smaller. Evaluating each element with every other element in the array would be computationally difficult, and the point of this project is to ignore things that are insignificant. If a random number between zero and one is less than this probability, the element remains in its current state, otherwise, it is flipped.
A. Auto-Correlation
Auto-correlation is the sum of dot products between an element and other elements along some predictable function (usually lines) summed for every element divided by the total number of elements and the total number of functions used. This is used to find locally magnetic regions that could be canceled out by other such regions, preventing them from showing up in the magnetization. The number of dimensions only affects how the elements are counted and compared.
B. Convergence
The first issue is with the convergence of the data, and dealing with how long the random walk takes. When the temperature is changed, the system will require some time to evolve into the new equilibrium state at the new temperature. This is because going through every element once only allows each element the ability to flip once, and each run of this makes the magnetization become closer to what it should be, as the chances to flip increase. This process is equivalent to allowing more than one possible location in the random walk to be used. As a result, this oscillation becomes insignificant after some number of run-throughs, just as the random walker eventually wanders close to the most probable state. If not enough sweeps of the array are made, there is no guarantee that the states recorded are close to the most probable state.
Convergence studies must be done before actual data runs. The goal was to find an approximate number of sweeps through the array that would give the correct state. The initial results were taken with a fifty by fifty array, which was initialized randomly. The results of these convergence studies are as follows: Page 208 Helbers, Jordan, Petkun
Fractional Magnetization for n = 50*50, T = 2 and F = 50
1
0.9
0.8
0.7
0.6
0.5
0.4 Magnetization 0.3
0.2
0.1
0 0 102030405060 Sweep Number
Figure 5: Calibrating the Sweep Number
F is the number of times each element in the array was allowed to flip, T is the temperature, and n is the number of elements given in such a way that the dimensions are shown. Figure 5 shows that F = 50 is not a high enough value, as there is a significant number of sweeps before the magnetization levels out. So, F = 50*3, or F = 150 was tried next, as that seemed like a good lower bound for the convergence length in Figure 5. Journal of the PGSS Gambling for the Truth Page 209
Fractional Magnetization for n = 50*50, T = 2 and F = 50*3
0 0 102030405060 -0.1
-0.2
-0.3
-0.4
-0.5
-0.6 Magnetization -0.7
-0.8
-0.9
-1 Sweep Number
Figure 6: Calibrating the Sweep Number
As this result (Figure 6) has the same problem, F = 200 and F = 250 were examined. Page 210 Helbers, Jordan, Petkun
Fractional Magnetization for n = 50*50, T = 2 and F = 50*4
1
0.9
0.8
0.7
0.6
0.5
0.4 Magnetization 0.3
0.2
0.1
0 0 102030405060 Sweep Number
Figure 7: Calibrating the Sweep Number Journal of the PGSS Gambling for the Truth Page 211
Fractional Magnetization for n = 50*50, T = 2, F = 50*5
1
0.9
0.8
0.7
0.6
0.5
0.4 Magnetization 0.3
0.2
0.1
0 0 102030405060 Sweep Number
Figure 8: Calibrating the Sweep Number
As a result, since Figure 5 suggested six trials of fifty sweeps were needed, and all of the other sweep calibration graphs say that less are needed, for a safe estimate, it was determined that F = 50 * 6 or F = 300 was appropriate. As the convergence length varies with temperature, modest overestimation is prudent. As the temperature approaches the critical temperature at which the magnetization decreases to almost zero, convergence time increases. This effect is known as “critical slowdown,” and many algorithms have been devised for speeding up the convergence at the critical temperature, which is unfortunately also the most important point of interest. The above charts were used for a good approximation for conducting other experiments. Figure 9 shows a sample graph of the convergence graphed for each run-through of the array. Page 212 Helbers, Jordan, Petkun
Fractional Magnetization for T = 2.0 vs. Sweeps
0.2
0 0 200 400 600 800 1000 1200 -0.2
-0.4
-0.6 Magnetization
-0.8
-1
-1.2 Sweeps
Figure 9: Calibrating the Sweep Number
Fractional Magnetization for T = 1.25 vs. Run- Throughs
0 0 200 400 600 800 1000 1200
-0.2
-0.4
-0.6
Magnetization -0.8
-1
-1.2 Sweeps
Figure 10: Calibrating the Sweep Number Journal of the PGSS Gambling for the Truth Page 213
By making several of these graphs and visually finding the number of run-throughs of the array before the magnetization stabilizes, we were able to construct Figure 11.
Sweeps Required for Convergence vs. Temperature
800
700
600
500
400 Sweeps 300
200
100
0 00.511.522.5 Temperature (J/k)
Figure 11: Required Sweep Number vs. Temperature
Since Figure 11 is created by visual inspection, it does not show a definite function, but it can be trusted for a general trend. In addition, each data point is taken from looking at single graphs for various temperatures like those in Figures 9 and 10.
Above the critical temperature, the magnetization becomes random, and averages to zero. Figure 11 shows that roughly three hundred sweeps were appropriate for most temperatures. As the temperature is being slowly changed, the simulation of the next temperature is closer to that temperature’s state than the completely random state that was used for constructing the above graph. One can anneal the temperature of this simulation by simulating one temperature, then changing the temperature slightly, and starting the next simulation from that state. This is helpful because when two temperatures are close to one another, the magnetization differs less than if the temperatures were far apart. In the random walk explanation, this is similar to starting the random walk at a point close to the most probable state. We start at temperature 0 or above the critical temperature because those are the only temperatures where we know what the configuration should look like via physical argument.
Page 214 Helbers, Jordan, Petkun
III. Results
A. Magnetization vs. Temperature
Starting at a temperature of 0.1 J/k, incrementing by .1 each time, and saving the results in a file each time, one trial gave Figure 12.
Fractional Magnetization vs. Temperature - One Experiment
1.2
1
0.8
0.6
0.4 Magnetization
0.2
0 012345678910 -0.2 Temperature (J/k)
Figure 12:
This result was taken from a fifty by fifty two-dimensional model. There seems to be a drastic change in magnetization between 2.1 and 2.6. To improve statistics, multiple trials were taken and averaged together. This also follows the idea that it would be best to examine the most probable states. The above only evaluates one possible state for each position.
Total magnetization can be negative or positive, and only whether the elements are different changes the probability of a state (Equation 3). Thus, the absolute value of the magnetization is taken for each trial before being averaged. To allow comparisons between different-sized arrays, the fractional magnetization was taken by dividing by the number of elements. Note that in Figure 12, the data points are relatively sparse near the location of interest, due to there being a high derivative at roughly T = 2.3, and the points being uniformly distributed along the x-axis. To compensate, more points were taken between T = 2.0 and T = 3.0. Journal of the PGSS Gambling for the Truth Page 215
Average Absolute Value of Magnetization vs. Temperature
1
0.9
0.8
n 0.7
0.6
0.5
0.4
Absolute Magnetizatio 0.3
0.2
0.1
0 012345678910 Temperature
Figure 13: Magnetization vs. Temperature for 2D- 50 by 50
Again, it seems that the critical temperature was approximately between 2.0 and 2.3 J/k. However, since Figure 13 takes into account three thousand numbers instead of one hundred, this graph is likely to be more accurate. The literature value for two dimensions is T = 2.269 J/k, which is in this range.3
B. Visualization of the Model
In an attempt to visualize the magnet, an animation and graphic user interface were used to show the magnet. The temperature and dimensions could be controlled, and each element was represented as a blue or red block, depending on orientation. Figures 14, 15, and 16 are pictures of the results of visualizing the array. Page 216 Helbers, Jordan, Petkun
Figure 14: T = 3.5
Figure 15: T = 2.0
Journal of the PGSS Gambling for the Truth Page 217
Figure 16: T = 0.5
Below the critical temperature, the magnet begins to become ordered, and becomes completely ordered at a low enough temperature most of the time, and above it, the array is completely random.
C. Alternate-Dimensional Models
1. One-Dimensional Model
Every result so far has related to the two-dimensional version of the Ising model. The one- dimensional model has an interesting nuance to it. When the Ising model was originally created, a proof was made for the one-dimensional model that shows that no phase transition occurs.3 This conclusion was thought to apply to higher dimensions. The above results show that this is not so. The critical temperature for the one-dimensional model should be exactly zero, so no phase transition occurs, as there is never a second phase in which the array may exist. The two- dimensional model shows a slight decrease in magnetization up to a critical temperature, at which the magnetization suddenly falls to almost zero. The results of a single simulation of a one- hundred element one-dimensional model for magnetization and temperature are shown by Figure 17. Page 218 Helbers, Jordan, Petkun
Fractional Absolute Average Magnetization vs. Temperature
1
0.8
0.6
0.4
Magnetization 0.2
0 012345 Temperature (J/k)
Figure 17: Magnetization vs. Temperature for 1D- n = 100
Note that the drop-off isn’t sudden, as it is in the two-dimensional model. In the two-dimensional model, the derivative appears go to almost negative infinity, while in this graph, the derivative does not appear to act as such.
For comparison, Figure 18 shows the derivatives for the two models plotted against each other. Journal of the PGSS Gambling for the Truth Page 219
- d
8
7
6
5
4 1D 3 2D
- d
1
0 0 0.5 1 1.5 2 2.5 3 -1
-2 Temperature J/k
Figure 18: Comparison of Derivatives for 1D and 2D
There is something fundamentally different about how the two drop-off to zero. There is no discontinuity in the derivative of magnetization in the one-dimensional model, while there is such a discontinuity in the two-dimensional model, leading to the one dimensional model having no critical temperature, while the two-dimensional model does. Since such a thing occurs in comparing two models that differ in one dimension, it would be interesting to examine a three- dimensional model.
2. Three-Dimensional Model
The three-dimensional model currently has no exact solution, but there is no reason not to continue by applying the same techniques for three dimensions. Figure 19 results for sixteen and eight cube models. Page 220 Helbers, Jordan, Petkun
Average Absolute Magnetization vs. Temperature for 3D Model
1.2
1
0.8
16 Cube 0.6 8 Cube
Magnetization 0.4
0.2
0 024681012 Temperature (J/k)
Figure 19: Magnetization vs. Temperature for 3D
It is difficult to determine whether a phase transition occurs on this graph, so the derivative is also graphed. The correct value has been estimated to be approximately twice that of the two- dimensional model.3 Just as when examining the two-dimensional graph, it is possible to extract a range. The critical temperature is between 4.0 and 7.6 J/k for the three-dimensional Ising model. The autocorrelation in Figure 21 shows similar results for the range, and the derivative graph shows that a detectable phase transition does occur, as there is a discontinuity.
Journal of the PGSS Gambling for the Truth Page 221
-d
3.5
3
2.5
2
1.5 16 Cube
1 8 Cube -d
0 0 2 4 6 8 10 12 -0.5
-1 Temperature (J/k)
Figure 20: Derivative of Magnetism for 3D
Fractional Autocorrelation Length vs. Temperature
1
0.8
0.6
16 Cube 0.4 8 Cube
0.2 Correlation Length
0 024681012
-0.2 Temperature (J/k)
Figure 21: Correlation Length for 3D- 16 by 16 by 16
D. Summary of Results
Page 222 Helbers, Jordan, Petkun
The results for the plot of magnetization versus temperature were as expected: at low temperatures, the material was almost perfectly magnetized. As the temperature increased, the magnetization slowly decreased until the critical temperature, at which the magnetization sharply dropped to just above zero. At this temperature, the derivative of magnetization with respect to temperature had a discontinuity, signifying a phase transition. This critical temperature was found to be 2.3 J/k. For auto-correlation, it also began to decrease as the temperature increased, but with a steadier fall. After it passed the critical temperature, the auto-correlation also hovered around zero.
In other dimensions, the results came out a bit differently. For the 1-D simulation, no critical temperature was found. Instead, the magnetism decreased in a smooth manner. The derivative of this graph did not have any sharp peak, as the 2-D simulation had at its critical temperature. The 3-D model, on the other hand, did prove to have a critical temperature, as the discontinuity in the derivative of magnetization as a function of temperature strongly indicated a possible phase transition. The magnetization decreased sharply (though not as sharply as the 2-D model) at around 7.2 J/k.
IV. Acknowledgments
Pennsylvania Governor’s School for the Sciences, for allowing us to further our scientific education this summer
Carnegie Mellon University, for devoting its resources to us during the program
Frankie Li, for being an awesome project leader (and for taking us on necessary coffee breaks)
Aaron Hernley, TA, for being an inspiration to us all
Ernst Ising, for giving us such an interesting topic to research
V. References
1 Introduction to the Hrothgar Ising Model Unit, http://oscar.cacr.caltech.edu/Hrothgar/Ising/ intro.html 2 Richard P. Feynman, Robert B. Leighton, and Matthew Sands, The Feynman Lectures on Physics: Commemorative Issue, Volume 2, Addison Wesley, 1989, p. 6-5 Journal of the PGSS Gambling for the Truth Page 223
3 David Chandler, Introduction to Modern Statistical Mechanics, Oxford University Press, New York, 1987, p. 122-123, 168-175 4 Tug Sezen, The Basics of Monte Carlo Simulations, http://folding.stanford.edu/education/ montecarlo/index.html 5 Frankie Li, Lecture, July 2007 6 Walter Greiner, Ludwig Neise, and Horst Stöcker, Thermodynamics and Statistical Mechanics, Springer, New York, 1995, p. 418
Journal of the PGSS An Investigation of Chaos in One and Two Dimensions Page 225
An Investigation of Chaos in One and Two Dimensions
Diya Das, Ankur Goyal, Andreea Manolache, Natalie Morris, Nancy Paul, and Varun Sampath Advisors: Jared Rinehimer, Barry Luokkala, PhD
Abstract This investigation sought to document, analyze, and compare dimensionally different forms of chaotic systems and the fractals that they create when graphed. We conducted two independent, distinct experiments for exploration of chaos in multiple dimensions of motion. The first experiment featured a damped driven mechanical Duffing Oscillator provided by the Pennsylvania Governor's School for the Sciences and the physics department at Carnegie Mellon University. This apparatus allowed us to generate chaos in one dimension by means of a steel reed hanging between two magnets and driven by a sine wave generator. We analyzed two different sets of chaotic motion generated by the Duffing Oscillator with an oscilloscope and the LabVIEW program. From the first set we recorded 265,063 data points and found a fractal dimension of 1.772 ± 0.026. The second set contained 40,892 points with a fractal dimension of 1.761 ± 0.002. The second phase of our investigation consisted of constructing and analyzing a two- dimensional oscillator that was driven from two sides, allowing motion along the x- and y- axes. We achieved two-dimensional chaotic motion and qualitatively observed the apparatus. We recorded the motion with a video camera, but due to time restraints, we were unable to quantitatively analyze the collected data.
I. Introduction to Concepts/Background
A. Chaos
1. Conceptual History Until the revolution of quantum theory, physicists had held that the entire universe was deterministic. This concept, known as Laplacian determinism, received its name from the French mathematician Pierre Simon de Laplace. Laplace maintained that as long as he knew the velocity and position of every particle in the universe, he could state exactly where all those particles would be at any given time1. Of course, Laplace based this belief on the assumption that every system in the universe was governed by Newtonian mechanics, and it was not until Heisenberg postulated his uncertainty principle that physicists rethought their absolute ability to measure a particle’s exact position and velocity.
Even so, scientists remained skeptical of Heisenberg’s postulate and argued that one should be able to observe a particle’s motion and position well enough to approximate its path in the past and future. A response to these queries emerged in 1903 when French mathematician Henri Poincaré postulated that, “A very small cause which escapes our notice determines a considerable effect that we cannot fail to see, and then we say that the effect is due to chance…. But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena.”2 Although it was not termed as such yet, Poincaré was talking about a chaotic system, one that is highly sensitive to its initial conditions. It was not until the work of American meteorologist Edward Lorenz that scientists began to realize that chaos has a profound Page 226 Das, Goyal, Manolache, Morris, Paul, Sampath, Rinehimer, Luokkala effect upon almost all systems. In 1961, Lorenz, who had been experimenting with computer models of weather patterns, stumbled across a startling result: if his input parameters were varied even in the slightest way, the resulting weather model varied drastically after a short period of time.3 Until this discovery, most scientists assumed that a slight change such as rounding off a number would result in negligible changes, which would smooth out over time. Lorenz showed through further analysis that a non-periodic system such as a weather pattern would never be predictable due to its enormous sensitivity to initial conditions.
However, Lorenz also showed that such a system produced a map that implied some sort of order. Regardless of the randomness of the direct path of the system, when plotted over time, the paths stayed within a certain boundary, now known as a Lorenz attractor. Figure 1 depicts Lorenz’s attractor pattern of a system with three input parameters in chaotic motion.
Figure 1: The Lorenz Attractor4
2. Chaos Defined
Chaos is a state of a nonlinear, dynamical system that is highly sensitive to its initial conditions. A nonlinear system is one in which the output is not proportional to the input5: therefore the equation of motion cannot be determined, and one can only approximate the behavior of the system when the nonlinear terms are ignored. The phase space of a system consists of the entire range of possible position and velocity coordinates of a given point in the system. Since the system is dynamical, recorded data points create a trajectory in phase space.6
The point or pattern towards which trajectories converge is called the attractor. In a non-chaotic system such as an oscillating pendulum under the force of friction, the attractor is a single point, since eventually friction will slow the oscillation to a halt. A system can also have a periodic attractor, where the trajectory repeats itself after a certain number of cycles. In the case of chaotic systems such as our pendulum, the periodic attractor is termed a strange attractor. Although the period is actually infinite, there is still an attractor pattern that keeps the system within a certain set Journal of the PGSS An Investigation of Chaos in One and Two Dimensions Page 227 of phase space. If a system were truly random, the phase space would have no attractor within it; it would be completely filled.
In physics, chaos does not imply complete disorder. Herein exists the essential difference between chaotic motion and random motion. Chaos is thought to be an intermediate dimension between order and randomness, called the fractal dimension. A system is generally considered chaotic if its attractor is a fractal. Although the chaotic system does not appear to have symmetry regarding its motion or position, it does have certain symmetry of scale that produces the fractal dimension, as is explained below. Simply put, the fractal dimension is interpreted as a measurement of how chaotic the system is.
B. Poincaré Plots and Fractal Dimension
1. Poincaré Plots
To plot the trajectories of an object in one-dimensional chaotic motion, we must plot a three- dimensional graph containing velocity, displacement, and time information. A Poincaré plot is the cross-sectional graph of the trajectory graph, where the time axis is eliminated. The points on the Poincaré plot are determined by the displacement and velocity coordinates of the various trajectories after every period of the driving function. Poincaré plots are an effective way in which to map chaotic motion, since the distribution of the intersections allows us to determine the fractal dimension of the system. Essentially, the Poincaré plot is a two-dimensional graph that represents the trajectories of the object in chaotic motion, but does not tell the time at which the object obtains these coordinates, because we wish to analyze the chaotic behavior of the trajectories independent of time.7 The accuracy of the Poincaré plot depends on how many points are included, because although the attractor pattern it represents is finite, the trajectories are infinite. This becomes an issue in the analysis of data, as described in Section II.C.4.
2. The Fractal Dimension
A fractal is a geometric figure commonly defined using the principle of self-similarity, wherein a magnified section of the figure resembles the original figure.8 As geometric objects, fractals have their own geometry, separate from the standard Euclidean geometry, appropriately called fractal geometry. Fractal geometry is necessary to describe irregular surfaces such as those encountered in the study of chaotic motion, explain their existence, and to understand their properties.9
Since the beginning of the study of fractal geometry, there have been multiple definitions of the so- called fractal dimension. In the early 1990s, Bunde and Havlin claimed that one type of fractal dimension is insufficient to accurately describe all fractals, and that there must be at least ten types of fractal dimension.10 For the sake of simplicity and time, we have chosen to analyze our data by calculating only one type of fractal dimension. The fractal dimension we will hereafter refer to in our analysis is commonly known as the similarity or self-similarity dimension.11 Page 228 Das, Goyal, Manolache, Morris, Paul, Sampath, Rinehimer, Luokkala
We can define the dimensions of regular geometric objects using the principle of self-similarity. In one, two, and three dimensions, the objects that adhere to the principle of self-similarity are a line, a square, and a cube, respectively.
Suppose we divide a line of length l into an arbitrary number of segments n , all of the same length l n . To return the original length l , we must magnify any one segment by a factor of n , the total number of segments. Therefore, a line is composed of n self-similar fragments with magnification factor n .
In a similar fashion, we can divide a square with edge length l and area l 2 into an arbitrary 2 number of small squares n 2 , each of area l . We must magnify each length of a small square n 2 by a factor of n to return the area of the original square. Because the magnification occurs along both perpendicular lengths of the square, the n is squared in the process of magnification, returning an area of l 2 . Thus, a square is composed of n 2 self-similar fragments with the same magnification factor of n .
Using calculations similar to those used in the determination of the number of self-similar fragments and magnification factors for lines and squares, we can show that a cube is composed of n3 self- similar fragments with a magnification factor of n .
