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Started by Hasdeo.2013.05.31
This is Moya's logbook for Nano Japan project from 2012.6.2-2012.7.31.
* Daily schedule [#daily]
- 09:00-10:00 Discussion (1)
- 10:00-12:00 Solve the problem and write the progress on pukiwiki
- 12:00-13:30 Lunch
- 13:30-14:30 Discussion (2)
- 14:30-16:30 Solve the problem and write the progress on pukiwiki
- 16:30-17:30 e-mail report and problems
* Goal of the project. [#goal]
- To describe the thermal conductivity of Nanotubes and graphene. The output will be an animation of atomic vibration and its propagation in Mathematica
- Keywords: Thermal conductivity, carbon nanotubes, Differential equation, Fourier Transform, Solid-state physics, Mathematica
* Schedule for discussion [#discussion]
|Time and Day|Mon|Tue|Wed|Thu|Fri|
* Division of tutors [#v0d44b28]
|Name|Lesson to teach|Misc|
|Hasdeo|Phonon in Nanotubes| How to make a presentation, and Japanese class|
|Tatsumi|Nanotubes| Printer and Copy machine|
|Thomas|Physical Mathematics| Library|
|Mizuno|Physical Computation| Computer in Lab|
|Nugraha|Phonon in Nanotubes| |
- Tutorial takes place in Coffee room, since visitor room will be occupied with many people
* Questions and Answers [#QA]
- This section is for posting questions from Moya-san and answers from other group members.
- Please list here with some simple reasons or details.
- For every problem, give a tag double asterisks (**) in the code so that it will appear in the table of contents.
- For the answer, give a tag triple asterisks (***) in the code below the problem in order to make a proper alignment.
- List from new to old.
* Report: [#d6196c2a]
** June 3: [#k93c5c39]
Today, Hasdeo-san picked me up from Urban Castle Kawauchi (UCK) with Thomas-san. I learned how to get to the lab by campus bus and have been introduced to the lab, Saito-sensei and the secretaries. Hadeo-san taught me about coupled oscillators, where I learned how to solve for the motion of two masses connected by springs. The goal is to model heat in CNT's as atoms vibrating with springs. Thomas-san also lectured me and taught me some math. He taught me how to solve the 2-dimensional Laplacian using separation of variables. He also provided some insight on the applications solving the laplacian. So far, everyone seems very nice and I am very excited to start working on thermal conductivity, which was a surprise.
** June 4: [#p63feebc]
Today, I walked from Urban Castle Kawauchi to the lab and managed to find my way, but I might have strayed a bit! I came early to skype my family back home, but began my lessons with Tatsumi-san at 9:00. He taught me about the graphene and carbon nanotube lattice and how vectors compose the unit lattice to form the Chiral Vector and Translational Vector. He also showed me formulas which describe the nanotube lattice. For homework, I was assigned to build a carbon nanotube out of paper and I have already drawn the Chiral Vector, but have not finished the job as I need to borrow scissors and tape. I also spoke with Saito-sensei and discussed the oscillator between two walls. I have a much better understanding of this situation and Saito-sensei assigned me to work on a similar model including a drag force. I believe I understand this problem, but am not to sure about the graphs which show resonance.
After lunch, Mizuno-san introduced me to Mathematica. I do not understand it that well, but for homework I am to research it online and learn different functions. I also spent some time playing with the functions and getting the hang of working with differential equations and plotting. I am also supposed to work on an animation, but this seems rather difficult at the time. Saito-sense also showed me how to calculate different aspects of the CNT today. I learned a lot from our meeting and am very excited to work on the problems he sets up for me. I am to work on a time dependent oscillating function for homework tonight and hopefully I can get this one right and continue to expand.
** June 5: [#gf3952b1]
I walked to the lab again today and met with Mizuno-san again to discuss Mathematica. He taught me about Rules and Patterns in programming and showed me several functions in mathematica. For homework I am to study graphics and attempt to make a figure of a particle between springs. I am also to study animation and manipulation, so I can modify this graphic. These are not very easy tasks as I do not know much about programming and I plan on looking for material to help me on mathematica. I have not begun the homework yet as I met with Saito-sensei to discuss more on a particle between two springs with an applied force. I believe I am starting to get the hang of solving these differential equations as I am beginning to recall what I learned last semester in my Differential Equations class. Homework from Saito-sensei involved me recalculating the solutions to the equations and plotting them on mathematica. I think I have successfully plottted them, but plan to ask Saito-sensei if it is a correct graph. I will try to solve the differential equation of the particle with friction and later try to derive the calculations for the properties of the CNT that Saito-sensei showed me yesterday.