We can derive an equation to explicitly define dimension using the physical definition of dimension and the calculations of self-similarity above. Using the example of the square, we must find a relationship between the number of self-similar boxes or cubes and the magnification factor so that the calculated dimension equals 2 or 3 respectively. (Using the calculations for the line will not work, since a linear relationship, which could satisfy the requirements for a line, would not give an appropriate calculated dimension for squares and cubes.) The dimension D of an object, as calculated according to the principle of self-similarity, is
log (number of self similar pieces ) D = log( magnification factor ) .
For example, a square would have dimension
log (number of self similar pieces ) log(n 2 ) 2log(n ) D = = = = 2 log( magnification factor ) log(n ) log(n ) ,
as expected.
Similarly, a cube would have dimension Journal of the PGSS An Investigation of Chaos in One and Two Dimensions Page 229
log (number of self similar pieces ) log(n3 ) 3log(n ) D = = = = 3 log( magnification factor ) log(n ) log(n ) , also as expected.
Although we determined the expression of dimension for self-similar objects of Euclidean geometry, we can also apply it to fractals, since fractals follow the principle of self-similarity. Thus, the mathematical definition of the fractal dimension DF is
log (number of self similar pieces ) D F = log( magnification factor ) .
Physically, the fractal dimension is a measure of the relative complexity of a fractal, compared to the objects of the neighboring whole-number dimensions on either side. For example, for a one- dimensional system, a fractal dimension DF between one and two would indicate that the number of points in the set of the fractal is between the number of points contained in the set of a line ( D = 1 ) and the number of points contained in the set of a plane ( D = 2 ). In other words, the fractal dimension is a measure of the relative number of points a fractal contains. The fractals of one-dimensional chaos that we are exploring have a fractal dimension between one and two, between the dimension of a line and the dimension of planar randomness. For the two-dimensional chaos, the fractal dimension is usually less than 4, the difference between dimensions arising because in two dimensions there are two independent coordinates used to express both the velocity and the position of a point in the dynamic set.
3. The Box-Counting Dimension as an Approximation of the Fractal Dimension
Although there are various methods of calculating the number referred to as the box-counting dimension,12 many produce the same result. We used the method of determining the box-counting dimension described below as an approximation of the fractal dimension.
To determine the box-counting dimension of a Poincaré section F , we overlay the graph with a grid of boxes (squares for two-dimensional plots, which correspond to one-dimensional motion, and cubes for three-dimensional plots, which correspond to two-dimensional motion), each of side length s . We then count the number of boxes N()s that contain points in the set. The number of boxes containing elements of the set is a measure of the irregularity of the set at the various scales s . Plotting N()s for boxes of various lengths s , we define the box-counting dimension as the negative slope of the linear fit of the graph of log N()s over log 1/ s . The box-counting dimension then becomes the "logarithmic rate at which N()s increases as s approaches zero," or less technically, the rate at which randomness develops in a set, with respect to scale.13 Page 230 Das, Goyal, Manolache, Morris, Paul, Sampath, Rinehimer, Luokkala
The box-counting dimension described above is especially useful, compared to the mathematical definition of the fractal dimension, because the calculation does not require knowledge of the number of self-similar pieces or of the magnification factor of the fractal, allowing us to determine an approximation for the fractal dimension of a set of points without any additional data. Physically, the box-counting dimension is a measure of the randomness, or the spread, of a set. But because the sizes of the boxes are varied in the calculation of the box-counting dimension, the box-counting dimension also measures the self-similarity of a set. Similarly, the fractal dimension, which is primarily a measure of self-similarity, is also a measure of the spread of the dataset, because of its use as a measure of the relative number of points the set contains. Thus, in our investigation, we used the box-counting dimension as an excellent approximation of the fractal dimension.
C. The Duffing Oscillator
The emergence of nonlinear dynamics in the 1800’s led to increased research in mechanical motion centered about forced oscillators and their connection to chaotic motion. The Duffing Oscillator, studied in 1918 by Georg Duffing, is a simple one-dimensional chaotic oscillator defined by the general equation, 14
15 3 2 &&xx++σβω & ( x ±0 x) = γcos() ω t
In this equation, σ represents the damping constant, γ represents the amplitude of an external driving force, and ω represents the angular frequency of the same external driving force. The term enclosed in parentheses corresponds to the potential energy of the oscillator. The apparatus typically consists of a thin steel reed attached at one end to a solid structure, oscillating freely between two adjustable magnets, where an external force drives the reed. This force sustains its chaotic motion by compensating for the system’s energy losses due to friction and eddy currents. In addition to chaotic motion, the apparatus can display periodic motion under certain driving parameters. This quality, along with the apparatus’s relative simplicity and derivable formula makes it ideal for a study of chaos and its properties.16
II. Our Experiments
A. Purpose The goal of this investigative experiment was to observe and analyze chaos in one- and two- dimensional motion. We aimed to find and record chaos in the one-dimensional Duffing Oscillator, and then apply what we learned about one-dimensional chaotic motion to build our own two dimensional driven oscillator. By combining our observations, we hoped to gain a greater understanding of the chaos and fractal dimensions present in both dimensions of motion. Journal of the PGSS An Investigation of Chaos in One and Two Dimensions Page 231
B. Apparatus of Duffing Oscillator
1. Diagram of One-Dimensional Duffing Oscillator
Sine Wave Oscilloscope Generator
driver Strain Gauges (measure position & velocity on
oscilloscope)
reed
magnets
Figure 2: Diagram of 1-Dimensional Duffing Oscillator
2. Computing Programs
In the process of data accumulation and analysis, we used several computing products. In order to collect data from the oscillator, we used a personal computer fitted with a National Instruments PCI- 6035E DAQ and an analysis program using LabVIEW software, which accrued position and velocity data into a text file. We then, via a Microsoft Office Excel workbook (created by the Modern Physics Laboratory at CMU), analyzed subsets of the data (Excel could not handle the entire 1D oscillator data set at once). We felt, however, that it was necessary to analyze the whole data set at once, so we imported the data in Mathematica 5.1.1 and applied a box counting function.17 To keep our analysis consistent, we then reanalyzed all the subsets in Mathematica as well. Page 232 Das, Goyal, Manolache, Morris, Paul, Sampath, Rinehimer, Luokkala
C. Procedure
1. Initial Conditions/Presymmetrization
The apparatus was stored with the magnets centered under the rod and maximally separated to prevent magnetization of the rod. Once the apparatus was turned on, the magnets were moved to 1.905 cm of separation, a distance that the physics department historically found to be optimal for finding and sustaining chaos. Once this separation was established, we used the alignment knobs to visually adjust the magnets until the rod hung directly between them. After we established a visual approximation of symmetry in the system, we used the oscilloscope to more accurately center the magnets beneath the rod and symmetrize the potentials.
2. Symmetrizing the Potentials
Symmetrizing the potentials of the wells is an essential step for establishing enduring chaos within the confines of the Duffing Oscillator. If the rod is attracted to one particular well or another, then that well will serve as an attractor and draw the system out of chaos. The symmetrization process determines the resonant frequency of each well, or the frequency at which the decay frequency is equivalent to the oscillation frequency. Knowing the resonant frequencies of each well, it is possible to adjust the alignment of the magnets to equilibrate the resonant frequencies of the two wells.
To begin symmetrizing the potentials, we first increased the amplitude and frequency of the sine wave produced by the generator, creating a "bow tie" shape on the oscilloscope which indicated that the rod oscillated equally between the two wells. Establishing this visual approximation of symmetry between the wells sufficiently equalized the potentials, creating a system conducive to sustaining chaos.
3. Finding and Recording Chaos
Once symmetrization was complete, we proceeded to attempt to place the oscillator in chaotic motion. Beginning with the "bow tie" state, by adjusting amplitude and frequency pairings of the sine wave generator, we found sustainable motion above the wells. Then, by slowly adjusting one parameter or the other, normally the amplitude, the motion of the rod could be altered so that it skimmed just over the top of the wells and launched into chaotic motion. After the chaos was visually determined to be sustainable, we began recording data points into the LabVIEW program for analysis. The strain gauges recorded data points at each period of the oscillation, eliminating the time variable and making them ideal for graphing on a Poincaré plot. These data points could be plotted directly as displacement versus velocity and a Poincaré plot was obtained, showing us a representation of our fractal produced by the oscillator.
4. Process for Data Analysis Journal of the PGSS An Investigation of Chaos in One and Two Dimensions Page 233
The primary goal of our data analysis was to determine the fractal dimension for a data set. Initial data analysis involved splitting the data into appropriate pieces and importing it into a Microsoft Office Excel workbook designed by the Carnegie Mellon University Modern Physics Laboratory. Each sheet in the workbook corresponded to a portion of the data and displayed a Poincaré plot for each set. It also used a macro to perform box counting. Using the macro, we chose the box size r to start at a particular size and decrement in equal steps until reaching our own set minimum value. However, when this is graphed on the box-count plot, it is entered as log 1 , meaning the points r on our graph were not evenly spaced as they increased. We found that as the box size decreased, the line of points leveled off, and as the box size increased, the line of points became more lumpy and random. This is because as box size increases, is encompasses a greater number of points which will possess a smaller probability of being found in similar places in phase space and will appear more random. We therefore selected only the points that formed the most linear pattern and took the slope of the resulting best-fit line as the fractal dimension.
In order to analyze the entire data set however, we transitioned to Mathematica. In Mathematica, we imported the data and, using a box counting function, reanalyzed all the data. This new box counting algorithm differed from the Excel macro in that it did not decrement the box size by a constant, but rather divided it by a factor in each iteration. This presented an advantage over the Excel method in that the data points would be evenly spaced in the log-log plot. Using the function, we set an initial and minimum box size along with a value of r. We then took the antilog of the box size and the log of the number of points within the box. From this graph, we determined where the graph tended towards a horizontal asymptote, as opposed to the more linear section. The asymptote occurs because as the antilog of the size of the box increases (box size decreasing), each box will eventually contain at most one point. Therefore, the number of boxes containing a point will reach the number of points, where at that point the plot cannot increase any longer, leading to the asymptote. This problem can only be resolved by an infinite number of data points; thus using a box counting function will lead to the graph leveling off at some size of the box. To get a better approximation of the fractal dimension, we selected only the linear points and graphed the best-fit line. The slope of this resulting line is the fractal dimension.
Now that we had the fractal dimension of both pieces of the data set and the entire data set itself in Mathematica, we could compare these values, seeing if they fell within 1 standard deviation from each other. By doing so, we could prove the fractal properties of the data, as the subdivisions should be as self-similar as the entire set. We averaged the fractal dimension values of the pieces and then calculated the standard deviation of the data in order to prove this concept, thus concluding data analysis.
D. Data
1. Initial Chaotic Recording: 265,063 points
We found stable chaos in our Duffing Oscillator for the first time at an amplitude of .0466 volts and a period of 77545.0 microseconds for the driver. We collected 265,063 points representing the Page 234 Das, Goyal, Manolache, Morris, Paul, Sampath, Rinehimer, Luokkala reed’s path through its phase space, which we graphed as a Poincaré plot of displacement (volts) vs. velocity (volts) in Figure 3. The velocity scaling in volts is due to a differentiator circuit taking the time derivative of the voltage measurement of from the strain gauge and sending the velocity data to the oscilloscope in volts. We then divided the first 250,000 points into sets of 50,000, found the fractal dimension for each set in Excel, averaged the fractals together, and compared this value to the value of the fractal dimension for all the points analyzed at once, as in Table 1. By showing that the average value fell within one standard deviation of the overall value, we showed that the fractal was the same across all the points we collected, and also was a reliable measurement.
Figure 3: Poincaré Plot 265,063 points
Data Set Fractal Dimension
0-50,000 1.78451
50,000-100,000 1.77335
100,000-150,000 1.80454
150,000-200,000 1.77581
200,000-250,000 1.769
Average 1.78144
All 1.772± 0.026
Table 1: Comparing Fractal Dimension from a 265,063 point set Journal of the PGSS An Investigation of Chaos in One and Two Dimensions Page 235
Within each set of 50,000 and for the set of all the points, we used the box-counting method to find the fractal dimension. At first, using the Excel program, we chose the box size r to start at 1.3 units and decrease by .01 units until r reached .01. We selected only 45 points that formed the most linear pattern and took the slope of the resulting best-fit line as the fractal dimension. However, in order to get a more accurate representation of the fractal dimension, we knew the points on our box-count plot ought to be evenly spaced so as to avoid the disturbances on the extremes. We therefore used the program in Mathematica18 where we set r equal to 1/.9n where n was an integer running from 0 to 65. Again, we broke the data into sets of 50,000, averaged the fractal dimension, and compared it to the overall dimension value. The linear graph of this data, which plots log of box size versus the log of the number of boxes, confirms and provides an alternate display of the fractal dimension of this chaotic system in Figure 5. In this figure, the dashed line is the actual graph of the data and the solid line is the best fit line whose slope is the fractal dimension.
Figure 5: Logarithmic Graph of Fractal Dimension 265,063 points
2. Secondary Chaotic Recording: 40,892 points
Using the same Duffing Oscillator on a different occasion, we were able to record another instance of chaotic motion, this time at an amplitude of .0600 Volts and a period of 72513.0 microseconds. We recorded 40,892 data points in approximately fifty minutes before the system fell into periodic motion. We divided this data up into sets of 10,000 points for comparison and calculated the fractal dimension for each set as indicated in Figure 5. Then we determined the mean fractal dimension Page 236 Das, Goyal, Manolache, Morris, Paul, Sampath, Rinehimer, Luokkala and determined deviation for each set. Finally, we combined all of the data and comparatively displayed the fractal dimension in Table 2.
Data Set Fractal Dimension
0-10,000 1.70749
10,000-20,000 1.70749
20,000-30,000 1.70543
30,000-40,000 1.70543
Average 1.70646
All 1.761 ± 0.002
Table 2: Fractal Dimension comparison from 40,892 point set
Figure 7: Logarithmic graph of Fractal Dimension from 40,892 point set
E. Apparatus of Two Dimensional Oscillator
Our initial plans for constructing a two dimensional chaotic oscillator were far from flawless and required much alteration to accomplish their ultimate goal. The apparatus we initially envisioned utilized two cubic neodymium magnets as attractors and a copper rod connected to a metallic sphere as a pendulum. The components were extremely adjustable to ensure quick and precise changes were possible that could better accommodate chaos. We were able to alter the height of Journal of the PGSS An Investigation of Chaos in One and Two Dimensions Page 237 the apparatus by sliding one of two component boxes up or down and fitting them together with bolts along predrilled holes. Both drivers were held on stands and able to be moved up or down to match the height of the apparatus. Finally, and possibly most importantly, the device allowed for adjustable distances between attractors through use of a system of rotatable plexiglass circles.
Although the adjustability of the system was very valuable to our experiment, since our hope of finding chaos rested mostly on trial and error, we were forced to sacrifice some of this adjustability for practicality in our initial setup. To simplify the system, we found that is was essential to build one box with an adjustable clip rather than two sliding boxes. After completing the initial setup, we encountered several other difficulties with our design. Among our greatest concerns was that the system seemed to be losing a large amount of energy through its loose connections, which were essentially only loops of copper wire attaching together various parts of the pendulum and the drivers. The weight of the metallic sphere used in the pendulum also prevented the device from functioning efficiently, as did the excessive distance between the magnets used as attractors. These factors, along with the dissipation of the driver energy in the device’s connections made chaos all the more elusive.
We replaced the metallic sphere at the base of the oscillating rod with a smaller metallic nut, which yielded a little more motion, but still not enough to allow much interaction with the magnets. In addition, in order to redress the energy dissipation, we duct taped several of the joints together, which again yielded little improvement. We also eliminated the circular magnet positioning system, replacing it with an array of 4 cube magnets in a diamond position. However, the motion was still far too small to observe.
We then tried a completely different approach, using string to minimize energy dissipation. However, this failed to substantially improve results. We realized that energy dissipation and the complexity of the apparatus was far too high to provide sustainable chaos. Using alligator clips instead of masses of duct tape, the copper rods held more tightly to the mechanical vibrator as well as the oscillating rod. The clips attached to the oscillating rod as high as possible as it provided the best movement. We changed the magnet configuration as well, removing the nut attached to the oscillating rod to a clip with one of the cube neodymium magnets attached. On the base are four weaker magnets attached to a metal plate with the same polarity arranged in a diamond pattern. The cube magnet was attracted to these base magnets, permitting chaotic motion as the cube could be tugged by each of the magnets.
Through the new design without the magnets at the base, we had consistent elliptical motion. However, as the motion was still small, we increased the amplitude by doubling the volts from peak to peak in the function generator through High Z load impedance. We now had a radius in excess of 1 centimeter. We then inserted the magnet plate, and after rotating it several times, we had sustainable chaos that we videotaped, though, due to time constraints, analysis was not possible. A representation of our two-dimensional oscillator can be seen in Figure 8. Page 238 Das, Goyal, Manolache, Morris, Paul, Sampath, Rinehimer, Luokkala
Alligator clamps
Figure 8: Our Two Dimensional Oscillator IV. Conclusion
A. One-Dimensional Duffing Oscillator
Between the two chaotic situations that we recorded with our one-dimensional Duffing Oscillator, we were able to establish a fractal dimension in both instances that was approximately 1.8. This dimension lies nearer to randomness than periodic motion. It is interesting to note that even though we started with distinctly different conditions in numerous respects -- including amplitude, frequency, and magnet arrangement -- in both trials, we found very similar fractal dimensions, each within two hundredths of the other. The fractal dimension for our set of 265,063 points was 1.8 ± .026 and the fractal dimension for our set of 40,892 was 1.8 ± .002 This similarity may be attributed to the fact that both systems existed within the confines of our Duffing Oscillator which, while it contains adjustable variables, will nevertheless tend towards certain attractors and thus will be conducive to a certain fractal dimension, apparently somewhere within the realm of 1.8.
B. Two Dimensional Self-Built Oscillator
From our qualitative observations of two-dimensional chaos, a number of conclusions can be drawn. Perhaps the primary, and most definitely the obvious, observation is the realization that controlling chaos is just as difficult as removing its influence completely. Fortunately, after a Journal of the PGSS An Investigation of Chaos in One and Two Dimensions Page 239 significant amount of tinkering, we were able to achieve sustainable chaos. It is quite easy, if the magnets are positioned correctly, to achieve temporary chaos; however, this motion is quickly dampened. The difficulty in controlling chaos comes with driving it in a manner such that it is free enough to move chaotically but controlled enough to be driven.
The other significant difficulty in controlling chaos involves the number of variables involved with a two-dimensional oscillator. A one-dimensional Duffing Oscillator has a limited number of variables: frequency, amplitude (controlled by gain), the position of both magnets relative to the reed, and the position of both magnets relative to each other. The two-dimensional oscillator, however, introduces a number of new variables including the relative position of each (of four) magnet to the other, the angular orientation of the magnets relative to the frame, the relative amplitudes and frequencies of the perpendicular drivers, and the phases of each as well.
V. References
1 James P. Crutchfield, J. Doyne Farmer, Norman H. Packard and Robert S. Shaw, “Chaos,” Nonlinear Physics for Beginners, ed. Lui Lam, (World Scientific Co., River Edge, NJ, 1998), 93.
2 James P. Crutchfield, J. Doyne Farmer, Norman H. Packard and Robert S. Shaw, “Chaos,” Nonlinear Physics for Beginners, ed. Lui Lam, (World Scientific Co., River Edge, NJ, 1998), 94.
3 James Gleick, Chaos: Making a New Science, (Penguin Books, New York, NY, 1987), 17.
4 James Gleick, Chaos: Making a New Science, (Penguin Books, New York, NY, 1987), 114.
5 Nonlinear Physics for Beginners, ed. Lui Lam, (World Scientific Co., River Edge, NJ, 1998), 6.
6 James Gleick, Chaos: Making a New Science, (Penguin Books, New York, NY, 1987), 50.
7 James Gleick, Chaos: Making a New Science, (Penguin Books, New York, NY, 1987), 142-143.
8 Nonlinear Physics for Beginners, ed. Lui Lam, (World Scientific Co., River Edge, NJ, 1998), 11.
9 Jean-Francois Gouyet, Michel Rosso, and Bernard Sapovel, “Fractal Surfaces and Interfaces,” Fractals and Disordered Systems, 2nd ed., ed. Armin Bunde and Shlomo Havlin, (Springer-Verlag, New York, NY, 1996), 23.
10 H. Eugene Stanley, “Fractals and Multifractals: The Interplay of Physics and Geometry,” Fractals and Disordered Systems, 2nd ed., ed. Armin Bunde and Shlomo Havlin, (Springer-Verlag, New York, NY, 1996), 37.
Page 240 Das, Goyal, Manolache, Morris, Paul, Sampath, Rinehimer, Luokkala
11 Garnett P. Williams, Chaos Theory Tamed, (Joseph Henry Press, Washington, D.C., 1997), 304.
12 The box-counting dimension is also known as Kolomogorov entropy, entropy dimension, capacity dimension, metric dimension, logarithmic density and information dimension. Information regarding this and other methods of calculating the box-counting dimension can be found in Kenneth Falconer, Fractal Geometry: Mathematical Foundations and Applications, 2nd ed., (Chichester, West Sussex, England: John Wiley and Sons, 2003).
13 Kenneth Falconer, Fractal Geometry: Mathematical Foundations and Applications, 2nd ed., (Chichester, West Sussex, England: John Wiley and Sons, 2003), 42.