After lunch, I worked on the plot of the solution on mathematica and went online to learn different functions of mathematica. I learned some shortcuts in typing and how to edit graphics, but I still have no idea how to make good graphics or animations.I met with Tatsumi-san and he explained some basic CNT properties like chiralities and metallic/semiconductor conditions. After this we discussed relationships between energy and temperature and energy and wavelength. We then went on to attempt to solve the Heat Equation. I understood the steps to solve the heat equation up until we had to solve for the constants. Solving for the constants required some ugly integrals with summations and trigonometry tricks that I do not know. I hope to look into this more when I go home and get a better understanding of this. Afterwards I continued to study the motion of the particle and work on its plot in Mathematica. I showed this plot to Saito-sensei and he explained the physical interpretation of the graph which dealt with beats and periods. he also went on about the imaginary part of damped oscillations. I have some homework for tonight on the imaginary solution as the imaginary part is key to understanding the transformation of heat in these situations.
** June 6: [#e24a9280]
Today I arrived at the lab early and worked on solving the damped oscillator equation and plotting the solution in mathematica. I successfully plotted the solution and was able manipulate the variables, so I could study the physical event. However, Saito-sensei would later explain this graph in more detail giving me a much better understanding. I meant with Thomas-san and he had me solve the 3-D Laplacian with boundary equations. This problem worked into a real ugly system of equations, but I understood the concept behind it. I continued to reteach myself some differential equations with Thomas-san and he later explained his presentation to me. He taught me about the solving the Laplacian in spherical coordinates, but I couldn't quite understand it and I plan on looking into it again. I then attended to 'group meeting' and I got to learn from Thomas-san's and Nugraha-san's presentations.I learned about the tough standards Saito-sensei has for presentations and a little about CNT's that Nugraha-san talked about. Later on, Saito-sensei explained the damped oscillator and has assigned me to study a coupled oscillator. I think I successfully modeled the non-damped homogeneous solution, but I have had some trouble solving the driven equation and plan to ask Hasdeo-san or Saito-sensei for help later. The goal is still to understand the transfer of energy through the springs.
** June 7: [#wae9b44d]
Today I met with Nugraha-san and he helped me solve the driven coupled oscillator that Saito-sensei assigned me. He taught me a better trial solution to use which let me understand the phase difference. However, I think we set up our differential equations wrong and I plan on solving them again. Nugraha-san also introduced infinite coupled oscillators. We have not yet introduced a force to them, but have worked on solving the differential equations. He explained the wavevector and lent me a book to that will help explain what I learned. He assigned me to study an infinite series of different coupled masses. I plan to work on this over the weekend and show it to him when I have time. I met with Saito-sensei as well and he showed me the error in our differential equations for the coupled oscillator and introduced some concepts behind the wavevector and optical/acoustic phonons.
** June 10: [#a77f1b0a]
Today,Hasdeo-san helped me finish the homework that Nugraha-san had assigned. This was to solve for the displacement of particles (coupled by springs) in an infinite lattice where the lattice cell is constructed of two particles of different mass as opposed to one. We found the solutions to the system of differential equations and he introduced the concept of zone boundaries and which phonons are acoustic or optical. I understand the math behind these conclusions, but am still rather confused by the physical meaning of frequency as a function of the wavevector. After discussing the acoustic and optical modes, Hasdeo-san introduced the model of thermal conductivity for a crystal lattice. He taught me how the transfer of energy is related to the phase shift in the two particles. I still do not completely understand this concept and plan on studying it more. After lunch I went with Thomas-san and he introduced the laplacian in polar coordinates. We had some trouble converting the initial definition, but I asked Saito-sensei for helped and he helped me work my way through. With some help from the internet, I understand the process of converting it and will begin to studying solving the differential equation. Hasdeo-san also came to my room again and discussed the thermal conductivity again and how it relates to the velocity of the center of mass. For homework, I am to study how heat transfer relates to the phase of different atoms in the atomic lattice.