14 Werner Ebeling and Igor M. Sokolov, Statistical Thermodynamics and Stochastic Theory of Nonequilibrium Systems, Series on Advances in Statistical Mechanics, Vol. 8, http://www.worldscibooks.com/phy_etextbook/2012/2012_chap01.pdf.
15 Wolfram Research, 1999. http://mathworld.wolfram.com/DuffingDifferentialEquation.html.
16 J.E. Bergera and G. Nunes Jr., “A Mechanical Duffing Oscillator for the Undergraduate Laboratory”, American Association of Physics Teachers, (Department of Physics and Astronomy, Dartmouth College, 1997), 841-846.
17 Compliments of Pasquale Nardone, Universiti Libre de Bruxelles
18 Pasquale Nardone, MathGroup Archive 1995, December 9, 1995, http://forums.wolfram.com/mathgroup/archive/1995/Dec/msg00401.html.
Journal of the PGSS A Verification of Wave-Particle Duality in Light Page 241
A Verification of Wave-Particle Duality in Light
Peter Alexakos, Todd Harder, Matthew Kapelewski, Matthew Long, Silvia Manolache, Cassie Miller, Omar Rahman, Nishant Rastogi, Ashley Reichardt, and Nick Rhinehart
Abstract
This paper discusses the dual wave-particle nature of light. We have examined the history of the study of light from the Ancient Greeks, through the Arabic World, past the queries of Sir Isaac Newton, into Maxwell and Hertz’s experiments, ending in Niels Bohr’s complementarity. Several experiments discussing whether light is a wave or a particle are also contained in this paper, including Measurement of the Speed of Light, Young’s Double Slit Experiment, and the Photoelectric Effect. Our measurement for the speed of light was not within our calculated error, Young’s double slit experiment yielded the interference patterns that were expected, and the photoelectric showed a weak correlation to the predicted results.
I. INTRODUCTION:
The nature of light has been studied for centuries and has intrigued generations. It has played both a controversial and critical role in the advancement of modern physics and the understanding of the universe. However, one particular aspect of its nature has been the most widely debated and has arguably offered the most vital question in the field to date: does light act as a particle or as a wave? Today, it has been concluded that while light exhibits properties of both waves and particles, it never exhibits properties of both concepts at the same time.
A. Ancient History
The study of light can be traced back to the Ancient Greeks, in particular to Empedocles, who theorized that all matter was composed of four elements: fire, air, earth, and water. His theory provided the basis for the theory of sight which said that a fire within the eye sends out a beam that provides sight to people. In the year 300 BCE, a Greek mathematician named Euclid made a mathematical study of light, which he titled Optica, the first in-depth analysis of light and its properties. Euclid said that light travels in straight lines and further theorized about how the Page 242 Alexakos, Harder, Kapelewski, Long, Manolache, Miller, Rahman, Rastogi, Reichardt, Rhinehart eye receives light. He concluded that Empedocles’ theory of sight was inaccurate unless the “beam” from the eye traveled infinitely fast. Around the year 55 BCE, a Greek physicist named Lucretius developed another theory of light which stated that light and heat from the sun are composed of minute atoms that shoot around the atmosphere.1
Around 1000 CE, Islamic mathematician al-Haytham proposed that light was merely an outside force that entered the eye. He noticed that when light passed through a pinhole onto a wall, the image was inverted. This led him to believe that light was made of tiny particles moving about in straight lines, enabling people to see. He was also one of the first people ever to experiment with the separation of white light into various colors, using light phenomena such as rainbows and shadows. Furthermore, Al-Haytham theorized about the speed of light, saying that although light travels at very great velocities, its speed must be a finite value, contrary to Lucretius’ theory. These theories were major advances toward a more modern theory of light. Unfortunately, Europe and the rest of the Western World remained oblivious to al-Haytham’s findings until they conducted research of their own starting in the late 1500s.1
B. Modern History
In the late 17th century in England, Newton became the first physicist to convince the world of his findings due to his new level of experimentation and greatly respected reputation. During this time, Newton dedicated himself to uncovering the mysteries of light.
In 1672 Newton had published his first scientific paper, but his claim that light was composed of different particles was immediately rejected by scientists Robert Hooke and Christian Huygens, who argued that light was a wave.1 Fortunately for Newton, his book, his reputation and the peoples’ confidence in his scientific method made Particle Theory of Light the accepted theory in society.
Newton’s first published book on light came in 1704. The book, known as Opticks, was composed of four different volumes and more than 30 different queries about light. It was at this time when Newton’s Particle Theory, or Corpuscular Theory, became widespread.2
The text included, not only guesses at light’s properties, but also descriptions of light particles, information on the reflection of light, the color of light, refraction experiments, and the idea of an ether force. In explaining light, Newton characterized it as consisting of tiny particles, or corpuscles. Journal of the PGSS A Verification of Wave-Particle Duality in Light Page 243
His belief in this idea was strengthened by two fairly simple experiments he carried out. First, Newton noticed that opaque objects when hit by light would create a shadow. If light were a wave, such as sound, the waves would diffract around the object and not be hindered. Since light was a particle, an opaque object affected it more, concluded Newton. Secondly, Newton performed an experiment in which he detected that light could move in a vacuum. Since particles were expected to do be able to travel through a vacuum, but waves weren’t (at this time), this enhanced his hypothesis. With these two basic observations he formed the beginning of his Corpuscular Theory.2
Next, Newton conducted experiments to understand the refraction and reflections properties of light. To explain refraction, Newton compared the result of light hitting water with that of cannon balls hitting water.
Figure 1: Newton’s Explanation of Refraction
He knew that cannon balls hitting water slowed down and bent away from the normal line. Light, on the other hand, bent toward the normal (See Figure 1). To explain this, Newton stated that there was a net pull near the surface of the water that bends the light particles towards the normal vertical component. Newton used a similar theory to explain reflection as well. He believed reflection occurred because of a reflecting force which pushed the light particles away.3 His observations and conclusions are best stated in his Query 29:
“Nothing more is requisite for putting the Rays of Light into Fits of easy Reflexion and easy Transmission, than that they be small Bodies which by their attractive Powers, or some other Force, stir up Vibrations in what they act upon…”2
Perhaps the most significant discovery made by Newton about light came from his prism experiment. Before Newton, western philosophers had believed white light was a singular, Page 244 Alexakos, Harder, Kapelewski, Long, Manolache, Miller, Rahman, Rastogi, Reichardt, Rhinehart simple, homogenous entity; however, when Newton passed light through a glass prism he discovered the visible spectrum. Newton concluded that white light is actually composed of different types of rays and the colors come because these rays are refracted at different angles.3 Again in his Query 29 of Opticks he writes:
The least of which may take violet the weakest and darkest of the Colours, and be more easily diverted by refracting Surfaces from the right Course; and the rest as they are bigger and bigger, may make the stronger and more lucid colours, blue, green, yellow, and red, and be more and more difficultly diverted."2
This theory explained some properties of light but failed to explain light diffraction and interference, which can only be explained if light acts as a wave. Interference is the interaction of waves that have similar frequencies. Interference had been detected in light and could not be described by particles traveling in a line. Diffraction cannot be explained by Corpuscular Theory either, even though Newton inadvertently studied it with his experiments concerning prisms. Since, light exhibits properties of waves, it cannot be accurately described by Corpuscular Theory alone.
C. Young’s Double Slit Experiment
1. History of the Experiment
One of the most important experiments in optics during the nineteenth century proved to be Young’s Double Slit Experiment. Since the public considered light as a particle, this experiment conflicted with general thinking. It brought the Wave Theory of Light back into the mainstream of science.
Young conducted the original double slit experiment in the early 1800s in an attempt to conclude the debate. Young passed a beam of light through two parallel slits in an opaque screen. If light acted as a particle, after passing through the slits, only a single spot would be seen. However, the double slit experiment showed an interference pattern after the beam was shot through the screen, meaning light had to act like a wave.
2. Theory of the Experiment
Wave interference involves the interaction of coherent waves according to the Principle of Linear Superposition. Coherence is essential for interference to occur, meaning all of the waves must be both of equal wavelength and of the same initial phase.4 If this is achieved, the addition of two or more waves results in a new wave pattern with an amplitude that depends upon the phase difference between the two added waves. If the two added waves are in phase, with the Journal of the PGSS A Verification of Wave-Particle Duality in Light Page 245 troughs and peaks of each lined up, the interference is constructive. This means that the resultant wave will have a higher amplitude, which corresponds to a higher intensity.5 In contrast, destructive interference occurs when the two added waves are out of phase, or the troughs and peaks do not line up. In this case, the resultant wave has a smaller amplitude, and thus a lower intensity.
Interference of waves can be demonstrated with Young’s double-slit experiment. This experiment involves a coherent light source that diffracts though two narrow slits to produce an interference pattern on a distant screen. Both types of interference, constructive and destructive, occur to create a pattern of alternating dark and bright fringes on the screen (See Figure 2). The experimental interference pattern is quantified with intensity measurements made by a photodetector as a function of position. A central bright fringe is obtained, and other high intensity fringes occur at certain distances from the center where light waves from both slits arrive in phase.
Figure 2: Young’s Classic Double Slit Experiment Set-Up
Such bright fringes require that the path difference between the two rays is a multiple, n, of the wavelength of the light. Therefore, the locations of the maxima are given by the equation:
nλ x = d L Page 246 Alexakos, Harder, Kapelewski, Long, Manolache, Miller, Rahman, Rastogi, Reichardt, Rhinehart
Figure 3: Equation for Location of Maxima in Double Slit Experiment where L is the distance from the slits to the detector or screen, d is the separation of the slits, and x is the distance between the bright fringe and the central maximum.
Several types of photodetectors can be used in the double-slit experiment in place of the screen to quantitatively observe the interference pattern. The measurements can be made using a photodiode, a semiconductor device that generates current when light excites electrons from the valence band to the conduction band. The resulting photocurrent is then displayed on an oscilloscope. It is important to note that for a photodiode detector, there is one-to-one ratio of the number of photons incident on the screen and the number of photoelectrons generated within the photodiode.6 The charge flow of these electrons is conserved throughout the circuit and determines the current measured through the detector.
Measurements can also be made to witness the quantum theory of light using a photomultiplier tube. The detector relies on the photoelectric effect, which relies on the particle nature of light. In the photocathode of the detector, a light beam interacts with atoms and liberates individual electrons. These single photoelectrons then trigger a cascade effect that releases many more electrons within the multiplier region of the tube. Ultimately, enough electrons are released to generate a current pulse that is large enough to be detected with an electronic counter. Each pulse counted corresponds to the release of individual electrons from the photocathode and to a single photocount.7 A photocount rate is obtained by recording counts over set time intervals. Despite the difference in treating light as a particle as opposed to a wave, both types of detectors, the photodiode and photomultiplier, produce equivalent results in double- slit experimentation.
D. Electromagnetism
1. James Maxwell
Maxwell is most widely known for his expansion and mathematical formulation of Faraday’s electricity and magnetic lines of force theories. Maxwell was responsible for making it known that a few simple equations could be used to convey the behavior of electric and magnetic fields.
Maxwell’s equations show how electric charges produce electric fields. Furthermore, they show how magnetic fields are produced by electric currents and changing electron fields. Lastly, these useful equations show that electric fields are produced by changing magnetic fields.8 Journal of the PGSS A Verification of Wave-Particle Duality in Light Page 247
Figure 4: Maxwell’s Equations
These four equations must be looked at more closely in order to understand the applications.
Through some mathematical manipulation, Maxwell’s equations predicted that electric and magnetic fields act as waves that move at approximately 300,000,000 meters per second. This gave rise to the fact that light waves are nothing but oscillating electric and magnetic fields. Consequently, Maxwell’s work and equations predicted the speed of light through his experimentation.9
Maxwell explains the application of his mathematical equations in more depth in his book A Dynamic Theory of the Electromagnetic Wave. Maxwell states, “The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic waves.”9 Thus, it is possible to show that the electromagnetic force moves at the speed of light in a vacuum giving credence to light being an electromagnetic wave.
2. Hertz
Heinrich Hertz proved Maxwell’s theory of electromagnetism in 1886 by conducting an experiment that detected light moving as an electromagnetic wave. Page 248 Alexakos, Harder, Kapelewski, Long, Manolache, Miller, Rahman, Rastogi, Reichardt, Rhinehart
Figure 5: Hertz’s Apparatus
In this experiment, Hertz set up a circuit (Figure 8) consisting of a high-voltage conducting coil, a capacitor, and a gap between two poles that allowed for a spark to jump in between and emit electromagnetic radiation. One end of the wire was pointed and the other had a small brass ball. The distance between the ball and the pointed end was adjustable although usually only by micrometers. When he ran the experiment, a spark appeared between the two ends of the copper circle because the radiation jumped from the initial circuit to the second and caused a spark, indicating that light had electromagnetic properties.10
E. The Photoelectric Effect
The photoelectric effect is a phenomenon which occurs when electrons are emitted from a mass after a large amount of energy is absorbed in the form of electromagnetic rays. Its discovery was first published in 1886 by Heinrich Hertz in the journal Annalen der Physik.11 Hertz came upon the discovery while performing experiments to confirm the existence of electromagnetic waves, which had been proven mathematically by James Clerk Maxwell in 1864, and later published in his book Untersuchungen Ueber Die Ausbreitung Der Elektrischen Kraft, Investigations on the Propagation of Electrical Energy.12 During Hertz’s experiment, he discovered the spark was more powerful when exposed to ultraviolet light, a light of a higher frequency than visible light, which had enough energy to alter the results of the experiment. He had observed the photoelectric effect
1. Lenard’s Experiment
In 1902, Lenard startled the scientific community with his discoveries about the relationships between the intensity, frequency, and energy of light. His apparatus involved using Journal of the PGSS A Verification of Wave-Particle Duality in Light Page 249 a carbon arc light that would radiate light on a piece of metal. The radiated light would cause electrons to eject onto a receiving plate, which would send the collected electrons through an ammeter for the purpose of measuring the current produced by the photoelectrons. Lenard calculated the energy with which the photoelectrons were colliding with the collecting plate by giving the collecting plate a negative charge. This prevented any photoelectrons without enough energy to break through the electromagnetic field from adding to the current. Lenard found that there was a minimum value of voltage at which no photoelectrons would get by the magnetic field. Following this, he discovered that as intensity increased the number of electrons ejected increased proportionally instead of the energy of the electrons increasing. This was revolutionary because most physicists theorized that the energy would proportionally increase with the intensity of light. Also, he found that although a stronger oscillating field produced more photoelectrons, it interestingly resulted in the same maximum kinetic energy as that of the weaker fields. Lenard further noted that when the intense light from the carbon arc light was split into different colors there was a positive correlation between the frequency and the number of photoelectrons produced. Unfortunately, Lenard did not have an explanation for this phenomenon. Einstein did.13
2. Einstein’s Theory
In 1905, Albert Einstein, explained his theory of how the effect was formed in his Nobel Prize winning paper, On a Heuristic Viewpoint Concerning the Production and Transformation of Light.13 Although the theory that radiation existed as “quantum” had existed since Max Planck had introduced it in 1901, no real proof of the truth of this theory had been previously uncovered before Einstein’s paper.13
In the effect, light was shined on a specific metal in an attempt to produce an electric current, kicking electrons out of their atomic orbits. According to wave theory, the intensity of the light should have correlated to the amount of current being produced; however, only certain colors of light produced any effect at all. Although an extremely bright, intense red produced no reaction from the metal, dim blue light kicked off electrons, and an electric current was formed. The data concluded that there was a correlation between the kinetic energy of the ejected electrons and the frequency of the light hitting them. It seemed there was a frequency threshold that produced current. Once that threshold was reached the light followed the principle that was assumed, the more intense the light, the higher the current. However, if the frequency was below that threshold frequency, the intensity did not matter; a current would never be produced. By theorizing that only certain amounts of energy could knock an electron out of orbit, the effect could be explained. Electrons left the surface of the metal in less than 1 * 10-9 seconds after a photon was shot, proving that light could behave as a particle.13 In order for light to behave in the Page 250 Alexakos, Harder, Kapelewski, Long, Manolache, Miller, Rahman, Rastogi, Reichardt, Rhinehart way it did, it would need to behave as a packet of energy, or a quantum. It could not have behaved in the way in which it did if it were simply a wave and had no particle properties.
Figure 6: Diagram of Photoelectric Effect
The frequency of light was directly related to the energy emitted by the equation
Figure 7: Photoelectric Effect Relation
E is energy, f is frequency of the light, and h is Planck’s constant. Planck’s constant is a specific figure that describes sizes of the specific quanta of energy
His findings, like Planck’s, were not immediately accepted. They went against far older theories, particularly Maxwell’s theory of electromagnetic waves, which had been widely accepted.
F. Complimentarity
Since, light was shown to exhibit both wave and particle properties, it was proposed that light was both. Niels Bohr, however, developed a theory to explain why light cannot be measured as a particle and wave at the same time. His final analysis led to the development of his theory of complementarity which explains this phenomenon. Bohrs’ agreed with the wave-particle duality theory, but physics at the time was unable to account for this impossibility. Bohr first suggested that space and time cannot exist on a practical level with regards to light. He went on to clarify that there is a complementary relation between space time and causality, meaning that one could not look at space time coordinates of light and the results of the light at the same time. Therefore, the basic principle presented by Bohr is the fact that while light can exhibit properties Journal of the PGSS A Verification of Wave-Particle Duality in Light Page 251 of both a wave and a particle, it is impossible to measure both its location and effect at the same time and therefore it is impossible to prove that it is both a light and a wave at the same time.14
Bohr’s theory is effective because instead of trying to explain the wave-particle duality of light with what were considered traditional physical ideas, Bohr created a new idea suggesting that distinct physics ideas can coexist in one form or another if one accepts that both ideas cannot be measured at the same instant. While Bohr’s theory of complementarity is widely accepted by physicists today, it is still certainly not the only theory that is available. Some scientists have claimed to have measured both properties simultaneously, but their claims have been highly contested and are not widely accepted. It is fairly certain that the duality of light paradox will never be explained in terms of just one of the two. Therefore, one must either accept the mutual exclusivity of space time and causality with respect to light or devise a new theory that can somehow combine these two distinct ideas in physics.
II. EXPERIMENTS
A. Measuring the Speed of Light.
This experiment consisted of measuring polyethylene BNC cables length and, based on the distance between two oscilloscope peaks, measuring the time it took the light to travel the length of the connected cables. A pulse of light was sent through cable and then returned to the oscilloscope. We began by attaching one cable to the oscilloscope, and used the formula v=d/t to find the speed of light.
Figure 8: Apparatus to Determine the Speed of Light
The distance was double the total length of the attached cables because the light had to travel the length of the cable twice, once going away from the oscilloscope and once returning to it. We Page 252 Alexakos, Harder, Kapelewski, Long, Manolache, Miller, Rahman, Rastogi, Reichardt, Rhinehart could use these two numbers to generate a linear graph to determine the slope for the average rate for the experimental speed of light.
B. Young Double Slit (Photodiode, Counter)
The apparatus used for testing the interference pattern involved aligning a laser, a coated glass plate with slits in the coating, and a photon counter that could be adjusted perpendicularly to the alignment of the apparatus. For the first test, a red class II laser was used; the second test used a green class III laser. Once the apparatus was set up, the laser and the photon counter were turned on while the lights were turned off. Furthermore, the area in which the tests were performed was entirely enclosed by black curtains to prevent outside interference.
Figure 9: Young’s Double Slit with Red Laser
Figure 10: Young’s Double Slit with Green Laser
The diffraction pattern was blocked from the photon counter using an experimenter’s hand while the photon counter was reset to zero and the proper measuring point was set using the position adjusting knob. The experimenter then moved his hand simultaneously starting a stopwatch. After thirty seconds elapsed, the experimenter again blocked the laser with his hand, ceasing the count. The number of photons detected was recorded. Finally, the apparatus was moved by 0.1 mm in accordance with the progression of data collection. This was repeated until Journal of the PGSS A Verification of Wave-Particle Duality in Light Page 253 the full 25 mm range was tested. In some cases, the experiment was ended when sufficient data was collected and the photon counter could no longer detect any photons.
C. Photoelectric Effect (Gels, Mercury Spectrum)
1. Gels
In order to determine the energy of the emitted electrons at various frequencies of light, an appropriate apparatus was set up. First a light source was used and covered so that light would pass through a narrower opening. Our first measurement was that of just white light without any gels. Next, a “flame red” colored gel was tested because red light should block all electrons from being emitted. After doing these two measurements, other colored gels were used to find a correlation between frequency and current. The data was read by a photodiode which measures current in amps and this indirectly told us the electrons’ kinetic energy.
Figure 11: Photoelectric Effect Apparatus (Gels)
The higher the kinetic energy, the higher the current will be. Twenty-one gels were used and the entire visible spectrum was represented.
2. Mercury
A mercury lamp was used as the light source. The light passed through a spectrometer, and as a result emission lines were seen. The emission lines of mercury in the visible spectrum are at wavelengths [690.75 nm(red), 576.959, 579.065 nm(yellow), 546.074 nm (green), 407.781nm, 404.656nm (blue)]. The spectrometer was then adjusted to focus specifically on one of those emission lines while the current was read on an electrometer. To determine the stopping potential, a potentiometer was used to increase the negative charge in the loop where the electrons were collected. This potential was then read and graphed versus frequency.
Page 254 Alexakos, Harder, Kapelewski, Long, Manolache, Miller, Rahman, Rastogi, Reichardt, Rhinehart
Figure 12: Apparatus for Photoelectric Effect with Mercury
III. Discussion & Error Analysis
A. Measuring the Speed of Light
1. Discussion
Using an Oscilloscope, measurements were obtained for the time it took light to travel through a wire of some known length. We were then able to graph Time (m) vs. Distance light travels (s). This graph produced a straight line and the slope of that line is equivalent to the speed of light in polyethylene, the insulator of the cable. This variable, called n in the experiment, could then be used to find the speed of light in a vacuum by multiplying the speed of light in polyethylene by the index of refraction of polyethylene, which we obtained through research. In this experiment, the speed of light in polyethylene was found to be 1.91 x 108 meters per second. When these two values are multiplied together, one more factor must be taken into account, the error must also be included. A Least Squares Fit found the error of the speed of light in polyethylene was found to be 3.80 x 105 m/s. This value must also be multiplied by the index of refraction to obtain the speed of light in a vacuum. Then this new product must be subtracted from the value that has already been obtained. Thus, our experimental value for the speed of light in a vacuum is 2.92 x 108 meters per second.
2. Error Analysis
There are many possible ways in which data collected in this experiment could have been skewed. The primary source error is in the human measuring of the cable length. A tape Journal of the PGSS A Verification of Wave-Particle Duality in Light Page 255 measure was used to measure the length of the cable, however, it was much too short and had to be moved along the cable to complete the measurement. This accounted for substantial error since the tape measure couldn’t be placed perfectly at the exact measurement each time. Furthermore, the cables connect together, causing overlap and complicating the measurement process. Reading the data on the oscilloscope was also a possible source of error, since the peaks were chosen by a person, not the oscilloscope itself.
B. Double Slit Experiment
1. Discussion
In examining Young’s double-slit experiment, it was found that the data correlated with the predicted interference patterns. Each trial of the experiment produced a graph of intensity versus position that consists of a pattern of alternating high intensity and low intensity fringes (See Appendices A through E). Since this double-slit interference pattern is a wave property, the results confirm the wave nature of light.
According to calculations made with Young’s formula, the observed positions of the maxima tend to fall within the range of error. A basic error analysis was done to determine the uncertainty made in the measurement of the distance between maxima for each experimental trial (See Appendix F). Unfortunately, a solid confirmation of Young’s formula based on error analysis is not possible due to experimental limitations. The sensitivity of the photodetectors used in recording the intensity of the light was only sufficient enough to detect three to five of the brightest maxima. Since individual error calculations can only be carried out for each maximum, there was a lack of data to analyze effectively using least squares regression.
In order to observe the experiment from all angles, a certain aspect was varied in each trial. These variables include the separation of the slits, the wavelength of the laser, the distance from the slits to the detector, and the type of photodetector. Overall, three different types of detection methods were used: manual measurement, a photodiode, and a photomultiplier tube. Within the first set of experiments, performed by manual measurement, the distance from the slits to the detector and the slit size were varied over several trials. Other than making the manual measurement more error-prone, this variable had no effect on the results (See Appendix A).
The next series of experiments were performed using a photodiode to measure the intensity of the light in terms of voltage. The use of the photodiode was more accurate than manual measurements, as it was used to make voltage measurements over small distance intervals of half-millimeters, as seen in the results in Appendices B and C. The photodiode trials Page 256 Alexakos, Harder, Kapelewski, Long, Manolache, Miller, Rahman, Rastogi, Reichardt, Rhinehart included variation of the separation of the slits. The results of these trials show a relationship in accordance with Young’s formula: the width of the slits is inversely proportional to the distance between the fringes of the interference pattern.
The final set of double-slit experiment trials utilized a photomultiplier tube to record intensity in terms of individual photons. Despite the fact that the detector relies on the particle nature of light for the photoelectric effect, the results from the device still produced an interference pattern analogous with the wave picture. The photomultiplier was used for two separate trials in which the wavelength of the light source was altered. Based on the results, it can be concluded that the wavelength of the light is directly proportional to the distance between the fringes of the interference pattern. When comparing the accuracy of the photomultiplier to that of the photodiode, it can be noted that the photomultiplier relies more on human limitations. While the photodiode provides instantaneous values, the photomultiplier tube requires recording accumulated photocounts over a set time interval, making it more prone to human error. In addition, the results show that the photodiode was much more sensitive than the photomultiplier (See Appendices D and E).
2. Error Analysis
There are many ways in which a double slit experiment can provide skewed results. Even the slightest bumping or moving of any part of the apparatus can result in all data to that point becoming null and void. The only solution to this problem is redoing all data collection up to and including that point, because all data is irrelevant due to the fact that it won’t correlate with any data collected after the moving of the apparatus. Although this problem is easily corrected just by spending large amounts of time redoing all previous data collection, it is very time consuming and most likely the worst problem one will face with this experiment. However, there are various other problems that may distort results. Inexact measurements, such as the distance from the laser to the slits can also affect the way in which data is analyzed. a. Photomultiplier
Photomultipliers are very sensitive instruments and could be easily disrupted by a number of things. Light entering the photon counter from outside could change the data significantly, especially if the light was sustained. Several other errors that could cause unusual data include improperly or inexactly timing, not having the laser light, the slits, and the photon counter exactly perpendicular to each other, or failure on the part of the experimenter who blocks the laser to completely block the light or block it with the right timing. All of these possibilities could contribute to errors in an experiment involving the photon counter. Journal of the PGSS A Verification of Wave-Particle Duality in Light Page 257 b. Photodiode
Photodiodes are devices that produce current in response to light. More specifically, the photons from the incident light excite the electrons of the receptor, which generates a current that can be measured. There are many important characteristics of photodiodes that influence performance. The accuracy of the device is measured in terms of quantum efficiency, or the percentage of incident photons that contribute to current produced. The responsivity is the output photocurrent divided by the input optical power. This is related to the quantum efficiency, which is the percentage of incident photons that contribute to current produced. The other characteristics include: the size of the active area, which is the light sensitive area, the maximum rate of photon detection, which is measured by photocurrent, the dark current, which is the photocurrent caused by background radiation, and the speed, which is the number of data points taken over a certain time period. In photodiodes the active area is usually very small and the responsivity is usually around 95%, but the other characteristics vary with the material used in the photodiode. Silicon diodes are known to have high speed, low dark current, and good sensitivity in the range of 400-1000 nm. Photodiodes composed of germanium have high dark current, low speed and good sensitivity in the range of 600-1800 nm. Indium gallium arsenide photodiodes are very sensitive in the range of 800-1700 nm. The photodiode we used was a silicon diode, because it can detect in the range of 400-1000 nm, which includes the visibile spectrum. They also have low dark current and a very high speed.15
C. Photoelectric Effect
1. Discussion
Light of different frequencies was made incident upon a potassium metal plate which produced photoelectrons. These photoelectrons included a current in a loop of wire inside the photodiode. This current was measured using an ammeter. The current for each tested frequency was recorded. These points were graphed in ordered pairs of the following format (current in amps, frequency in hertz). This produced a scatter plot when it should have been a straight line going through the x-axis due to the positive correlation between frequency and current (Appendix G). The reason the line would come through some x-intercept and not zero is due to the fact that photoelectrons are not emitted until the threshold frequency is reached. A correlation was not observed because the gels did not block all frequencies of light and therefore the light was not monochromatic. Monochromatic light needs to be constant so as to isolate the frequency that is to be tested instead of the unwanted light that the gels allowed to pass. This unwanted light brought extra energy and skewed the results. What makes it worse is that each gel let through a different amount of unwanted light so the skew was not constant and therefore Page 258 Alexakos, Harder, Kapelewski, Long, Manolache, Miller, Rahman, Rastogi, Reichardt, Rhinehart could not be entirely accounted for. The data for the mercury lamp experiment was far more accurate because the mercury spectrum was composed of monochromatic light emissions. The line of fit of the data had a positive slope, as we had expected the data from the photoelectric effect to have. Although, the first point, which represented a red emission line, did not have a stopping point of zero. In comparison to the first experiment, however, the data was far more accurate to the results expected.
2. Error Analysis
There are several ways in which the data and results of the photoelectric effect could be misrepresentative of the actual data. The most important part of this experiment is selecting a variety of gels that represent various wavelengths of light. The selection of gels is crucial to the success of the experiment because the data will not be correct if varied wavelengths are not being used. The variety of wavelengths should include all wavelengths with steep peaks. There are other possible sources of error, such as the equipment not being exact enough to provide accurate results. Other possible sources of error include bumping the apparatus, misreading the current, and the instability of the gel being held.
When this experiment was performed using a mercury bulb in place of the gels, different sources of error were possible. These sources of error included misjudgment in reading exactly where the emission lines were. The arrangement of the mirror and the slit may also cause problems because if the light is not lined up properly, the emission lines will not appear. Furthermore, inaccurate zeroing of the electrometer would result in a miscued reading of the voltmeter.
Journal of the PGSS A Verification of Wave-Particle Duality in Light Page 259
IV. Wrap-Up
Appendix A: Young’s Double Slit (Manual Measurements)
Experiment 1: Red Laser, Narrow Slits Wavelength Trial 1: L (cm): 267.5 ± 1.0 (nm): 632.8
n: 11 2 2 x (cm): 0.84 0.87 1.93 1.93 ± 0.1 d (mm): 0.201517 0.194568 0.175413472 0.175413472 Avg d (mm): 0.186728
Wavelength Trial 2: L (cm): 160 ± 1.0 (nm): 632.8
n: 11 x (cm): 0.63 0.65 ± 0.1 d (mm): 0.160711 0.155766 Avg d (mm): 0.158239
Wavelength Trial 3: L (cm): 300 ± 1.0 (nm): 632.8
n: 11 2 2 x (cm): 1.16 1.12 2.13 1.95 ± 0.1 d (mm): 0.163655 0.1695 0.178253521 0.194707692 Avg d (mm): 0.176529
Avg for Slit 1 (mm): 0.173832
Table 1: Young’s Double Slit (Red Laser-Narrow Slits)
Page 260 Alexakos, Harder, Kapelewski, Long, Manolache, Miller, Rahman, Rastogi, Reichardt, Rhinehart
Experiment 2: Red Laser, Wide Slits Wavelength Trial 1: L (cm): 300 ± 1.0 (nm): 632.8
n: 11 2 2 x (cm): 0.55 0.57 1.06 1.11 ± 0.1 d (mm): 0.345164 0.333053 0.358188679 0.342054054 Avg d (mm): 0.344615
Wavelength Trial 2: L (cm): 248 ± 1.0 (nm): 632.8
n: 11 2 2 x (cm): 0.44 0.5 0.86 0.89 ± 0.1 d (mm): 0.356669 0.313869 0.364963721 0.352661573 Avg d (mm): 0.347041
Wavelength Trial 3: L (cm): 172.5 ± 1.0 (nm): 632.8
n: 11 2 2 x (cm): 0.31 0.32 0.62 0.64 ± 0.1 d (mm): 0.352123 0.341119 0.352122581 0.34111875 Avg d (mm): 0.346621
Wavelength Trial 4: L (cm): 123 ± 1.0 (nm): 632.8
n: 11 2 2 x (cm): 0.23 0.22 0.45 0.45 ± 0.1 d (mm): 0.33841 0.353793 0.345930667 0.345930667 Avg d (mm): 0.346016
Avg for Slit 2 (mm): 0.346073
Table 2: Young’s Double Slit (Red Laser- Wide Slits)
Journal of the PGSS A Verification of Wave-Particle Duality in Light Page 261
Appendix B: Photodiode with Red Laser (Wide Slits)
Experiment 3 Trial 1: Red Laser, Photodiode, Wide Slits
Position (mm) Voltage (V) Position (mm) Voltage (V) 0.0 0.90 13.0 6.30 0.5 1.28 13.5 5.86 1.0 1.58 14.0 5.10 1.5 1.94 14.5 4.46 2.0 2.12 15.0 3.14 2.5 2.24 15.5 2.22 3.0 2.26 16.0 1.32 3.5 2.14 16.5 0.54 4.0 1.88 17.0 0.28 4.5 1.42 17.5 0.20 5.0 1.10 18.0 0.36 5.5 0.64 18.5 0.66 6.0 0.38 19.0 1.04 6.5 0.22 19.5 1.46 7.0 0.32 20.0 1.78 7.5 0.72 20.5 2.00 8.0 1.24 21.0 2.08 8.5 2.28 21.5 2.04 9.0 3.04 22.0 1.88 9.5 4.30 22.5 1.72 10.0 5.12 23.0 1.42 10.5 5.96 23.5 1.12 11.0 6.44 24.0 0.84 11.5 6.96 24.5 0.62 12.0 7.10 25.0 0.36 12.5 6.80
Table 3: Photodiode (Red Laser-Wide Slits) Trial 1 Page 262 Alexakos, Harder, Kapelewski, Long, Manolache, Miller, Rahman, Rastogi, Reichardt, Rhinehart
Experiment 3 Trial 1: Red Laser, Photodiode, Wide Slits Voltage vs. Position
5.00
4.50
4.00
3.50
3.00
2.50 Voltage (V)
2.00
1.50
1.00
0.50
0.00 0.0 5.0 10.0 15.0 20.0 25.0 30.0 Position (mm)
Figure 13: Photodiode (Red Laser-Wide Slits) Trial 1
Journal of the PGSS A Verification of Wave-Particle Duality in Light Page 263
Experiment 3 Trial 2: Red Laser, Photodiode, Wide Slits
Position Voltage Position Voltage (mm) (V) (mm) (V) 0.0 0.5 14.5 4.0 0.5 0.4 15.0 2.0 1.0 0.3 15.5 1.2 1.5 0.5 16.0 2.0 2.0 0.6 16.5 3.2 2.5 0.6 17.0 3.8 3.0 0.5 17.5 3.2 3.5 0.4 18.0 1.2 4.0 0.3 18.5 0.9 4.5 0.3 19.0 1.5 5.0 0.4 19.5 1.7 5.5 0.5 20.0 1.7 6.0 0.5 20.5 1.0 6.5 0.4 21.0 0.6 7.0 0.6 21.5 0.5 7.5 1.0 22.0 0.6 8.0 1.6 22.5 0.6 8.5 1.6 23.0 0.5 9.0 1.0 23.5 0.4 9.5 0.9 24.0 0.4 10.0 2.0 24.5 0.5 10.5 3.4 25.0 0.6 11.0 4.0 25.5 0.7 11.5 3.3 26.0 0.6 12.0 1.8 26.5 0.4 12.5 1.3 27.0 0.4 13.0 2.6 27.5 0.5 13.5 4.8 28.0 0.7 14.0 5.1
Table 4: Photodiode (Red Laser-Wide Slits) Trial 2
Page 264 Alexakos, Harder, Kapelewski, Long, Manolache, Miller, Rahman, Rastogi, Reichardt, Rhinehart
Experiment 3 Trial 2: Red Laser, Photodiode, Wide Slits Voltage vs. Position
6.0
5.0
4.0
3.0 Voltage (V)
2.0
1.0
0.0 0.0 5.0 10.0 15.0 20.0 25.0 30.0 Position (mm)
Figure 14: Photodiode (Red Laser-Wide Slits) Trial 2
Journal of the PGSS A Verification of Wave-Particle Duality in Light Page 265
Appendix C: Photodiode with Red Laser (Narrow Slits)
Experiment 4 Trial 1: Red Laser, Photodiode, Narrow Slits
Position Voltage Position Voltage (mm) (V) (mm) (V) 0.0 0.90 13.0 6.30 0.5 1.28 13.5 5.86 1.0 1.58 14.0 5.10 1.5 1.94 14.5 4.46 2.0 2.12 15.0 3.14 2.5 2.24 15.5 2.22 3.0 2.26 16.0 1.32 3.5 2.14 16.5 0.54 4.0 1.88 17.0 0.28 4.5 1.42 17.5 0.20 5.0 1.10 18.0 0.36 5.5 0.64 18.5 0.66 6.0 0.38 19.0 1.04 6.5 0.22 19.5 1.46 7.0 0.32 20.0 1.78 7.5 0.72 20.5 2.00 8.0 1.24 21.0 2.08 8.5 2.28 21.5 2.04 9.0 3.04 22.0 1.88 9.5 4.30 22.5 1.72 10.0 5.12 23.0 1.42 10.5 5.96 23.5 1.12 11.0 6.44 24.0 0.84 11.5 6.96 24.5 0.62 12.0 7.10 25.0 0.36 12.5 6.80
Table 5: Photodiode (Red Laser-Narrow Slits) Trial 1
Page 266 Alexakos, Harder, Kapelewski, Long, Manolache, Miller, Rahman, Rastogi, Reichardt, Rhinehart
Experiment 4 Trial 1: Red Laser, Photodiode, Narrow Slits Voltage vs. Position 8.00
7.00
6.00
5.00
4.00 Voltage (V)
3.00
2.00
1.00
0.00 0.0 5.0 10.0 15.0 20.0 25.0 Position (mm)
Figure 15: Photodiode (Red Laser-Narrow Slits) Trial 1
Journal of the PGSS A Verification of Wave-Particle Duality in Light Page 267