** June 11: [#ib956120]
Today Tatsumi-san was absent, but I managed to work on the problems that were assigned to me yesterday. First I worked on plotting the dispersion plot where frequency is a function of wavevector for a two atom periodic lattice. While I attained the right plot, I am still rather lost behind the meaning of the plot. I am still confused as to the strict definition of a phonon and how the modes are determined at various k and w values. After achieving this graph, I began to work on Hasdeo-san's homework which consisted of calculating a constant which describes heat transfer in the lattice. I managed to calculate this lattice,but am still confused as to how the relationship is attained. I managed to plot the relationship of the constant vs frequency and learned the characteristics of this relationship. However, I do have some questions regarding the imaginary part of the solution and what it means for the conductivity to approach 'negative infinity'. Next I want to attempt to solve for this constant with forces at both walls and then to work on achieving this constant for a periodic lattice. Afterwards, I met with Mizuno-san and he taught me some more about Mathematica. I learned about rules and different functions like Flatten, Transpose, Real, and Refine. These different functions can be really useful for manipulating data in Mathematica. He also helped me work on the periodic lattice problem I had been considering. I also met with Saito-sensei today and he forced me to present what I have learned. He was pretty picky, but I think I did rather well for coming up with it on the spot and barely understanding it myself. I wish I could have talked to Saito-sensei more about science, but I understand the importance of presenting what I know. For tomorrow, I am to work on a presentation of a single atom infinite lattice and learn about boltzmann's constant and heat transfer. I also want to ask Hasdeo-san about the meaning of the velocity of the center of mass and how it relates to energy.
**June 12: [#s4419843]
Today, I met with Mizuno-san and he continued to show me different functions of Mathematica. I learned different input methods for basic functions and how to handle matrices and vectors.I also learned how to manipulate solutions of differential equations with NDSolve. For homework, I was assigned to look up Graphics and understand them. I looked up different graphics and have began understanding the various how to use lines, polygons, circles and how to edit their style. I first want to make a solid graphic of a single oscillator and then I want to try to animate it. I hope to ask Mizuno-san for help on animating the object. Afterwards, I met with Saito-sensei and he introduced the relationship of energy to our model of springs. He showed me various laws and equations that relate energy, motion, heat, and conductivity. Today, I will study and attempt to understand these various equations.
I continued to work on diffusion equations and I have a strong understanding of their derivation and physical implications. I continued to work on understand these and began to research the relationships between phonon modes, energy, and temperature. I stumbled upon the Debye and Einstein models, which Saito-sensei later encouraged me to look at. They involve some difficult math and physics I don't quite understand, but I may need some information that it has to offer. I met with Hasdeo-san to ask him various questions about the wave vector, energy equations and heat relationships. I now have a better understanding of the wave vector and the concept of phonons in solid materials. I also met with Saito-sense again and he has advised to attempt to solve the Heat equation with certain boundary conditions and to study the Debye model.
** June 13: [#zbf07e54]
Today I met met with Thomas-san in the morning and we worked on solving the 2D polar laplacian. I understand what we are trying to do and can solve the equation in a general method, but am having trouble with the Fourier series solution. I borrowed Thomas-san's book on Fourier series and plan to study the solution to this differential equation. On my own time, I attempted to study some basic statistical mechanics, but am having a decent amount of trouble with the subject and plan on asking Hasdeo-san and Nugraha-san for help on it tomorrow. I particularly stuck on the derivation of Boltzman's factor and the partition function. I want to understand these topics so that I can understand the Debye Model which provides a relationship between phonon modes, heat, and energy. Hasdeo-san has provided me the assignment of learning to derive the distribution functions, but I am having trouble with this as well. I hope to ask Hasdeo-san for help tomorrow and hopefully I can receive a better understanding of the subject.
** June 14: [#r357f51e]
Today I met with Nugraha-san in he morning and he introduced the expansion of coupled oscillators to containing N oscillators. He showed me how to solve for the frequency and amplitudes of movement for the oscillators. In order to solve them, I was instructed to use Mathematica. I have attempted to solve the matrix, but am having some trouble with obtaining the final solution. The program provides several solutions, that I don't quite understand. I also found the relationship of N coupled oscillators with a damping constant in the springs. This was not very difficult, but I still cannot solve the matrix in mathematica, which I will have to learn to do.
** June 17: [#cf8661e7]
I spent the first part of my day discussing my project with Hasdeo-san. We talked over the basic ideas behind the project and tried developing a simple model describing energy transfer in a system of oscillators. We left the room confused as we could not understand how to relate the temperature of the wall vibrations and force of the wall. After lunch I met with Hasdeo-san again and we may have created a simple way to observe heat transfer. This method consists of comparing the amplitudes of vibration to energy for specific 'points'. These points are atoms in the lattice which may provide a general structure for a plot fit. We plan to use the general solution to the heat equation to fit the data. I will want to use Mathematica to generate the exact solution. I will begin working on this simple model right now and plan to extend this model to multiple atoms later. I also want to consider using phonons in the model in time.
I met with Thomas-san today and we worked on general solutions and fourier series of the heat equation.
** June 18: [#oe7b407b]
Tatsumi-san wasn't here to lecture me, so I worked on my project on my own. My initial idea with Hasdeo-san has hit a road block as the 'spring constant' appears in the relationship of the amplitudes. I do not know if there is a way to solve for this constant, or any assumptions I can make to eliminate the constant. I will have to talk with Hasdeo-san later to see if we can find a way to complete the relationship between the amplitudes of movement. I am also interested in determining the frequency of movement of the wall. We are assuming the wall vibrates at a set frequency, but is there anyway to determine frequency? I also studied the heat equation and using Fourier series to find a solution. I also continued to study phonons and have began looking into the quantum nature of phonons, but this topic seems to be to difficult to allow easy application in our model.
In the afternoon, Mizuno-san taught me more about mathematica. He showed me how to work with graphics and animations and we created a small animation of the 'balls between springs'. This animation is not that great, but its a start! I think I understand the process very well and he has also taught me several functions as well. I also continued to work on my project, and talked to Hasdeo-san about several problems I had. I think we have a running model, but I will have to input values for the 'spring constant' and the 'frequency of vibration'. I also found errors in some of my derivations, so I will have to find these relationships again. I plan to work on finishing a plot tomorrow and to continue expanding the idea to many atoms.
** June 19: [#f92b7927]
This morning I meant with Mizuno-san and we continued to work on the animation of the atoms between springs. We managed to get a realistic animation for the case with several atoms, but are now trying to extend the model to multiple atoms. After showing me how to use Tables, and Block, Mizuno-san assigned me to generate a code that would produce the animation for 'n' atoms. I've been working on this for a while and have hit a problem where I cannot assign the equations of motion to the animated objects. I will need to ask Mizuno-san for help later on.
I spent the rest of my day working on my project. I used mathematica to try to solve for 'lambda' which represents the thermal diffusivity of the solid. I managed to solve for constants representing the general solution of the heat equation for my two atom system, but lambda is both negative and imaginary. I do not know if this makes physical sense and will need to ask Hasdeo-san for his opinion on the matter. Hasdeo-san also showed me some software on flex that may help fitting a plot to the points I obtained.
** June 20: [#keffb9a7]
This morning I began looking into Fourier transforms as they keep appearing in the quantum definition of phonons. I met with Thomas-san and we worked through some differential equations using Fourier series and we derived the fourier transform for the Gaussian distribution. I understand that Fourier transforms are used to make a time dependent function, only dependent on the frequency of a periodic function, but I need to look into the derivation of fourier transforms and their application physics. However, Thomas-san showed me some interesting applications that Fourier Transforms have in imaging.
I spent the afternoon continuing to work on my project. Hasdeo-san and I came to the conclusion that Boltzmann's constant was to small and affecting the calculations on my computer, so we placed Boltzmann's constant=1. This should hopefully keep the values on a larger magnitude which makes it easier to relate. I used this value and obtained a much better list plot of the three points which seems to make physical sense, but I cannot find an acceptable best fit line using the general solution to the heat equation. I keep receiving graphs whose data is way off point. I think I may need may points and I may have to take the time dependence of x along with time to use more points for a better fit. However, I would not know how to fit this into Mathematica and may need to ask for help. I believe that these points are correct, but the general solution of the heat solution is not the correct line that we should be fitting to. Hasdeo-san also began teaching me about Carbon Nanotubes and he reviewed the basic structure of the CNT and began teaching me about energy dispersion and the relationship to the brillioun zone. I hope to learn more about the carbon nanotube and to understand phonons in the CNT so I can later model it.