Experiment 4 Trial 2: Red Laser, Photodiode, Narrow Slits
Position Voltage Position Voltage (mm) (V) (mm) (V) 0.0 1.2 14.5 1.6 0.5 1.4 15.0 1.7 1.0 1.5 15.5 1.5 1.5 1.5 16.0 1.25 2.0 1.4 16.5 1.2 2.5 1.1 17.0 1.0 3.0 0.8 17.5 0.95 3.5 0.6 18.0 0.85 4.0 0.6 18.5 0.75 4.5 1.0 19.0 0.6 5.0 1.7 19.5 0.5 5.5 2.6 20.0 0.4 6.0 3.5 20.5 0.3 6.5 4.0 21.0 0.3 7.0 4.4 21.5 0.3 7.5 4.5 22.0 0.4 8.0 4.2 22.5 0.45 8.5 3.6 23.0 0.45 9.0 3.0 23.5 0.45 9.5 2.1 24.0 0.45 10.0 1.3 24.5 0.4 10.5 0.85 25.0 0.35 11.0 0.7 25.5 0.35 11.5 0.7 26.0 0.35 12.0 1.0 26.5 0.35 12.5 1.2 27.0 0.3 13.0 1.5 27.5 0.3 13.5 1.7 28.0 0.3 14.0 1.7
Table 6: Photodiode (Red Laser-Narrow Slits) Trial 2
Page 268 Alexakos, Harder, Kapelewski, Long, Manolache, Miller, Rahman, Rastogi, Reichardt, Rhinehart
Experiment 4 Trial 2: Red Laser, Photodiode, Narrow Slits Voltage vs. Position
5
4.5
4
3.5
3
2.5 Voltage (V)
2
1.5
1
0.5
0 0.0 5.0 10.0 15.0 20.0 25.0 30.0 Position (mm)
Figure 16: Photodiode (Red Laser- Narrow Slits) Trial 2
Journal of the PGSS A Verification of Wave-Particle Duality in Light Page 269
Appendix D: Photon Counting (Red Laser)
Experiment 5 Trial 1: Red Laser, Photomultiplier
1000000
800000
600000
400000 Photons
200000
0 0 5 10 15 20 25
-200000 Position (mm)
Figure 17: Photon Counting (Red Laser)
Page 270 Alexakos, Harder, Kapelewski, Long, Manolache, Miller, Rahman, Rastogi, Reichardt, Rhinehart
Experiment 5 Trial 1: Red Laser, Photomultiplier
Position Position Position Position (mm) Photons (mm) Photons (mm) Photons (mm) Photons 7 0 10.1 0 13.2 0 16.3 311803 7.1 0 10.2 0 13.3 1 16.4 344627 7.2 0 10.3 0 13.4 1 16.5 246711 7.3 1 10.4 0 13.5 1 16.6 357002 7.4 1 10.5 0 13.6 13 16.7 495217 7.5 0 10.6 0 13.7 406 16.8 583951 7.6 3 10.7 0 13.8 12014 16.9 645638 7.7 0 10.8 0 13.9 79489 17 691171 7.8 0 10.9 0 14 251589 17.1 682902 7.9 0 11 1 14.1 470246 17.2 632541 8 1 11.1 1 14.2 662842 17.3 584712 8.1 1 11.2 0 14.3 746360 17.4 474683 8.2 0 11.3 0 14.4 828133 17.5 342595 8.3 0 11.4 2 14.5 901039 17.6 237991 8.4 0 11.5 1 14.6 923874 17.7 111307 8.5 0 11.6 7 14.7 916784 17.8 40299 8.6 0 11.7 14 14.8 854928 17.9 8632 8.7 0 11.8 125 14.9 756602 18 944 8.8 0 11.9 365 15 593178 18.1 73 8.9 0 12 908 15.1 396186 18.2 16 9 0 12.1 1442 15.2 198366 18.3 11 9.1 0 12.2 1731 15.3 53568 18.4 46 9.2 0 12.3 1472 15.4 8670 18.5 342 9.3 0 12.4 879 15.5 414 18.6 2527 9.4 0 12.5 354 15.6 13 18.7 15552 9.5 0 12.6 72 15.7 12 18.8 45749 9.6 1 12.7 10 15.8 4 18.9 97368 9.7 0 12.8 1 15.9 78 19 181630 9.8 3 12.9 4 16 2943 19.1 273289 9.9 0 13 1 16.1 21349 19.2 326810 10 1 13.1 1 16.2 113415 19.3 375298
Table 7: Photon Counting (Red Laser 7mm-19.3 mm)
Journal of the PGSS A Verification of Wave-Particle Duality in Light Page 271
Position Position (mm) Photons (mm) Photons 19.4 396121 22.5 59 19.5 416580 22.6 9 19.6 372553 22.7 3 19.7 366404 22.8 3 19.8 289329 22.9 5 19.9 207206 23 1 20 135572 23.1 1 20.1 75660 23.2 4 20.2 26712 23.3 1 20.3 7459 20.4 957 20.5 174 20.6 7 20.7 2 20.8 2 20.9 1 21 1 21.1 3 21.2 34 21.3 76 21.4 373 21.5 1351 21.6 2153 21.7 4554 21.8 5078 21.9 5523 22 4953 22.1 4873 22.2 2057 22.3 961 22.4 289
Table 8: Photon Counting (Red Laser 19.4 mm-23.3)
Page 272 Alexakos, Harder, Kapelewski, Long, Manolache, Miller, Rahman, Rastogi, Reichardt, Rhinehart
Appendix E: Photon Counting (Green Laser)
Double-slit experiment (Green Laser)
140000
120000
100000
80000
60000 Photons Photons
40000
20000
0 0.0 5.0 10.0 15.0 20.0 25.0
-20000 Position (mm)
Figure 18: Photon Counting (Green Laser)
Journal of the PGSS A Verification of Wave-Particle Duality in Light Page 273
Experiment 5 Trial 2: Green Laser, Photomultiplier
Position Position Position (mm) Photons (mm) Photons (mm) Photons 0.0 0 3.0 0 6.0 0 0.1 0 3.1 0 6.1 0 0.2 0 3.2 0 6.2 0 0.3 0 3.3 0 6.3 0 0.4 0 3.4 0 6.4 0 0.5 0 3.5 0 6.5 0 0.6 0 3.6 0 6.6 0 0.7 0 3.7 0 6.7 0 0.8 0 3.8 0 6.8 0 0.9 0 3.9 0 6.9 0 1.0 0 4.0 0 7.0 0 1.1 0 4.1 0 7.1 0 1.2 0 4.2 0 7.2 0 1.3 0 4.3 0 7.3 0 1.4 0 4.4 0 7.4 0 1.5 0 4.5 0 7.5 0 1.6 0 4.6 0 7.6 0 1.7 0 4.7 0 7.7 0 1.8 0 4.8 0 7.8 0 1.9 0 4.9 0 7.9 0 2.0 0 5.0 0 8.0 0 2.1 0 5.1 0 8.1 0 2.2 0 5.2 0 8.2 0 2.3 0 5.3 0 8.3 0 2.4 0 5.4 0 8.4 0 2.5 0 5.5 0 8.5 0 2.6 0 5.6 0 8.6 0 2.7 0 5.7 0 8.7 0 2.8 0 5.8 0 8.8 0 2.9 0 5.9 0 8.9 0
Table 9: Photon Counting (Green Laser 0mm-8.9mm)
Page 274 Alexakos, Harder, Kapelewski, Long, Manolache, Miller, Rahman, Rastogi, Reichardt, Rhinehart
Position Position Position (mm) Photons (mm) Photons (mm) Photons 9.0 0 12.0 29384 15.0 17103 9.1 0 12.1 47964 15.1 3520 9.2 0 12.2 67396 15.2 241 9.3 0 12.3 80230 15.3 2 9.4 0 12.4 74542 15.4 2 9.5 1 12.5 67877 15.5 1 9.6 0 12.6 55633 15.6 1 9.7 1 12.7 41264 15.7 3 9.8 2 12.8 23698 15.8 225 9.9 35 12.9 9672 15.9 2385 10.0 252 13.0 1624 16.0 13255 10.1 922 13.1 73 16.1 35119 10.2 1963 13.2 1 16.2 52075 10.3 2598 13.3 1 16.3 58590 10.4 3038 13.4 1 16.4 92959 10.5 2553 13.5 1 16.5 91935 10.6 1620 13.6 1 16.6 76217 10.7 667 13.7 57 16.7 66821 10.8 161 13.8 1542 16.8 59798 10.9 5 13.9 12871 16.9 37252 11.0 1 14.0 31692 17.0 16357 11.1 1 14.1 61112 17.1 5409 11.2 0 14.2 83093 17.2 702 11.3 0 14.3 103755 17.3 22 11.4 1 14.4 114269 17.4 1 11.5 1 14.5 113106 17.5 1 11.6 5 14.6 101838 17.6 0 11.7 252 14.7 85234 17.7 1 11.8 3280 14.8 61568 17.8 1 11.9 13009 14.9 38882 17.9 1
Table 10: Photon Counting (Green Laser 9 mm-17.9 mm)
Journal of the PGSS A Verification of Wave-Particle Duality in Light Page 275
Position Position Position (mm) Photons (mm) Photons (mm) Photons 18.0 5 21.0 0 24.0 0 18.1 45 21.1 0 24.1 0 18.2 207 21.2 0 24.2 0 18.3 531 21.3 0 24.3 0 18.4 1188 21.4 0 24.4 0 18.5 1770 21.5 0 24.5 0 18.6 1752 21.6 0 24.6 0 18.7 1292 21.7 0 24.7 0 18.8 489 21.8 0 24.8 0 18.9 83 21.9 0 24.9 0 19.0 6 22.0 0 25.0 0 19.1 2 22.1 0 19.2 1 22.2 0 19.3 0 22.3 0 19.4 0 22.4 0 19.5 0 22.5 0 19.6 0 22.6 0 19.7 0 22.7 0 19.8 0 22.8 0 19.9 0 22.9 0 20.0 0 23.0 0 20.1 0 23.1 0 20.2 0 23.2 0 20.3 0 23.3 0 20.4 1 23.4 0 20.5 0 23.5 0 20.6 0 23.6 0 20.7 0 23.7 0 20.8 0 23.8 0
Table 11: Photon Counting (Green Laser 17.9 mm-25 mm)
Page 276 Alexakos, Harder, Kapelewski, Long, Manolache, Miller, Rahman, Rastogi, Reichardt, Rhinehart
Appendix F: Error Analysis of Double Slit Experiment
Experiment 1 Red Laser - Manually measured - Trial narrow slits 1
n (m) (m) x (m) x (m) L (m) L (m) dtheo (m) d (m) d (m) 1 6.328E-07 5.000E-10 0.0084 0.001 2.68 0.01 0.00020 0.000180 2.40E-05 1 6.328E-07 5.000E-10 0.0087 0.001 2.68 0.01 0.00019 0.000180 2.24E-05 2 6.328E-07 5.000E-10 0.0193 0.001 2.68 0.01 0.00018 0.000180 9.11E-06 2 6.328E-07 5.000E-10 0.0193 0.001 2.68 0.01 0.00018 0.000180 9.11E-06
Red Laser - Manually measured - Trial narrow slits 2
n (m) (m) x (m) x (m) L (m) L (m) dtheo (m) d (m) d (m) 1 6.328E-07 5.000E-10 0.0063 0.001 1.60 0.01 0.00016 0.000180 2.55E-05 1 6.328E-07 5.000E-10 0.0065 0.001 1.60 0.01 0.00016 0.000180 2.40E-05
Red Laser - Manually measured - Trial narrow slits 3
n (m) (m) x (m) x (m) L (m) L (m) dtheo (m) d (m) d (m) 1 6.328E-07 5.000E-10 0.0116 0.001 3.00 0.01 0.00016 0.000180 1.41E-05 1 6.328E-07 5.000E-10 0.0112 0.001 3.00 0.01 0.00017 0.000180 1.51E-05 2 6.328E-07 5.000E-10 0.0213 0.001 3.00 0.01 0.00018 0.000180 8.39E-06 2 6.328E-07 5.000E-10 0.0195 0.001 3.00 0.01 0.00019 0.000180 1.00E-05
Experiment 2 Red Laser - Manually measured - wide Trial slits 1
n (m) (m) x (m) x (m) L (m) L (m) dtheo (m) d (m) d (m) 1 6.328E-07 5.000E-10 0.0055 0.001 3.00 0.01 0.00035 0.000360 6.28E-05 1 6.328E-07 5.000E-10 0.0057 0.001 3.00 0.01 0.00033 0.000360 5.84E-05 2 6.328E-07 5.000E-10 0.0106 0.001 3.00 0.01 0.00036 0.000360 3.38E-05 2 6.328E-07 5.000E-10 0.0111 0.001 3.00 0.01 0.00034 0.000360 3.08E-05
Red Laser - Manually measured - wide Trial slits 2
n (m) (m) x (m) x (m) L (m) L (m) dtheo (m) d (m) d (m) 1 6.328E-07 5.000E-10 0.0044 0.001 2.48 0.01 0.00036 0.000360 8.11E-05 1 6.328E-07 5.000E-10 0.0050 0.001 2.48 0.01 0.00031 0.000360 6.28E-05 2 6.328E-07 5.000E-10 0.0086 0.001 2.48 0.01 0.00036 0.000360 4.25E-05 2 6.328E-07 5.000E-10 0.0089 0.001 2.48 0.01 0.00035 0.000360 3.97E-05
Red Laser - Manually measured - wide Trial slits 3
n (m) (m) x (m) x (m) L (m) L (m) dtheo (m) d (m) d (m) 1 6.328E-07 5.000E-10 0.0031 0.001 1.73 0.01 0.00035 0.000360 1.14E-04 1 6.328E-07 5.000E-10 0.0032 0.001 1.73 0.01 0.00034 0.000360 1.07E-04 Journal of the PGSS A Verification of Wave-Particle Duality in Light Page 277
2 6.328E-07 5.000E-10 0.0062 0.001 1.73 0.01 0.00035 0.000360 5.68E-05 2 6.328E-07 5.000E-10 0.0064 0.001 1.73 0.01 0.00034 0.000360 5.33E-05
Red Laser - Manually measured - wide Trial slits 4
n (m) (m) x (m) x (m) L (m) L (m) dtheo (m) d (m) d (m) 1 6.328E-07 5.000E-10 0.0023 0.001 1.23 0.01 0.00034 0.000360 1.47E-04 1 6.328E-07 5.000E-10 0.0022 0.001 1.23 0.01 0.00035 0.000360 1.61E-04 2 6.328E-07 5.000E-10 0.0045 0.001 1.23 0.01 0.00035 0.000360 7.69E-05 2 6.328E-07 5.000E-10 0.0045 0.001 1.23 0.01 0.00035 0.000360 7.69E-05 Experiment 3 Red Laser - Photodiode - wide slits Trial 1 σL x theo n λ (m) σλ (m) d (m) σd (m) L (m) (m) (m) x (m) σx (m) 1 6.328E-07 5.000E-10 0.00036 0.000005 2.85 0.01 0.00501 0.0050 7.19E-05 1 6.328E-07 5.000E-10 0.00036 0.000005 2.85 0.01 0.00501 0.0050 7.19E-05 2 6.328E-07 5.000E-10 0.00036 0.000005 2.85 0.01 0.01002 0.0100 1.44E-04
Red Laser - Photodiode - wide slits Trial 2 σL x theo n λ (m) σλ (m) d (m) σd (m) L (m) (m) (m) x (m) σx (m) 1 6.328E-07 5.000E-10 0.00036 0.000005 1.74 0.01 0.00306 0.0030 4.60E-05 1 6.328E-07 5.000E-10 0.00036 0.000005 1.74 0.01 0.00306 0.0030 4.60E-05 2 6.328E-07 5.000E-10 0.00036 0.000005 1.74 0.01 0.00612 0.0060 9.21E-05 2 6.328E-07 5.000E-10 0.00036 0.000005 1.74 0.01 0.00612 0.0060 9.21E-05
Experiment 4 Red Laser - Photodiode - narrow slits Trial 1 σL x theo n λ (m) σλ (m) d (m) σd (m) L (m) (m) (m) x (m) σx (m) 1 6.328E-07 5.000E-10 0.00018 0.000005 2.85 0.01 0.01002 0.0090 2.81E-04 1 6.328E-07 5.000E-10 0.00018 0.000005 2.85 0.01 0.01002 0.0090 2.81E-04
Red Laser - Photodiode - narrow slits Trial 2 σL x theo n λ (m) σλ (m) d (m) σd (m) L (m) (m) (m) x (m) σx (m) 1 6.328E-07 5.000E-10 0.00018 0.000005 1.74 0.01 0.00612 0.00625 1.74E-04 1 6.328E-07 5.000E-10 0.00018 0.000005 1.74 0.01 0.00612 0.00625 1.74E-04
Experiment 5 Red Laser - Photomultiplier - wide slits Trial 1 σL x theo n λ (m) σλ (m) d (m) σd (m) L (m) (m) (m) x (m) σx (m) 1 6.328E-07 5.000E-10 0.00036 0.000005 1.37 0.01 0.00241 0.0024 3.78E-05 2 6.328E-07 5.000E-10 0.00036 0.000005 1.37 0.01 0.00482 0.0049 7.57E-05 3 6.328E-07 5.000E-10 0.00036 0.000005 1.37 0.01 0.00722 0.0073 1.13E-04
Green Laser - Photomultiplier - wide slits Trial 2
n λ (m) σλ (m) d (m) σd (m) L (m) σL x theo x (m) σx (m) Page 278 Alexakos, Harder, Kapelewski, Long, Manolache, Miller, Rahman, Rastogi, Reichardt, Rhinehart
(m) (m) 1 5.435E-07 5.000E-10 0.00036 0.000005 1.37 0.01 0.00207 0.0021 3.25E-05 1 5.435E-07 5.000E-10 0.00036 0.000005 1.37 0.01 0.00207 0.0020 3.25E-05 2 5.435E-07 5.000E-10 0.00036 0.000005 1.37 0.01 0.00414 0.0040 6.50E-05 2 5.435E-07 5.000E-10 0.00036 0.000005 1.37 0.01 0.00414 0.0041 6.50E-05
Table 12: Error Analysis of Double-slit Experiments Appendix G: Photoelectric Effect
Photoelectric Effect
Current % Peak Frequency Color (amps) Transmittance (nm) (GHz) Mauve 1.1 4.14 430 697674.4 Special Med. Lavender 1.6 5.86 450 666666.7 Dark Lavender 1.9 6.61 450 666666.7 Lagoon Blue 1.65 25.36 480 625000 Dark Steel Blue 2.35 30.07 450 666666.7 Slate Blue 2.35 24.78 440 681818.2 Daylight Blue 2.15 20.04 440 681818.2 Evening Blue 1.9 12.5 455 659340.7 Tokyo Blue 0.65 1 450 666666.7 Jade 1.35 32 505 594059.4 Dark Green 0.85 29.71 505 594059.4 Clear 2.85 95 530 566037.7 Spring Yellow 1 84.14 530 566037.7 Apricot 1.24 52.97 650 461538.5 Pale Red 0.9 24.98 675 444444.4 Plum 1.4 19.4 700 428571.4 Bright Rose 0.75 14.44 705 425531.9 Bright Pink 1.5 13.72 675 444444.4 Flesh Pink 1.85 34.85 620 483871 Pale Salmon 1.9 64.9 620 483871 Light Salmon 2.25 54.86 630 476190.5 White Light 2.9 100 Flame Red 0.1 17.97 680 441176.5
Table 13: Photoelectric Effect
Journal of the PGSS A Verification of Wave-Particle Duality in Light Page 279
Figure 19: Photoelectric Effect- Current vs. Frequency
Stopping Potential VS Frequency
2.5
2
1.5
1
0.5 Stopping Potential (V) Potential Stopping
0 3.8E+14 4.3E+14 4.8E+14 5.3E+14 5.8E+14 6.3E+14 6.8E+14 7.3E+14 Frequency (Hz)
Figure 20: Photoelectric Effect- Mercury
Page 280 Alexakos, Harder, Kapelewski, Long, Manolache, Miller, Rahman, Rastogi, Reichardt, Rhinehart
Appendix H: Measuring the Speed of Light
Speed of Light in Polyethylene
140
120 y = 2E+08x - 0.2576 100 R2 = 0.9993 80
60 Length (m) Length 40
20
0 0.00E+00 1.00E-07 2.00E-07 3.00E-07 4.00E-07 5.00E-07 6.00E-07 7.00E-07 8.00E-07 Time (s)
Figure 21: Graph determining the Speed of Light in Polyethylene
VI. References
1 Baienlein, Ralph. Newton to Einstein: the Trail of Light. Middletown, Connecticut: Cambridge
UP, 1992. 33-45.
2 L6, David. "Is Light a Wave Phenomenon or Particle Phenomenon?" Science Magazine. 25 July 2007
3 Newton, Sir Isaac. Greenwich 2000. 2006. Greenwich Grid. 23 July 2007
4 Mauldin, John H. Light, Lasers and Optics. Blue Ridge Summit: Tab Books Inc., 1988. 88-94. 5 Goldin, Edwin. Waves and Photons: an Introduction to Quantum Optics. New York: John Wiley & Sons, 1982. 64-67. 6 Fox, Mark. Quatum Optics. Oxford: Oxford UP, 2006. 89-94. 7 Loudon, Rodney. The Quantum Theory of Light. 3rd ed. Oxford: Oxford UP, 2000. 117-119. 8 Nave, C R. "Maxwell's Equations." HyperPhysics. 2005. Georgia State University. 24 July 2007
10 Rubin, Julian. Heinrich Hertz: The Discovery of Radio Waves. Sept. 2006. 25 July 2007
11 “photoelectric effect.” Encyclopedia Britannica. 2007. Encyclopedia Britannica Online. 24 Jul 2007
12 Fowler, Michael. “The Photoelectric Effect.” The Photoelectric Effect. 1997. University of Virginia. 24 Jul 2007
13 Jones, Andrew. “The Photoelectric Effect” About.com: Physics. 2007. The New York Times. 24 July 2007
14 Folse, Henry J. The Philosophy of Niels Bohr: the Framework of Complementarity. Amsterdam: North-Holland Personal Library, 1985 15 Paschotta, Rudiger. “Photodiodes.” Encyclopedia of Laser Physics and Technology. 04 May 2007. RP Photonics. 24 Jul 2007
Journal of the PGSS Wil-ber-force be with you: An Analysis of the Wilberforce Pendulum Page 283
Wil-ber-force Be With You: An Analysis of the Wilberforce Pendulum
Ryan Cook, James Han, Liyun Jin, Troy Menendez, Gopal Nataraj, Randy Stein, Eric Strom, Cecily Sunday, and XinChi Yang
Abstract
This project started with the following of goals in mind: to develop a driving mechanism, to design and build a number of Wilberforce pendulums. Towards these aims, we were very successful. Three pendulums were built during the course of the project, two of them via trial and error using the principles learned in the equations and concepts that govern the operation of a Wilberforce pendulum, and a third whose dimensions were derived entirely via mathematical processes. All of these pendulums were successful at demonstrating a characteristic beat pattern. The driving force was created via use of a solenoid connected to a DC power supply and is controlled via a 741 Light/Dark sensor. This motivator is highly successful at driving a Wilberforce pendulum’s normal mode, although it must be manually controlled in order to drive the characteristic beat pattern.
I. Introduction
A Wilberforce Pendulum consists of a bob attached to a coiled spring that will oscillate between longitudinal and rotational motion. The different oscillations are coupled and energy is transferred between the two modes. Adjustments to the placement of mass on the bob create equal longitudinal and rotational periods. When these two periods are equal, the superposition of the two normal modes gives the desired beat pattern.
In the ideal world of the physics classroom, most problems are considered in a frictionless world. Was this the case, a driving mechanism would be unnecessary as no energy would be lost from the system. In the real world, however, no system is frictionless and energy is always lost from the system. Because of this loss, a driving mechanism must be built to replace energy lost due to friction.
II. History
The Wilberforce Pendulum is named after Lionel Robert Wilberforce, its creator. In 1985, Wilberforce, a demonstrator of physics at the Cavendish Laboratory in Cambridge, exhibited a cylindrical mass on a helical spring that oscillated between rotational and longitudinal motion. Wilberforce was originally interested in determining accurate values for the Young’s modulus of various spring materials.
III. Previous Designs
The Wilberforce pendulum typically consists of a mass attached to the end of a loose, helical spring. The mass can be constructed of a variety of materials, but is usually made of metal. Ideal mass distribution depends upon the spring’s length, diameter and stiffness. Two or four outriggers are Page 284 Cook, Han, Jin, Menendez, Nataraj, Stein, Strom, Sunday, Yang usually connected to the sides of the mass to tweak its moment of inertia. Most bobs are cylindrical in shape, but there are working models with spherical and disk-shaped bodies as well.