** June 21: [#t0eb9e69]
Today Nugraha-san was busy with his thesis, so I continued to study various things on my own. I continued to study the brillioun zone and phonons in CNT's. I have begun to understand the process of analyzing phonons in CNT's but am a little lost when the math shows up. I will need to ask Hasdeo-san to explain some details to me. Nugraha-san also brought some sheets for me to read that taught me about modes of vibration in solid materials. In the afternoon, I met with Hasdeo-san and we continued to discuss the project. I told him how I had doubt in our heat solution and we checked it to find out the solution was in fact wrong. We found a new solution that I would use for a fit later. When I worked on my own, the fit worked much better and realistic lambda had been found!
** June 24: [#f62b8cf4]
Today I met with Hasdeo-san and we continued to talk about my project. I showed him the lambda that I had calculated using our new heat solution. We continued to analyze my mathematica program to make sure all details were taken accounted for and we discussed certain points of the model. These points included talking about 'complex temperature' and 'complex energy' which may not be reasonable. We also noticed that 'lambda' (our constant) changed with time. Since this constant may be dynamic, Hasdeo-san has assigned me to find the 'constant's' dependence on time. I will do this by trying to find multiple values at various t's and plotting for a dependence. I hope that we can find a simple relationship between the values. I have also been assigned to work on a presentation and to begin studying phonons in CNT's and graphene.
In the afternoon, I went with Thomas-san to go a library card and had Hasdeo-san connect me to the printer. I then began working on my powerpoint and got several introductory slides done. I then worked on my project some more. I managed to obtain a plot of \lambda vs time. There is a small deviation in the values over time, but most of the values are contained in a small (about 1 unit) area. Hasdeo-san wants me to make an animation where the plot will change with time accordingly, so we can monitor the time dependence. He also gave me a paper to read that discusses thermal properties of carbon nanotubes. I am also continuing to look into phonons in carbon nanotubes. I also would like to begin expanding the simple model to multiple atoms and then to carbon nanotubes in time.
**June 25: [#p83a0ed3]
Today, I spent my morning working on my powerpoint. I managed to finish all the slides, but will need to add graphs, tables, etc from the project I have been working on.I also continued to look up different details about phonons in graphene and carbon nanotubes. However, I am still a little stuck on some math parts that I do not understand. I will need to ask Hasdeo-san or someone to explain these concepts to me.
In the afternoon I worked with Mizuno-san on animations of connected atoms. We managed to create an animation on Mathematica of compounded oscillators. I used these skills to create different animations and graphs for my powerpoint. These graphs include simple pictures of the two atom system along with a '1D' object view that lets me label different point, which should make explaining facts simpler.
**June 26: [#k5e87d6a]
This morning I went with Mizuno-san to continue working on our animation of 'n' atoms connected by springs. The animation is working out pretty well, except that we have to chose our constants right so that the graph works out pretty. He also showed me some functions that will help me later on and began showing me how to analyze the motion of the ith atom in the series. We hope to translate the motion of the ith atom to energy in time. Afterwards, I continued to work on my powerpoint presentation and continued to study different things about the carbon nanotubes.
In the afternoon, I continued working on the powerpoint and went to Hasdeo-san for help in the afternoon. He showed me zone folding in the carbon nanotube and how phonons relate to raman spectroscopy. He then reviewed my powerpoint and I continued to study carbon nanotubes. Before I left I finished up m presentation.
**June 27 [#wbda2962]
This morning I went with Thomas-san and we began studying thermodynamic relations in crystal lattices. We looked at different relationships, and began to analyze a gas model that relates conductivity to mean free distance, etc. I then practiced my powerpoint afterwards and gave the presentation afterwards. I think I did a decent job, but it was very relax because no Professors were there and it was only the students. After my presentation, we discussed my project and what would be the next steps for getting there. I listened to the grad students discuss different topics from types of transport to how to relate heat to vibrational frequencies without quantum mechanics. We could not come to a simple answer and decided to separate to continue studying. I spent my time afterwards studying quantum transport and have a decent understanding, but will need to ask Hasdeo-san for help on specific topics later on. I also continued to study phonons in CNT's and zone folding. I think I'm beginning to understand teh derivation of phonons in CNT's, but I do not quite understand their physical relevance. For the time being, I want to continue studying quantum transport in materials.
** June 28 [#s3c962b1]
This morning, I met with Nugraha-san and he showed me some various programs on 'flex'. He gave me access to a file name 'phonons' which allows me to plot and visualize phonons in various nanotubes. This will probably be a very useful plot in time as I have continued to study phonons in CNTs. Nugraha-san also downloaded some plotting programs on my computer and explained how low energy phonons would be involved with my project. Using 'flex' is still a little complicated and I will need to practice with it more. Afterwards I have continued to study phonons in CNT's and the Boltzman Distribution Theory which we think may be useful in our heat transport equations.
In the afternoon, I went with Hasdeo-san to continue discussing how our simple model may be applied to carbon nanotubes. We are continuing to study phonons relationship to heat transfer. We has some trouble with the derivation of phonons as a free gas, but I think I figured it out the derivation later, but I do not understand how it may be applied to our situation. I understand the derivation of the thermal conductivity of phonons treated as a free gas, but I do not know if their reaction with the structure should be taken into account. I want to continue to study different transport methods being used today. Hasdeo-san also gave me some papers to read which discussed the modelling of phonons to calculate thermal conductivity. They seem to have successfully modeled thermal conductivity using phonons and the Landauer energy flux. I plan to study the Landauer energy flux equation as I do not quite understand what it says in that I do not understand the importance of its components. In this paper, the conclusion is taht thermal conductivity is quantized in the carbon nanotubes which is really interesting. I plan to continue studying transport models especially the boltzman distribution theory and Landauer's Energy Flux.
** July 1: [#i1e4ed4a]
This morning I met with Saito-sensei in the morning to show him the small model that we had began working on. He asked many questions and gave Hasdeo-san and I many new points to consider and things to try in our simple model. After our discussion, I was to reevaluate some minor points in model like the calculation of energy in the model along with observing the real and imaginary parts of position of the central atom. I managed to do this fairly quickly and have a decent graph which represents our new model, but I plan on expanding this model to include a customary frequency, but cannot find a way to animate the motion. I will have to ask Hasdeo-san or Mizuno-san on help for helping the loop. For the afternoon, I plan to work on adding a third atom to the simple model and later including the model to include 'n' atoms. I understand the process of achieving this goal, but I will have to find an appropriate way to program this challenge.
This afternoon I met with Thomas-san where we discuss the importance of transport theories in my project. We researched different papers and began to study intoductory transport equations, but many look difficult and we doubt their application in my project. Later, I continued to work on my project with Hasdeo-san and I believe we developed a decent model that describes three atoms fairly well. I think I have found a way to determine the thermal conductivity, but the graph oscillates with different magnitudes that depend on the frequency. I believe that when I add more atoms, the constant will approach smaller amplitudes. I must also look into the 'area' to appropriately solve for thermal conductivity.
** July 10: [#i4110eb5]
Today is my first day back from the Mid-Program meeting which came at a very decent time to help relax me and get me ready for work. I feel like I am a little behind schedule with little to present, but I think I found some answers to my questions that I left with. In order to find the thermal conductivity, I will attempt to take the time derivative of energy for a particular atom and relate it to the spatial derivative after I find an approximate equation for T(x). I need to discuss this with Hasdeo-san as I believe this method will work, but I will need some help accomplishing the programming as I am having some difficulty taking the respective derivatives. I will also have to look into the best way to approximate T(x). I also have to deal with the problem of parameters and all my functions expolding to infinity, but Mizuno-san gave me an idea that may cure this problem. This idea involves adjusting the time scale to see an appropriate relationship. I also want to check my damping factors as they seem to make little sense. I think the damping factors may in fact be a variable as well that we will have to solve for. For now, I need to find their relative magnitudes for successful implementation. Mizuno-san also spent some time helping me find the derivatives to these functions, but it was to little success. This afternoon, Kono-sensei will be visiting and I hope I can put up a decent conversation of my research as I am a little bit scared!
This afternoon, Kono-sensei was late and we had to rush to meet him at his hotel and then for dinner with the Sendai Crew. However, before I left I managed to meet with Hasdeo-san and workout a lot of kinks in our problem. We managed to fix the infinity problem and attained strong graphs of our data and I later managed to obtain a value for 'lambda' by using a heat equation solution. Some things I want to check is whether or not this solution can still be used and if this is the correct way of approaching the problem. I also want to work on a program that can determine 'lambda' for many t's and hopefully in time many different 'frequencies'. I have begun to wonder if all frequencies must be taken into account and if I can account for this in this simple model using mathematica.