IV. Theory
A. Longitudinal and Rotational Motion
The simplest form of harmonic movement is called simple harmonic motion (SHM), a type of periodic motion that describes the oscillation of various pendulums, including transverse, longitudinal, and rotational pendulums.
The movement of transverse pendulums, however, can only be described using SHM over a very small angular displacement, since, over a larger displacement, the component of the restoring force perpendicular to the pendulum bob deviates further and further from the horizontal.
However, SHM is also used as a term for a rotational or longitudinal pendulum, the two components of a Wilberforce pendulum. Since these pendulums do not incorporate a swinging movement as part of their harmonic motion, their movements can be predicted at any point during their periods with accuracy. In other words, SHM is periodic motion, movement that can be affected by several factors, including mass and moment of inertia of the spring/pendulum bob, and the longitudinal/rotational spring constants, depending on what sort of motion is being described, since the longitudinal period depends upon the spring’s mass and the longitudinal spring constant, and the rotational period depends upon the total moment of inertia and the rotational spring constant.
The period of a longitudinal pendulum’s motion (Tz ) can be described via equation (1):
M M + sp T = 2π 3 (1) z k
Where M is the mass of the pendulum bob, Msp is the mass of the spring (divided by three since the effective mass of the spring is 1/3 the entire mass of the spring), and k is the longitudinal spring constant. The equation for the rotational period (Tθ ) looks quite similar, save with moments of inertia being summed in the numerator of the radical and the rotational (as opposed to longitudinal) spring constant in the denominator:
I I + sp T = 2π 3 (2) θ δ Journal of the PGSS Wil-ber-force be with you: An Analysis of the Wilberforce Pendulum Page 285
As in equation 1, the spring’s moment of inertia is divided by three in equation 2 since only the equivalent of a third of the spring oscillates with the pendulum bob. However, since equation 2 concerns the rotational period, there are different variables that are referenced, such as moment of inertia (I) and the rotational spring constant (δ). In order to find the moment of inertia (a measure of the first law of the motion applied to rotating bodies), equations (3)-(7) are utilized depending on the type of figure being analyzed:
1 2 2 = ( + rrMI ) (3) Hollow cylinder 2 inner outer
ML2 I = (4) Thin rod, rotated about its center 12
Mr 2 I = (5) Solid Cylinder 2
2 = MrI 2 (6) Sphere 5
I = Mr 2 (7) Point Mass
As noted before, the rotational spring constant also has to be taken into account in order to find the rotational period. In order the find this, the frequency of a certain mass’s rotation must be found as well as its moment of inertia. Equation (8) shows the operations necessary to find the rotational spring constant using these data points:
2 (8) = ωδ θ I
B. Normal Modes
However, in order to predict the spring’s movements at any moment during its period, more information about its normal modes must be taken into account. Energy is transferred between the rotational and horizontal modes via the rotation of the pendulum bob and thus the stretch/compression of the spring. However, the frequency of the rotational motion varies depending on which way the spring is being twisted, subtly shifting the frequencies of the two normal modes (a combination of the two forms of SHM found in a Wilberforce pendulum: longitudinal vibration with a counter-clockwise or clockwise rotation). Page 286 Cook, Han, Jin, Menendez, Nataraj, Stein, Strom, Sunday, Yang
The motion that results from these normal modes has two degrees of freedom: one translational and one rotational. In order to predict its motion, a superposition of these two modes are required, since their slightly different frequencies generate an interference pattern between the two modes. The composition of the two frequencies exhibits this effect quite well, showing the formation of a beat pattern, the result of a weak coupling between the modes. An example of this can be seen below:
Figure 1: Superposition of normal modes and a beat pattern
The top set of waves represent the pendulum’s normal modes. These waves have slightly different frequencies which can be observed in a Wilberforce pendulum through the existence of a beat pattern in the pendulum’s oscillatory motion (as seen on the bottom graph). The main requirement for the normal modes to become well established and create such an interference pattern is that that the periods of rotational and longitudinal vibration must be equal to one another. The difference in frequency, as explained previously, will occur naturally, causing the beat. However, if only one normal mode is in operation (as can be caused by addition of a certain amount of energy into the system or starting under a certain set of initial conditions), no energy transfer between rotational/longitudinal motion will take place, and there will be no interference pattern and no coupling between modes.
A visualization of a beat pattern as opposed to SHM can be visually divided into regions of high and low amplitude, a technique that shows something known as the beat frequency, or how quickly the pendulum’s motion switches between regions of high and low amplitude. In a Wilberforce pendulum, this can be seen as a transition between longitudinal and rotational motion as energy is transferred between the longitudinal and rotational motion via the spring. At the areas of high amplitude, there is no rotational motion and extensive amounts of vertical movement, and, at where there is extremely low amplitude, virtually all of the motion is rotational. This is because of the beat pattern, a phenomena caused by the transfer of energy between the rotational and longitudinal movement.
Using the beat frequency, the frequencies of the pendulum’s normal modes can be found as described in equations (9) and (10). The variable ω in equations (9) and (10) is equal to the rotational and longitudinal frequencies: Journal of the PGSS Wil-ber-force be with you: An Analysis of the Wilberforce Pendulum Page 287
ω ωω += B (9) 1 2
ω ωω −= B (10) 2 2
Equation 6: Formulas for the Frequencies of the Normal Modes
Then, using the frequencies of these normal modes, the coupling constant, a force that tells the strength of an interaction, can be calculated as shown in equation (11). In a Wilberforce pendulum, this describes the interaction between the two normal modes, which is a weak interaction. This is exhibited in the beat pattern, where the normal modes interfere due to a slight difference in frequency.
2 2 1 −= ωωε 2 )( IM effeff (11)
C. Deriving the Coupling Constant
1. Vertical Motion
z = height = Start with force F ma = torsional spring constant F = − kz substitute in values and solve
− kz = m z && + = m z&& kz 0 kz z + = 0 && m
i ω z t guess : z = Ae Guessed a value for z in order to help solve Then take the derivatives: z = A[ cos (ω z t ) + i sin(ω z t )] i ω t z = A( i ω )e z = velocity & z 2 i ω t = acceleration z = A i ω e z && ( z ) 2 z = z ω z && kz z + = 0 && m 2 z = −ω z z && Substitute using the above force formula 2 kz − ω z z + = 0 m = longitudinal frequency 2 k ω z = m Page 288 Cook, Han, Jin, Menendez, Nataraj, Stein, Strom, Sunday, Yang
2. Torsional Motion: =− Iθδθ && Moment of inertia
ωθ ti θ = Be Guess a value
2 && −= θωθ θ
2 δ ωθ = I Substitute to get rotational frequency
3. Vertical and Torsional Motion:
Now we add a weak linear coupling constant, c, and write out the equations of motion:
Set the two equal
+ + θ = (1 ) m& z& kz c 0 Solve for z ( 2 ) I θ&& + δθ + cz = 0
− I θ − δθ z = && c
I δ Note : z = − θ&& − θ&& && c c
⎛ I δ ⎞ ⎛ I δ ⎞ 0 = m ⎜ − θ&& − θ&& + k ⎜ − θ&& − θ + cθ ⎝ c c ⎠ ⎝ c c ⎠ mI δ m Ik kδ − θ&& − θ&& − θ&& − θ + c θ = 0 c c c c
Plug in and simplify Journal of the PGSS Wil-ber-force be with you: An Analysis of the Wilberforce Pendulum Page 289
c ⎛⎛ mI ⎞ ⎛ δ −− Ikm ⎞ ⎛ δ +− ck 2 ⎞ ⎞ ⎜ && ⎜ ⎟ ⎟ ⎜⎜−− ⎟θ&& + ⎜ ⎟θ&& + ⎜ ⎟θ ⎟ = 0 mI )( ⎝⎝ c ⎠ ⎝ c ⎠ ⎝ c ⎠ ⎠ 2 && ⎛ δ k ⎞ ⎛ δk c ⎞ )3( θ&& ⎜ ++ ⎟θ&& ⎜ −+ ⎟θ = 0 ⎝ I m ⎠ ⎝ mI mI ⎠
guess :θ = Ae ωti Guess a value & = ()i θωθ Derivatives: && ()i 2 −== 2θωθωθ &&& = ()i 3θωθ && ()i 4 == 4θωθωθ ⎛ c 2 ⎞ 4 222 ⎜ 22 ⎟ ()()θ z ⎜ θ z −+−++ ⎟θωωθωωωθω = 0 ⎝ mI ⎠ ⎛ c 2 ⎞ 2224 ⎜ 22 ⎟ ()()θ z ⎜ θ ωωωωωω z −+−++ ⎟ = 0 ⎝ mI ⎠ ⎛ c 2 ⎞ 2224 ⎜ 22 ⎟ ()()θ z ⎜ θ ωωωωωω z −++− ⎟ = 0 ⎝ mI ⎠
2 −±− 4ACBB Quadratic formula 2 0 xCBxAx =⇒=++ 2A 2 2 ⎛ c ⎞ 22 22 ⎜ 22 ⎟ ()()θ z θ ωωωω z −+±+ ()14 ⎜ θ ωω z − ⎟ ⎝ mI ⎠ ω 2 = ()12
2 22 4 422 22 4c ()θ z θ 2 θ zz 4 θ ωωωωωωωω z +−++±+ 2 mI ω = 2 2 22 4 422 4c ()θ z θ 2 θ ωωωωωω zz ++−±+ 2 mI ω = 2 2 22 22 2 4c ()()θ z θ ωωωω z +−±+ 2 mI ω = 2 ⎡ 2 ⎤ 2 1 22 22 2 4c ⎢()()θ z θ ωωωωω z +−±+= ⎥ 2 ⎣⎢ mI ⎦⎥ Page 290 Cook, Han, Jin, Menendez, Nataraj, Stein, Strom, Sunday, Yang
1 c = ε 2
General Equation:
⎡ 2 ⎤ 2 1 22 22 2 ε ⎢()()θ z θ ωωωωω z +−±+= ⎥ 2 ⎣⎢ mI ⎦⎥
Now consider the specific case where : ω = ωθ = ω z
2 1 ⎡ 2 ε ⎤ 2 ⎢()()22 ωωωωω 22 +−±+= ⎥ Plug in and simplify 2 ⎣⎢ mI ⎦⎥
1 ⎛ ε 2 ⎞ 2 ⎜2ωω 2 ±= ⎟ ⎜ ⎟ 2 ⎝ mI ⎠
2 22 ε 1 ωω += 4mI ε 2 ωω 22 −= 2 4mI 1 ⎡ 2 ⎤ 2 2 ε 1 ⎢ωω += ⎥ ⎢ 4mI ⎥ ⎣ ⎦
1 ⎛ 2 ⎞ 2 n nn −1 ε Set equal and simplify () +=+ forbnaaba __ ⎜ ⎟ << ω ⎝ 4mI ⎠ 2 ⎛ 1 ⎞⎛ ε ⎞ ωω += ⎜ ⎟⎜ ⎟ 1 ω ⎜ 42 mI ⎟ ⎝ ⎠⎝ ⎠ ε 1 ωω += 4ω mI ε ω B = 2ω mI
ω B 1 ωω += 2 ω B 2 ωω −= 2 ω ω ωωω B ω +−+=− B 21 22
21 =− ωωω B Journal of the PGSS Wil-ber-force be with you: An Analysis of the Wilberforce Pendulum Page 291
D. Periods of the Pendulum
A period is a set time in which a certain event occurs. In the case of the Wilberforce pendulum, a period is the amount of time it takes for the weight to complete one cycle of up and down oscillation or left right rotation. These periods are coupled and one motion completely transfers its energy into the other. Coupled motion means that the amplitude of one decreases as the other increases. This is what makes the Wilberfoce pendulum different from other pendulums. .
The two motions of the Wilberforce pendulum include one that is strictly rotational and one that is strictly longitudinal. During the period of rotational movement, the pendulum transfers all of its energy into rotating and it will not move up and down. During the longitudinal period the pendulum transfers all of its energy into up and down motion with no rotation instead. Rotational inertia, or moment of inertia, is a measure of how easily an object can change its rotation.
Both the rotational and longitudinal motion of the pendulum must be equal in order for the energy to completely transfer. The various measurements affecting each of these periods are described in the following equations. Equation 1 describes the period of the longitudinal motion where Mbob is the mass of the pendulum bob in kilograms, Mspring is the mass of the spring in kilograms, and kz is the longitudinal spring constant in Newtons per meter (this constant was found empirically by testing the spring’s displacement by different weights).
Equation 2 is akin to the first though it describes the period of rotational motion. In the equation, Ibob stands for the bob’s moment of inertia in kilogram-meters, Ispring stands for the spring’s moment of inertia, and stands for the rotational spring constant in Newtons per radian. In order to determine an optimum mass and moment of inertia for the pendulum bob, these equations were set equal to each other. Solving for I, Equation 3 is produced.
⎛ M sp ⎞ δ ⎜M + ⎟ ⎝ 3 ⎠ I sp I = − (12) k 3
E. Spring Characteristics
A spring has the property that it exerts a resisting force when its shape is changed. The springs that are most commonly used are coil springs, a mechanical device that is used to store and release energy, usually to absorb shock or maintain a force between contacting forces. They are made of elastic material that forms a helix and returns to its natural length when unloaded. Coil springs also function as a special type of torsion spring. Torsion springs are close wound and increase in length when they are deflected. Basically, a torsion spring stores the energy when it is twisted and releases the energy when it is released, inducing a twist. The combination of its torsional and longitudinal properties makes a helical spring very useful on a Wilberforce pendulum. In order to pull a spring a Page 292 Cook, Han, Jin, Menendez, Nataraj, Stein, Strom, Sunday, Yang distance x from its equilibrium position, a stretching force F is required. F is proportional to the distance the spring is stretched.
F = -kx (13)
In the equation, k is the spring constant, in units of Newtons per meter. Therefore loose springs have low k values while stiffer springs have high k values. Hooke’s law is based on three variables: displacement, x; force, F; and the spring constant, k. In addition to the longitudinal spring constant, there is also a rotational spring constant.
2 = I (14)
I is the moment of inertia, the rotational frequency, and the rotational spring constant. It is the same basic idea as the longitudinal spring constant, except this time it measures how difficult it is to rotate the spring for some angle with an applied torque.
F. The Damping Force
The damping force can be caused by the nature of a viscous medium (air, water, etc) to take energy out of the pendulum system and, for small velocity, is proportional to the velocity of the pendulum through that medium. The system can be either over, under, or critically damped. In the critically damped case, the pendulum will stop its vibration in the smallest amount of time. In an overdamped case, the oscillator will reach zero amplitude in a larger time, but its damping constant is greater than in the under-damped case. Underdamping occurs when the amplitude reaches zero faster than the critically damped case, but continues to oscillate around zero; it is this sort of damping that is present in our pendulums (damping in our case is due mainly due to air resistance). This can be determined by finding the decay rate of a pendulum’s amplitude over time, an exponential decrease, a function shown in equation (15) where A is the peak amplitude in a period that has decreased since the previous period due to the damping force, C is the amplitude at t=0, b is the damping constant (also referred to as the damping coefficient), and t is time during the oscillation:
= CeA bt (15)
If b is less than the undamped resonant frequency (the beat frequency in a Wilberforce pendulum), the pendulum is underdamped; if it is equal to the resonant frequency, then the pendulum is critically damped, and if it is greater than the resonant frequency, the pendulum is overdamped.
V. Pendulum #1
A. Goal Journal of the PGSS Wil-ber-force be with you: An Analysis of the Wilberforce Pendulum Page 293
To design and build a prototype Wilberforce pendulum from a standard weight and clay and transform it into a metal pendulum that can be driven by an external force, most likely a magnet.
B. Procedure
The first pendulum that was built was centered on being small and light. The group believed that it would be easiest to work with and adjust a lighter pendulum. First, a spherical mass with clay around the circumference was tested. However the movement of the pendulum was very hard to tweak on this body, and the spherical mass was abandoned. Next a fifty-gram standard mass was tested out, tweaked using two clay masses attached opposite each other. On the bottom, more clay was added to slow the vertical period and build rotational inertia. It was observed during testing that even slight changes to the clay would bring significant changes to the two periods. Four clay masses instead of two were tested as well, but since there were no visible improvements, the idea was abandoned.
After creating a working pendulum with a fifty-gram weight and clay, dimensions were recorded and modified for the machinist to build a metal pendulum. Metal screws that had nuts on it to adjust the rotational inertia replaced the two antipodal clay masses. An aluminum disc was attached to the bottom of the pendulum to allow more nuts and washers to be added to account for the clay mass on the bottom on the pendulum previously. After the machinist produced the metal pendulum, few minor adjustments were made and a working pendulum was born.
Calculations were taken from this newly crafted metal pendulum. Calculations include moments of inertia, rotational spring constant, longitudinal spring constant, total mass, dimensions, coupling force, the beat frequency, the rotational frequency, the longitudinal period, and the rotational period. Error propagation was done for each calculation to track accuracy and precision. These calculations were taken and analyzed further.
The final step was to adapt the pendulum so that it could be powered by an electromagnetic field. Although the pendulum worked, it was neither massless nor frictionless and therefore required an external power to keep it going forever. Modifications were made to the pendulum so that a few magnets could be attached to the bottom of the pendulum and the driving mechanism could be used on this pendulum.
C. Equipment/Materials Pendulum #1
- Center mass: 50g pot metal, mainly Zinc oxide - 9 Hexagonal stainless steel nuts - Aluminum base - 3 Galvanized steel washers - 3 Stainless steel washers - 2 Iron threaded rods Page 294 Cook, Han, Jin, Menendez, Nataraj, Stein, Strom, Sunday, Yang
- 6 Neodymium magnets - 1 aluminum core
D. Data Analysis
Displacement vs. Time
0.7
0.6
0.5
0.4
0.3
0.2 Displacement (Meters)
0.1
0 0 102030405060 Time (Seconds)
Figure 2: Beat Pattern of Pendulum #1
Figure 2 shows the DataStudio plot for the pendulum. The data shows the beat typical of a Wilberforce pendulum. The beat period is the time it takes for the pendulum to complete one longitudinal period and one rotational period. In Figure 9, the beat period is from one large crest to the other. Also, Figure 2 shows the period of Pendulum #1. The period is the time it takes to complete one longitudinal circuit. In Figure 9, it is the time in between the smaller crests. This graph shows the beat period for Pendulum #1 to be (17.80 +/- 0.01) seconds. Additionally, the inverse of the rotational and longitudinal periods can be taken to determine the rotational and longitudinal frequencies.
The moment of inertia of the central mass was calculated to be (1.94 ± 0.49) x 10-6 kg m2. Because there were holes in the original mass, the moment of inertias of the “holes” had to be subtracted from the total moment of inertia of the cylinder. To start, the “negative” masses were calculated using the density of the material, and then those masses were used in the moment of inertia equations to find the “negative moment of inertias.” These were subtracted from the total moment of inertia, and this gave a value for the moment of inertia of the central mass. Journal of the PGSS Wil-ber-force be with you: An Analysis of the Wilberforce Pendulum Page 295
The moments of inertia were found using standard moment of inertia equations for various objects. The parallel-axis theorem was applied to objects like the magnets. The parallel-axis theorem is used to determine the moment of inertia about a parallel axis through the center of mass and perpendicular to the two axes. It states the moment of inertia of said object is its moment of inertia + MR2, where R is the distance from the center axis of rotation. All of the inertias were added up with the moment of 1/ inertia of the central mass. In addition, 3 of spring’s moment of inertia was added. It is only one- third because during movement, only one-third of the spring moves with the pendulum. Thus, the total inertia of the system was calculated to be (5.3 ± 1.8) x 10-6 kg m2.
2 The rotational spring constant was calculated using the formula = I, where I is the total moment of inertia and is the rotational frequency. The rotational frequency was determined by averaging several trials measured by a stopwatch and then inversing that number.
Using a cylindrical mass, the rotational spring constant was found to be (1.46 ± 0.19) x 10-4 Nm/rad. Using a spherical mass, it was found to be (1.75 ± 0.10) x 10-4 Nm/rad. Note that the spring constants found by each of these two methods were consistent within the accepted error range.
Figure 3: LSF program Used To Find Spring Constant k
A program in Microsoft Excel called LSF, or Least-Squares Fit, was used to determine the spring constant k. It graphed the Force versus its displacement, and the resulting line is the spring constant. It also calculated the error. A picture of the graph is shown in Figure 3.
The rotational and longitudinal periods were determined to check to see if they were identically correct. As shown Table 8, they are. The longitudinal period was calculated using equation (1) and the rotational period was calculated using equation (2). Page 296 Cook, Han, Jin, Menendez, Nataraj, Stein, Strom, Sunday, Yang
In addition to the data collected, error was calculated for its respective data. This was very useful in comparing the main pendulum’s data to those of other masses. Note that all values coincide with each other’s error ranges.
E. Conclusion
Since the rotational and longitudinal periods were the same in both pendulums, it was hypothesized that the moments of inertia must also be the same. Indeed, the above calculations show the moments of inertia were the same within their calculated error range. Also, the rotational spring constant found with the metal pendulum was compared with the one from the spherical mass. They were also the same within their calculated error range. Error propagation proved to be very useful in comparing values. Programs such as Datastudio and LSF were very beneficial in helping to understand the beat frequency and normal modes of the pendulum.
VI. Pendulum #2
A. The Early Prototype
Pendulum 2 was designed in an innately different style from the other two pendulums in that it is variable in both mass and moment of inertia. By allowing both factors to be changed, the longitudinal frequency and the rotational frequencies were variable, allowing for optimal results to be produced through experimentation.
An early prototype of the pendulum was created using a large bolt, copper wire, and clay. Using pliers, a copper wire was bent around the head of the bolt, creating a makeshift hook above the head. The entire system was secured and prepared for testing using hot glue. Trial and error quickly proved, however, that the rotational frequency was almost twice as high as its longitudinal counterpart. In other words, for each time it went up and down one period, it spun back and forth three half-oscillations.
We were excited by these results because it was an improvement from the hackneyed previous issue that we had grown tired of: the longitudinal frequency had been greater than the rotational frequency due to an excessive moment of inertia. By equations (1), (2), and (16), we concluded that our pendulum did not have enough of a radius and created a marginal moment of inertia, which alluded to a high rotational frequency: 1 f = (16) Tθ
As the radius of our earliest prototype was increased, the moment of inertia tended to increase as well. Since the rotational period Tθ was directly proportional to the square root of the inertia, it increased as well, while the frequency, which was inversely proportional to the period, decreased. Journal of the PGSS Wil-ber-force be with you: An Analysis of the Wilberforce Pendulum Page 297
Such a situation posed major issues because changing the radius of the system was a daunting, if not impossible, task. The alternative route, increasing the mass and thus reducing the longitudinal frequency, posed its own issues, mainly due to an inability to reliably adhere heavy masses in the symmetrical manner essential for such delicate procedures. All of these unnecessary difficulties precipitated our eventual abandonment of the original idea.
B. The Modified Rod-and-Arm Pendulum
Figure 4: Drawing and Picture of Pendulum #2
1. Procedural Methods
Our new rod-and-arm style pendulum tackled the problem we encountered using the 200g mass directly. The adjustable central rod had a high mass, yet a rather small radius, allowing for a low moment of inertia and, consequently, a high rotational frequency. Lowering the rotational frequency Page 298 Cook, Han, Jin, Menendez, Nataraj, Stein, Strom, Sunday, Yang was significantly easier than lowering the longitudinal frequency: we simply needed to insert a longitudinal arm of the necessary mass and inertia to increase the radius. Luckily, the initial clay-arm setup that we tested did not have to be tweaked excessively because the radius we added lowered the rotational frequency so that it matched the longitudinal analog almost perfectly.
Using our newfound information and an iota of logic, we gathered some scrap metal rods and asked Gary, our machinist, to construct a pendulum that was proportional to our prototype on a larger scale. Our final pendulum had holes drilled on the ends of the main body in order to add any weight necessary via additional screws. In addition, the arm passing horizontally through the main body was threaded on both ends so that extra masses could be screwed on to adjust the moment of inertia. Objects used for additional mass included steel washers, various nuts, and brass thumbscrews.
2. Graphs of Longitudinal Motion We used the program DataStudio along with its motion-sensing equipment to plot the displacement of our pendulum in action against time.
Short-Term Longitudinal Motion of Pendulum #2 0.12
0.1 0.08
) 0.06
0.04
0.02 0
-0.02 0 102030405060708090100
-0.04 -0.06 Distance from Equilibrium (m) -0.08
-0.1
-0.12 Time (s)
Figure 5: Beat Pattern of Pendulum #2 (Short Term) Journal of the PGSS Wil-ber-force be with you: An Analysis of the Wilberforce Pendulum Page 299
Long Term Longitudinal Motion of Pendulum #2
0.15 0.1 0.05 0 -0.05 0 100 200 300 400 500 Distance from Equilibrium (m) -0.1 -0.15 Time (s)
Figure 6: Beat Pattern of Pendulum #2 (Long Term)
4. Calculations a. Longitudinal Spring Constant (k)
A sub-derivation of the net force at equilibrium follows. Using the modified equation below, spring constant values were calculated and then averaged to estimate a rather good approximation of the spring constant.
mg k = (17) x
b. Longitudinal Period (Tz), Frequency (fz), and Angular Frequency ( z)
The graph was used to find the longitudinal frequency and period. This is calculated by measuring the distance between the crests of each individual oscillation that make up the large oscillation waves.
ωz = 2πf z (18)
Figure 15: Derivation of the longitudinal period (Tz), frequency (fz), and angular frequency ( z)
c. Torsional Period (T ), Frequency (f ), and Angular Frequency ( ) Page 300 Cook, Han, Jin, Menendez, Nataraj, Stein, Strom, Sunday, Yang
Putting our pendulum in action allowed us to visually see the full transfer of motion between rotational and longitudinal motions. We timed how long it took for the pendulum to rotate ten full periods and used that data to find the frequency and angular frequency. However, because the measurements were obtained by eyeballing the pendulum, considerable error is expected.
ωθ = 2πfθ (19)
d. Beat Period (TB) and Beat Frequency ( B)
The graph was also used to calculate the beat period and frequency. We estimated the trough of each beat oscillation and measured the distance between the troughs and used this data to find the beat period and frequency. The reason we estimated troughs instead of crests was because the crests were not definite enough. We had difficulty determining the location of the crest while the trough point was quite evident.
2π ω B = (20) TB e. Moment of Inertia
The total moment of inertia is calculated by adding the inertia of the pendulum, which is the sum moment of inertias for each individual piece of the pendulum and one-third the inertia of the spring. Equations (3) – (7) were used in this calculation.
f. Inertia of the Pendulum (Ipendulum)
The inertia of the pendulum cannot be calculated using one equation. Because each part of the pendulum has different dimensions and shapes, they each have different moments of inertias. Some parts, such as the screws, had to be broken down even further since their shape was not uniform. Using the equations defined above, we were able to calculate the inertias of each individual part and add them up to get the total inertia of the pendulum. The value for Ipendulum was found by summing the moments of inertia of the individual components. This value was found to be (9.20 ± 0.23) x 10-6 kg/m2
g. Inertia of the Spring (Ispring)
It is also necessary to calculate the inertia of the spring because it affects the total inertia of the system. The spring is assumed to be a hollow cylinder that spins about the center axis. The value for -7 2 ISpring was found to be (4.75 ± 0.16) x 10 kg/m
h. Total Moment of Inertia (Itotal) Journal of the PGSS Wil-ber-force be with you: An Analysis of the Wilberforce Pendulum Page 301
The total moment of inertia is calculated by adding the moment of inertia of the pendulum plus one- third the inertia of the spring. It was found to be (9.54 ± 0.17) x 10-6 kg/m2 i. Torsional Spring Constant ( )
With the total moment of inertia, the torsional spring constant δ can be calculated using two different equations.
I T = 2π total θ δ
2 ⎛ 2π ⎞ ⎜ ⎟ δ = Itotal ⎜ ⎟ ⎝ Tθ ⎠
δ = 1.15*10-4 N*m
Figure 18: Derivation & calculation of the torsional spring constant using the torsional period
2 δ ωθ = I total
2 = I totalωδ θ
δ = 1.15*10-4 N*m
Figure 19: Derivation of the torsional spring constant ( ) using the torsional frequency j. Weak Coupling Constant ( )
The equation we derived for the beat frequency includes the weak coupling constant , as well as the total mass and total inertia of the system. It also includes ω which is equal to the average of the rotational and longitudinal frequencies.
Using equations () and (), we find :
= 2 Bωωε mI
ε = −4 ± 10*49.010*12.4 −4 N
Figure 20: Derivation and calculation of the weak coupling constant ( ) Page 302 Cook, Han, Jin, Menendez, Nataraj, Stein, Strom, Sunday, Yang k. Damping Constant (b)
The damping constant, b, is a holistic account for the negative exponential pattern of energy loss. For the purposes of our experiment, it accounted mostly for air resistance. Calculation of the constant involved merely solving for the unknown after the best fit curve from the long-term graph above was extrapolated:
2 − mbt ( ) = max ekxtE
( ) = 0746.0 etE − 0061.0 t
kg b −4 ×±×= 10013.010866.7 −4 s
C. Analysis and Conclusions
The creation and mathematical modeling of Pendulum #2 was, in essence, largely a success, in that the pendulum was perfected to work with near-complete to complete energy transfers between the longitudinal and torsional modes. We attribute our success to the main advantageous aspect unique to this oscillator: the ability to tweak amounts and distributions of mass effectively and efficiently through several locations on the pendulum, including the top, bottom, and arms. Experimental testing was seen to obey all formulas and relationships between the various forces at play, with a respectable degree of precision on all horizons. With the degrees of error propagation in mind, our theoretical and experimental calculations matched quite well, proving that, given the correct conditions, Wilberforce Pendulums do indeed work.
Aside from the typical expected observations that were summarized through simple verification in the Calculations sections, certain not-so-familiar phenomena were also seen to take place. These unexpected observations will be focused upon.
Upon visualization of our graphs of longitudinal displacement from equilibrium versus time, it is facile to realize that Pendulum #2 was able to maintain a distinguishable beat pattern for a much longer period of time than either of the other pendulums. Although energy loss was evident through the steady decrease of amplitude between each beat, the longitudinal, torsional, and beat frequencies were observed to stay constant, thus still preserving the Wilberforce properties of the system. Unfortunately, however, the lower amplitudes that arise as the duration of oscillatory motion increases cause visual observation issues. In other words, the Wilberforce motion continues for a much longer period of time than that which the human eye can readily identify.
This loss of energy follows an exponential pattern of decay, as exemplified with the best-fit negative exponential curves used in the calculations section. Realizing that this amplitude decrease is caused by the damping force (primarily air resistance) as well as friction, some factor unique to our system Journal of the PGSS Wil-ber-force be with you: An Analysis of the Wilberforce Pendulum Page 303
must minimize these inhibitors in order to obtain the much longer durations of oscillation. A plausible explanation as to why this mechanism works so well is a balance in minimization of friction and air resistance. As the mass and moment of inertia are increased in a given system (such as in the relation between Pendulums #1 and #2), the relative effects of friction are decreased. In contrast, however, large masses and inertial moments also tend to increase the surface area upon which air resistance can take effect. Thus, according to such theory, an optimal middle ground of mass and inertia can be used to maximize the observational time of oscillation, minimizing the sum of damping and frictional effects.
Our theory is well-supported by the fact that the coupling interaction force of Pendulum #2, being in the 10-4 Newton range, is significantly smaller than both the coupling forces of the other two pendulums, which are in the order of 10-2 N, approximately one hundred times greater. Although coupling interactions are conservative and do not correspond directly to the damping and frictional forces, intuition suggests that greater interactions between the longitudinal and torsional modes of motion will naturally give rise to more resistance to move as well. Such a theory is by no means justified by this experiment, but the suggestion is supported and could probably be verified with extended research.
Finally, a brief section of error analysis can help provide grounds for modes of future extensions. In an abbreviated fashion, the major factor that must be considered is the degree of precision between the longitudinal and torsional measurements. While over 62,000 data points were taken in rapid succession by the DataStudio software and the motion sensor attachments, complimentary equipment for the torsional motion was not available for our usage. As a result, we were forced to use our senses for the rotational measurements of the experiment, which gave rise to much fewer significant figures and greater error. Although the mathematics behind the mechanics could be used to solve for these values, comparison with experimental data would not only increase the precision, but also provide further means of comparison between actual and experimental data. Such levels of analyses would cause our pendulum to display Wilberforce properties even closer to the ideal complete transfer of energy.
VII. Pendulum #3
A. Introduction
There are a variety of different styles of Wilberforce pendulums built around two basic principles: controlling the pendulum bob’s mass and varying its moment of inertia (fairly simple approach where the bob is easy to adjust) or operating at a relatively fixed moment of inertia and adjusting its mass. This is not to say that moment of inertia and mass are independent of one another, because they are related, but it is to say that it is possible to vary mass or moment of inertia without appreciably affecting the other. The bob with the fixed mass is by far the more common of the two, but examples of the latter type can be found. It is a bob of this more uncommon variety that we designed. This type was constructed because other groups were already in the process of designing or testing the more common variety of pendulum – one with a fixed mass and adjustable moments of inertia – and Page 304 Cook, Han, Jin, Menendez, Nataraj, Stein, Strom, Sunday, Yang thus a more unconventional approach was chosen, testing both types in case there might be a hidden advantage.
This sort of pendulum bob is generally constructed out of a number of concentric discs or washers stacked on top of one another. The very fact that they are discs makes their moment of inertia incredibly easy to calculate accurately, a fact that is untrue for many of the other materials available to us, especially since the sets of masses available in the lab are not perfect cylinders. The advantage to this design is that mass can be easily added if a threaded rod is connected to the longitudinal axis, and the total moment of inertia of the pendulum will not be varied appreciably; this invariability in the moment of inertia is in contrast to a pendulum with mass added to a transverse axis.
B. Procedure
A more mathematical rather than experimental approach was used to build the pendulum, an approach that allowed for the design of a pendulum bob of any mass for a given spring (though that mass is limited by the spring’s longitudinal spring constant). The equations for rotational (2) and longitudinal (1) period were set equal to each other and solved for the pendulum bob’s total moment of inertia (21):
M I M + sp I + sp = 33 (21) k δ
To solve this equation, several constants/data points (Appendix I) had to be found to satisfy equation (3), which would allow us to generate a number of total moments of inertia for various of pendulum bob masses. These data points included the intended spring’s rotational/longitudinal spring constants and the moments of inertia/masses of all of the parts of our pendulum in order to satisfy the equation for moment of inertia derived from the equations for finding rotational and longitudinal period.
We first determined our longitudinal spring constant – not only would this replace an unknown in (21), but it would also allow us to chose a mass of bob suitable for our spring. This was found via a simple experiment utilizing Hooke’s Law: F = -kx, where F is the force pulling on the spring, x is the negative displacement of the spring under that load, and k is the longitudinal spring constant, a measure of how much force is required to stretch the spring per unit displacement.
Our rotational spring constant was harder to find. In order to do this, the relationship between rotational frequency (ωθ ), moment of inertia ( I ), and the rotational spring constant (δ ) was utilized, as illustrated in equation (8). Journal of the PGSS Wil-ber-force be with you: An Analysis of the Wilberforce Pendulum Page 305
We found the rotational frequency with a one-kilogram pendulum bob, calculating the time it took for it to go through a full period. It was easy enough to find the total moment of inertia (approximating for the mass with a cylinder and using 1/3 of the spring’s calculated moment of inertia by using the equation for a hollow cylinder) and then to solve for the rotational spring constant from there. In order to solve for this, the measurement of the spring’s rotational frequency as well as certain dimensions of the bob must be substituted into the following equation derived from a substitution into (8) with the equation for moment of inertia of a solid cylinder:
ω 2 Mr 2 δ = θ (22) 2
With these values, it was possible to calculate the bob’s required moment of inertia at a variety of different masses. With those masses, it was possible to calculate the dimensions of a pendulum given certain materials (an aluminum disc to establish a moment of inertia and a 304 grade stainless steel rod to add weight along the longitudinal axis), plotting the effect of different disc radii on the length of the rod (which stayed at a constant radius). Using this information, we decided to design our pendulum bob to be 500 grams, something that we could build a bob for without taking a chance of overloading the spring nor constructing an extremely large pendulum bob.
Then, once we had selected 500 grams as an optimum mass to use for our spring, we plotted the effect of using discs of different radii on the length of their corresponding counterweight rod using the fact that moments of inertia are additive. We subtracted the disc’s moment of inertia from the total moment of inertia needed for the pendulum. Using the value for the required moment of inertia for the rod, we found the height of the rod needed to satisfy this value. However, this value does not take mass into account, so we checked our calculations against the mass of the required rod, searching for one that fulfilled the 500-gram requirement using the density of the rod material that we used. The optimum length that we found using this method was 0.2356 meters, a height found via equations (23) and (24) which describe the relationship between volume, density, and mass:
⋅ = MDV (23)
π 2 = MhDr (24)
Then, substituting (24) into the moment of inertia formula for a solid cylinder (5), we could solve for the height of the stainless steel rod that had the required moment of inertia as shown in (25):
2I = r (25) π 2 hDr )(
This rod was connected to the disc via an aluminum bolt. An aluminum bolt was chosen because it was known that the bolt’s head would protrude from the top of the disc, and so using less dense aluminum would have a lesser negative impact on our calculations. In fact, since it effectively Page 306 Cook, Han, Jin, Menendez, Nataraj, Stein, Strom, Sunday, Yang lengthened the pendulum bob, the added mass would compensate for the loss of stainless steel that was bored out for the hole needed to affix the disc to the rod. Also, drilling a small hole through the head of bolt allowed us to permanently attach the spring to the pendulum bob, stopping it from rotating free of the pendulum itself and losing energy as it did so.
However, after checking the pendulum’s rotational and longitudinal periods, it was determined that each longitudinal period was about 0.06 seconds shorter than each rotational period, a difference most likely caused by an error in calculating the pendulum’s rotational frequency. This was fairly easy to fix, though; to increase the longitudinal period and equalize the two frequencies, more mass needed to be added to the base of the pendulum (with as little an increase in moment of inertia as possible). This came about via the addition of a threaded rod and several nuts (six large (0.01662 kg/each) and five small (0.0320 kg/each), finally bringing the pendulum’s periods to equal out.
Now that the pendulum was in operation, the frequencies of the normal modes had to be determined in order to calculate the coupling constant, a measure of the interaction between two linked objects. The frequencies of the normal modes can be found by use of equations (26) and (27), since the beat frequency can be determined from a graph of the beat pattern, using the fact that frequency is the inverse of period:
ω ωω += B (26) 1 2
ω ωω −= B (27) 2 2
Once these frequencies are known, the coupling constant can be determined by equation (14):
2 2 1 −= ωωε 2 )( IM effeff (28)
The coupling constant of the pendulum was 6.935 x 10-4 N, which is much less than the forces in our scale, meaning that the two motions are weakly coupled. So, the rotational and longitudinal motions are not completely independent of each other but weakly related, since the spring also twists as it moves longitudinally. When the coupling is weak, the frequencies are almost identical, but not exactly so, producing the two normal modes and the beat pattern characteristic of a Wilberforce pendulum. The normal mode frequencies on our pendulum were found to be 3.916 and 3.837 radians per second, values that fit this description quite well.
E. Pendulum #3 Photographs Journal of the PGSS Wil-ber-force be with you: An Analysis of the Wilberforce Pendulum Page 307
Figure 7: Two normal modes combining as a beat pattern.
In Figure 7 the two normal modes of a Wilberforce pendulum combine to form a beat pattern due to a slight difference in frequency between the two modes. This beat pattern is responsible for the complete change between the two motions observed in the pendulum’s movement. However, in order for this to occur, the periods of those motions must be the same.
Small nuts (0.0032 kg each)
Threaded rod
(0.1145 m) Large nuts (0.01662 kg each)
Stainless steel rod (0.2356 m) Iron spring (.040 m)
Aluminum disk (0.04195 m radius)
Figure 8: Materials Page 308 Cook, Han, Jin, Menendez, Nataraj, Stein, Strom, Sunday, Yang
Figure 9: Bob of Pendulum 3
Beat Pattern of Pendulum #3
0.65
0.6
0.55
0.5 Displacement of Bob from Sensor (m)
0.45
0.4 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 Time (s)
Journal of the PGSS Wil-ber-force be with you: An Analysis of the Wilberforce Pendulum Page 309
Figure 10: Beat Pattern of Pendulum 3
In Figure 10, the y-axis is the vertical displacement of the pendulum, and the x-axis represents time. The vertical motion goes through periods of greatest and least amplitude. The points of maximum amplitude occur when the pendulum only oscillates up and down, while rotational motion is at a minimum. Likewise, the points of least amplitude represent the places where there is no up-and-down motion, only rotational. The beat period, as measured off the graph, is 79 ± 1 seconds.
VIII. Driving Mechanism
A. Introduction
In the idyllic haven of the physics classroom, most problems are considered in a world lacking any sort of force to oppose and object’s motion. If this were the case, there would be no need for pendulums to have driving mechanisms, since, if all the energy initially put into the system remained in the system, the pendulum would continue in motion for eternity. No pendulum, however, retains 100 percent of its energy. Energy is always lost to friction and air resistance. Due to this loss all pendulums require some mechanism by which the energy that has been lost to friction and air resistance can be replaced. The main purpose of this portion of the project is to construct a mechanism that will continually drive an oscillating pendulum.
B. Research
Before the construction of any sort of driving mechanism can be considered, existing designs for driving Foucault and torsion pendulums must be examined because a Wilberforce pendulum combines aspects of both these pendulums.
One design for driving a Foucault pendulum calls for the top of the pendulum to be attached to a speaker as shown in Figure 11. At the bottom of every swing the speaker pulls up two to three centimeters. The speaker then drops down to its previous height at the apex of its swing. When the speaker drops back to its former height, energy is added to the system, thus counteracting the negative forces of friction.
Page 310 Cook, Han, Jin, Menendez, Nataraj, Stein, Strom, Sunday, Yang
Figure 11: Driving a Foucault Pendulum with a Speaker
Only one means of driving a torsion pendulum was examined during this project. In this method, the top of the pendulum is attached to a chuck as shown in Figure 12. The chuck executes small oscillations by means of a lever and cam mechanism. The lever system consists of two circular discs, fastened one on top of the other such that the center of rotation of each disk is slightly off center with respect to the system as a whole. This results in the pendulum being forced to rotate in one direction and then the other at its natural frequency.
Figure 12: Driving a Torsional Pendulum
C. Mechanism Proposition
After completing research of how various types of pendulums are mechanically driven, it appears as though using some form of timed electromagnetic repulsion will be the best way to keep the Wilberforce pendulum in constant motion. Using an electromagnet or a solenoid to drive a Wilberforce pendulum is beneficial because such a circuit is easy to experiment with and readily adjustable where as using a purely mechanical device may be less efficient and harder to design. This is because from a purely mechanical approach, a very detailed construction plan will have to be drawn up and implemented before any sort of laboratory testing can occur.
By placing a solenoid underneath the system and an appropriate magnet at the base of the rotating mass, a magnetic force can be induced when the spring is at its maximum displacement as it approaches the solenoid, repelling the mass back upward while replacing the energy lost through friction and air resistance. Because the solenoid must be pulsed at appropriate intervals as to not interfere with the natural movement of the pendulum, some form of a sensor is critical for this mechanism to work. For example, a photogate may be place near the maximum displacement of the pendulum and worked into a circuit where the gate’s beam of light will act as a switch. When the beam of light in the photogate is broken, the circuit will close and the solenoid will pulse. As the spring recoils and the mass leaves the path of the beam, the circuit will disconnect again. Another Journal of the PGSS Wil-ber-force be with you: An Analysis of the Wilberforce Pendulum Page 311
option available to time the magnetic pulse includes a magnetic sensor. This device will detect the magnet at the base of the oscillating mass and will throw a switch closed when the mass is within a certain range. Regardless of the minor details essential to perfect such a circuit, this approach to driving the Wilberforce pendulum appears to be the most favorable idea to build upon.
D. Theory: Wire Solenoid
The basis for our proposed driving mechanism focuses on the use of a solenoid as the main repulsion force for the pendulum. A solenoid is a coiled wire that can act as a bar magnet and repel an approaching opposite charge.
In order to effectively incorporate a solenoid in this circuit, a Direct Current (D.C.) power supply must be used. Placing a solenoid in series with a D.C. power supply is beneficial because the current will always flow through the solenoid in a constant direction, therefore allowing the permanent magnets at the base of our mass to be oriented such that they will always be either repelled by or attracted to the solenoid.
Figure 13: An Electromagnetic Field around a Solenoid
E. Lab Testing: Part I
Using the previously proposed method for a driving mechanism, we began experimentation in the lab with an existing prototype of a Wilberforce pendulum along with an A.C. waveform generator as a power supply for the solenoid. We inserted a solid-state magnet onto the bottom of the pendulum and began testing by making adjustments to an A.C. waveform generator to find a frequency at which the pendulum naturally starts to oscillate. After testing multiple frequencies and observing no noticeable results, we decided to calculate the theoretical frequency of the pendulum, according equation (1).
The calculated frequency for the particular pendulum system used in this project is 0.946 Hz. When the A.C. waveform generator was set to imitate frequencies within this range, a small oscillation of the pendulum was observed, with maximum effects noted at 0.972 Hz. However, because the motion produced by the A.C. waveform generator was so minimal, it was decided that this piece of equipment did not put enough energy back into the system in order for it to be considered a useful method of driving a pendulum. Page 312 Cook, Han, Jin, Menendez, Nataraj, Stein, Strom, Sunday, Yang
After making little progress with the waveform generator as a power source for the solenoid, switching to a D.C. power supply seemed like the next move in continuing to test our proposed idea for a driving mechanism. Unlike alternating current, a direct current power source permits us to know which direction the magnetic field in the solenoid is pointing at all times. With this knowledge, we studied the repulsion and attraction of rare-earth magnets with the solenoid to determine which side repelled the force produced by the solenoid the most. This is the side of the magnet that would be best to place at the base of our mass in order to yield a maximum repulsion force. Next, we ran the D.C. power supply constantly with the magnets at the base of the pendulum and noted that the system moved more in a circular motion than anything else. What we did find, however, was that when we turned the power source on and off, the spring began to oscillate until eventually it reached its natural frequency. At this point, the pendulum began to transfer its longitudinal and rotational energy as desired, finally proving that a pulsating solenoid will indeed drive a Wilberforce pendulum.
Following this discovery, we ran a series of different test with the system while turning the power source on and off to try and help us decide what the timing and duration of each pulse should be. Turning on the power source every time the spring reaches its maximum displacement proves to be too frequent and sends the mass bouncing in an irregular circle rather than up and down; turning the power source on every other longitudinal period, however, does not interfere with the regular movements of the pendulum, but pulsing the solenoid every third longitudinal period does not put enough energy back into the system to keep the spring oscillating consistently. As for the duration of the pulse, the pendulum’s motion was found to be most consistent when turning on the electromagnet at the spring’s maximum displacement and then turning it off at its maximum compression. These observations will be very important to take into consideration when making the circuit to drive the Wilberforce pendulum. Journal of the PGSS Wil-ber-force be with you: An Analysis of the Wilberforce Pendulum Page 313
Figure 14: Basic Lab-Testing Set Up
F. Lab Testing: Part II
After discovering that a solenoid will actually drive a pendulum, finding a way to turn the solenoid off and on at appropriate intervals proved to be the next challenge of this project. To achieve this, we decided to construct an integrated circuit involving a photo resistor. When a laser beam is broken in front of a sensor, the resistance of this electronic component will drop almost to zero, letting current flow to the solenoid. However, when the beam of light to the photo resistor is not broken, the Page 314 Cook, Han, Jin, Menendez, Nataraj, Stein, Strom, Sunday, Yang resistance in the circuit will be so high that it will be like a switch in the circuit, preventing current flow. Using a 741 Light-Dark sensor and other electronic components, we built this circuit on a breadboard and made some adjustments so that it would include a D.C. power source and a solenoid. In addition, we used a voltmeter to check the continuity of our circuit and made appropriate modifications there after. Upon testing our driving mechanism for the first time with this sensor, we used a laser as a light source because the laser emits a focused and direct beam of light. However, multiple problems arose because of this light source. Since the laser beam was so exact, the pendulum did not always break the beam at appropriate times. As a result, the solenoid would pulse at irregular intervals, failing to keep the pendulum in the correct constant motion. Not only was the amplitude of the resulting motion not large enough, but also the mass at the base of the pendulum did not bounce up and down in a straight line, wavering instead in a circular motion. To try to solve this problem, we chose to replace the laser with an incandescent light bulb. This would hopefully create a larger area of light for the photo-resistor to monitor. This way, only when the main pendulum broke the beam, the solenoid would turn on. Constructing this switch had the results we hoped for, but did not perfect our driving mechanism completely. At this point, the main issue with both the laser and the incandescent bulb proved to be maintaining the natural motion of the Wilberforce pendulum. With both light sources, the pendulum demonstrated full longitudinal motion, but did not display the rotational motion we observed when experimentally pulsing the solenoid by hand. This problem may be a direct result of distance between the solenoid and the sensor, the amount of power going through the system, or the timing and duration of each pulse. Previously, it was observed that pulsing the solenoid with every other full displacement of the spring yielded the most natural results. However, incorporating this into our circuit is proving to be a challenge. Another complication that has been brought to our attention is that our solenoid is resisting the quick changes in current and is therefore delayed in repelling the mass. The D.C. power supply is also affected by these quick changing commands and has over heated, delaying experimentation. Our next step in perfecting our driving mechanism must focus on adjusting the ratio between power, pulse duration, and sensor distance from the solenoid. If we can achieve a balance between all of these factors, theoretically, our driving mechanism should work. Journal of the PGSS Wil-ber-force be with you: An Analysis of the Wilberforce Pendulum Page 315
Figure 15: The Solenoid and Sensor System
G. Lab Testing: Part III
The integrated circuit constructed to pulse a solenoid by using a photoresistor as a switch is comprised of various components. Refer to subsection I for a description of each piece of this circuit as seen in the schematic in Figure 32. The purpose of the Light Circuit is to sense the intensity of a light input and turn a solenoid on and off corresponding to the intensity of the light. When the pendulum is at the bottom of its vertical motion it will block the light source and thus trigger the solenoid being controlled by the circuit. Basically, when a light is striking the photo resistor, it has a very low resistance. There is therefore little difference between the voltages of the inputs at pins 2 and 3 of the op-amp. As an op-amp is a differential amplifier, it magnifies the difference in voltage between its two inputs. When there is little difference between the two inputs, the op-amp has a small output, and a relay remains open. When there is no light striking the photo resistor however, it has a very high resistance, resulting in a large difference between the inputs at pins 2 and 3 of the op-amp. The op-amp therefore has a large output, closing the relay, and turning on the solenoid. Page 316 Cook, Han, Jin, Menendez, Nataraj, Stein, Strom, Sunday, Yang
Figure 16: The Sensor Circuit
Figure 17: The Sensor Circuit
H. Material List
2 10K Resistors 2 470 Ω Resistors 1 1K Resistor Journal of the PGSS Wil-ber-force be with you: An Analysis of the Wilberforce Pendulum Page 317
1 10K Potentiometer 1 Photo Resistor 1 741 Op-Amp 1 2222A BJT 1 1N4001 Diode 1 Reed Relay 2 DC Power Supplies 1 Solenoid 1 Incandescent Bulb
I. Part Description
1. Resistor
The purpose of a resistor is to reduce the flow of current in a circuit. When added in series, it is possible to use Kirchoff’s rules along with Ohm’s law (V=IR) to show that the total resistance of a circuit is equal to the sums of the individual resistances. When added in parallel, it is once again possible to show by similar means that the reciprocal of the total resistance is equal to the sums of the reciprocals of the individual resistances. Another form of the resistor is the potentiometer, which is simply a variable resistor.
2. Capacitor
A capacitor is an electronic component that stores a charge between two oppositely charged plates. The largest charge that can be stored in a capacitor is equal to the capacitor’s capacitance multiplied by the voltage being applied to the capacitor.
3. Diode
A diode is utilized to effectively make the circuit a “one way street”. The diode allows current to flow in one direction while it resists all electron flow in the opposite direction.
4. BJT
A Bipolar Junction Transistor (BJT) is a specific type of transistor. Transistors are used primarily to either amplify a signal or as a switch.
5. Operational Amplifier
Operational Amplifier, abbreviated op-amp, is a differential amplifier that has a large voltage gain, high input impedance, and low output impedance. In other words an op-amp uses a differential in current from two inputs to amplify the voltage of one output.
6. Reed Relay Page 318 Cook, Han, Jin, Menendez, Nataraj, Stein, Strom, Sunday, Yang
A Reed Relay is an automatic switch that controls whether or not current can flow through a circuit. It is controlled by a coil, which becomes electromagnetic when current flows through it. When the coil has current flowing though it, the electromagnetic field pulls two metal plates together, connecting the circuit. When current ceases to flow through coil, the plates come apart, breaking the circuit.
J. Conclusion
While adjusting certain variables within the driving system, a set up to drive a pendulum in its normal mode was discovered. By placing the photo resistor sensor at the lowest point possible, the solenoid will turn on frequently enough to restore all lost energy. This is evident because the amplitude of the spring will remain constant with each longitudinal period of the pendulum. Due to the completion of a circuit driven solenoid, this portion of the project was successful in that we accomplished one of our goals, which was to find a way to restore lost energy in a pendulum system so that it will demonstrate constant motion.
Due to time constraints, we were unable to arrange this system so that it could drive a Wilberforce pendulum through its beat pattern motion. As we were adjusting the voltage and resistance through the circuit and the distances between the solenoid and the sensor, we discovered that regardless of the set up, the pendulum always lost so much energy in the rotational portion of its cycle that it never had enough energy left to transfer back into the longitudinal motion necessary to trip the sensor once again. The closer we moved the pendulum (when at rest) to the sensor, the more rotational movement was lost. The further apart we moved the resting pendulum and the sensor, the more longitudinal motion was lost. Finding a balance between these two variables is eventually what will be necessary to have a system that will drive a Wilberforce pendulum.
Part of the reason why we were able to drive a Wilberforce pendulum manually was because we could control our actions visually. Replicating exactly what we do by hand in a circuit would require a much more sophisticated circuit; one which time would not permit us to construct. If we could extend our project, we would try and build a circuit that could add more energy to the system when the light beam is first broken and less energy subsequent times that the beam is broken. This should then drive the pendulum while still not interfering with the beat pattern.
IX. General Discussion
A. Successes
This team project succeeded in constructing and analyzing three different working Wilberforce pendulums. We were able to both build the pendulum through a hands-on, trial and error approach, and also to build a pendulum based on the calculations and the theory behind the oscillations and periods. In addition, a working sensor circuit was built, and we were able to drive a pendulum through its normal mode. Journal of the PGSS Wil-ber-force be with you: An Analysis of the Wilberforce Pendulum Page 319
B. Difficulties
The only major difficulty encountered through this project was the construction of a driving mechanism that would be able to power a pendulum’s beat pattern. We sought to construct a driving mechanism that could power the pendulum through its entire beat period – through both purely rotational and purely longitudinal motion.
C. Conclusion
Summing up the team project, we managed to construct several Wilberforce pendulums and to study the theory behind their characteristic beat patterns and motion. Although we were unable to find a way to power the beat pattern of the pendulum, we did manage to power the pendulums’ normal mode (simultaneous longitudinal and rotational motion). The constructed pendulums were varied in dimensions, composition, and numerical characteristics (such as period), but all functioned correctly as Wilberforce pendulums. Finally, not only did we consider the pendulums to be Wilberforce pendulums due to their observable characteristic beat patterns, but also showed that they were Wilberforce pendulums from the mathematical calculations revealing that their rotational and longitudinal periods were equal.
X. Future Possibilities
With more time, a rotational sensor will be obtained to graph the rotational beat pattern in DataStudio so it can be at the same accuracy as the longitudinal beat pattern. Right now a driving mechanism can power a pendulum manually without losing energy and a driving mechanism can power a pendulum autonomously in the normal mode without losing energy. These two characteristics need to be combined so that an autonomous driving mechanism can power a Wilberforce pendulum to display longitudinal and rotational motion without losing amplitude.
After doing the calculations the conclusion has been reached that any shape or size of the bob is possible just as long as all the calculations line up for the longitudinal and rotational periods to be equal. Different shapes and materials can be tested and tweaked so that the two periods are equal. The entire system can also be put into different environments by varying the temperature and pressure. The pendulum would then need to be tweaked to adapt to these conditions. The system can also be immersed into different liquids to see the effects of added drag to the pendulum’s periods. A driving mechanism will be very useful here, as the pendulum will lose a lot more energy in a liquid.
XI. Acknowledgements
Amongst the variety of people who assisted in the success of the Wilberforce Team Project, special credit must be given to a handful of people who were central in the design-based, experimental, and analytical portions of the process. Page 320 Cook, Han, Jin, Menendez, Nataraj, Stein, Strom, Sunday, Yang
First and foremost, our thanks must be directed toward Ms. Michelle Hicks, our primary advisor, instructor, teacher, and friend. Her unmatched degree of effort was masked under a veil of distance to allow us to take the project in any desired direction, yet she was simultaneously very accessible, interactive, and helpful throughout the process.
A great deal of recognition must also be given to Howard Kim, a current student of Carnegie Mellon University and our Teaching Assistant. His patience and superior skills in education were accentuated by his perseverance to help, despite his battle against ailment during the final days of project completion.
A round of approbation is equally due for Gary “The Machinist,” a true mastermind in the hardware shop who was the driving force (no pun intended) in transforming our laundry lists of drawings and dimensions into tangible, working pendulums.
Equally deserving of appreciation, Mr. Eric Evarts contributed a sizable portion of time constructing the electromagnetic portions of this experiment. Indeed, his essential time facilitated the success of the challenging driving mechanism aspect to our project.
Expanding the horizon of assistance to those of indirect help, many thanks must be extended to Dr. Barry Luokkala, Ms. Maria Wilkin, the Residential Life Directors, and the entire Teaching Assistant staff for their direction and maintenance of the excellent program. We, the students, were blessed to be handed the top-notch facilities of Carnegie Mellon University under the guidance and care of those mentioned.
Finally, but by no means least importantly, our dearest gratitude must be conveyed to the Governor Edward Rendell for his funding of the Pennsylvania Governor’s School of Science, along with all others not mentioned for their collaboration to maintain such a wonderful foundation. We are indebted for the opportunity that was bestowed upon us, and have tried our very best through our work to fulfill the obligation of excellence.
XII. References
Bellis, Mary. All About Coil Springs
http://inventors.about.com/library/inventors/blsprings.htm
Berg. Richard E. and Marshall, Todd S. “Wilberforce pendulum oscillations and normal modes.” American Journal of Physics, 32-38, (January 1991).
http://scitation.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=AJPIAS000059000001 000032000001&idtype=cvips&prog=normal Journal of the PGSS Wil-ber-force be with you: An Analysis of the Wilberforce Pendulum Page 321
Chemistry Department, University of Illinois at Chicago. Techniques of Optical Spectroscopy in Analytical Chemistry
http://www.chem.uic.edu/tak/chem524/notes7/figureN_7.gif
Department of Physics, University of Toronto. The Wilberforce Pendulum http://faraday.physics.utoronto.ca/PHY182S/WilberforcePendulum.pdf
Eric Evarts, Graduate Student, Carnegie Mellon University
Frederick J. Bueche. Introduction to Physics for Scientist and Engineers. Page 95-96.
London’s Global University. Experiment M3: Study of an Oscillating Mechanical System Driven into Resonance
http://www.ucl.ac.uk/~ucapnsz/teachinglabs/2b40-m3.pdf
Malenbaum, Mike and Campbell, Peter J. Wilberforce Pendulum http://www.phy.davidson.edu/StuHome/pecampbell/Wilberforce/Setup%20and%20Procedure .htm
Marvin, Peter and Gray, Rachel. Physics & the Physics of Sound
http://ffden- 2.phys.uaf.edu/211.fall2000.web.projects/p%20marvin%20and%20r%20gray/page2.htm
Millersville University, Physics: Experiment of the Month http://muweb.millersville.edu/~physics/exp.of.the.month/34/index.html
“Moment of Inertia.” Hyperphysics
http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html
Thomas, H.A., M. Sc. “A New Relay and its Application to Sustaining Pendulum Vibrations”
http://www.iop.org/EJ/article/0950-7671/1/1/313/siv1i1p22.pdf
University of Guelph. Moment of Inertia. http://www.physics.uoguelph.ca/tutorials/torque/Q.torque.inertia.html
APPENDIX Journal of the PGSS Faculty Page 329
Christopher Borysenko Barry Luokkala, Director of PGSS Interdisciplinary Laboratories Dept. of Physics Carnegie Mellon University Carnegie Mellon University
Richard Butera Allison Marciszyn Dept. of Chemistry Dept. of Biological Sciences University of Pittsburgh Carnegie Mellon University
Benjamin Campbell Michael Picollelli Laser Technology Dept. of Mathematics Penn State Ectro-Optics Carnegie Mellon University
Jack Chen Gordon Rule Chemistry Laboratory Storeroom Dept. of Biological Sciences Carnegie Mellon University Carnegie Mellon University
Karl Crary Juan Schäffer Dept. of Computer Science Dept. of Mathematics Carnegie Mellon University Carnegie Mellon University
Carrie Doonan Robert Terwilliger Dept. of Biological Sciences Biology Instructor Carnegie Mellon University University of Pittsburgh and Carnegie Mellon University Mark Farrell Dept. of Natural Science & Engineering Patricia Zober Technology Physics Instructor Point Park University California, PA
Robert Frederking Dept. of Computer Science Carnegie Mellon University
Suzanne Gardner Chemistry Instructor Pittsburgh Public Schools
Barry Harris Dept. of Biological Sciences Ringgold High School
Michelle Hicks Dept. of Physics Carnegie Mellon University
Timothy Hoffman Dept. of Computer Science Carnegie Mellon University
Richard Holman Dept. of Physics Carnegie Mellon University
Journal of the PGSS Resident Life Staff Page 331
Resident Life Directors
Joseph Roberts Pennsylvania State University
Jennifer Zatorski Duquesne University
Teaching Assistants/Counselors
Ian Anderson Ryan Melnyk Physics Biology Carnegie Mellon University University of Pittsburgh
Laura Anzaldi Jessica Mink Chemistry Computer Science Duke University Carnegie Mellon University
Adnan Bashir John Paul Interdisciplinary Chemistry Pennsylvania State University New York University
Elizabeth DiCocco Jared Rinehimer Chemistry Physics Brown University University of Chicago
Jeremy Elser Seraphim Thornton Biology Mathematics Johns Hopkins University Pennsylvania State University
Pulak Goswami Leland Thorpe Mathematics Mathematics Carnegie Mellon University Carnegie Mellon University
Aaron Hernley Catherine Vinci Physics Chemistry Carnegie Mellon University Carnegie Mellon University
David Huston Eunice Wu Computer Science Biology Carnegie Mellon University Stanford University
Howard Kim Physics Carnegie Mellon University
Andrew McGuire Computer Science Carnegie Mellon University