Category Archives: Math

How the Universe Was Made

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How the Universe Was Made:
Before the Big Bang … and After

by Ken Roberts
24-July-2017

One day God was working on his latest project. He was making a perfect golden sphere. The sphere was so round, that there was nothing rounder. It was so smooth, that there was nothing smoother. And it was so pure, made of gold using only one type of atom, that there was nothing purer or more perfect.

You may be wondering why I am speaking of God as He. Well, it is a constraint of the English language that I am using to tell this story. God does not have a gender; God is He, She, It and everything else. God is plant, animal, rock, water and air. And time — God is past, present, and future, if those can be said to exist before the Universe. We will simply have to do the best we can in order to tell this story, using metaphors and frail vocabulary.

So … God had made a perfect sphere. It was almost done. Only a name was needed. The sphere was so perfect that only one bit of information would be needed to describe the sphere. God would call it 1. Or maybe 0. God hadn’t yet decided.

1 would be a good name, because it suggests One-ness, a Uni-Verse, a Poem that says everything. 0 would be a good name because it suggests the roundness of a perfect sphere, and the syllable Om. Either way, one bit would suffice for the sphere’s name.

While thinking about his decision, God took a coffee break. There was no rush … Infinite time, actually. After preparing his coffee, God returned to his workbench, where he had made the sphere. He set his coffee cup down on the workbench, and turned to examining the sphere, making sure it was perfectly smooth and pure.

Well, we’re here, so you know what happened. As God was concentrating on the sphere, his elbow nudged the coffee cup, which spilled coffee all over God’s workbench. God grabbed for the coffee cup, and the sphere, without his full attention, rolled to the edge of the workbench and fell onto the floor.

There was a Big Bang ! The perfect sphere was smashed into millions of bits. Coffee dripped off the workbench and mixed into the breakage, and created little bits of dark coffee and light cream. That became empty space and stary galaxies, and everything else in the Universe. “Oops!”, said God.

What happened after the Big Bang? Well, God is probably working on another sphere. Or, perhaps, She has a new project.

Solar Cells and the Lambert W Function

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Computations for the one-diode model for solar cells, if done using the exact formula with Lambert W function, are likely to produce arithmetic overflow or underflow. That is a constraint on the ability to implement such calculations in Fortran or C, or on microcontrollers. The solution: use a coordinate transformation of the computation problem. If the problem were being solved on graph paper, the coordinate transformation would be achieved by using log-log graph paper.

I gave a talk at a recent conference, “Celebrating 20 years of the Lambert W function”. Title: Solar Cells and the Lambert W function. Joint work with my colleague S. R. Valluri. The slides are available at Researchgate, at this URL:

https://www.researchgate.net/publication/305991463

Best wishes,
Ken R.
11-Aug-2016

Non-Harmonic Fourier Series

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2016 is the 200th anniversary of the publication of Joseph Fourier’s ideas for the solution of heat conduction and radiation problems using trigonometric series expansions. What we now call Fourier series. His ideas appeared in book form in 1822, but they first appeared in 1816 in a paper Theorie de la Chaleur (Extrait) which describes the book’s contents. It is appropriate to return to Fourier’s work. And there are gems to be found.

Chapter 5 of his book, The Analytical Theory of Heat (in English translation by Alexander Freeman), discusses the conduction of heat in a solid sphere. Fourier obtains a sine series which solves the differential equation. However, his series is not a harmonic series of the form of a weighted sum of terms sin(k*x) where the k are positive integers. Rather, Fourier’s solution is what we now call a non-harmonic series. It is a weighted sum of terms sin(b*x) where the b values are positive reals, moving steadily out roughly as do the integers. What are those values of b ? They are the solutions to an equation of the form b*cotan(b) = B. Those basic modes can be summed in a linear combination to match other constraints of a particular problem.

We have seen the equation b*cotan(b) = B previously. It is the solution for the bound state energy levels in a quantum mechanics problem, the one-dimensional finite square well.

This looks like fun. Fourier’s solution is very clever. William Thomson (Kelvin) worked on this topic also — it is the subject of Thomson’s first published paper. There is plenty to explore.

Just a heads-up, for anyone else who may be interested in this topic.

Best wishes,
Ken Roberts
08-Aug-2016

ps. Fourier’s book was republished by Dover. It is also online via the archive.org website.

Finding Good Recommendations

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There are many online services which attempt to recommend movies, books, webpages etc that someone will like. In some implementations, recommendations are based upon finding new items that are liked by other users: If you like A, and someone else likes A and B, then B is perhaps a good recommendation for you. The difficulty is that establishing your profile can be a tedious task, as you have to initially indicate several items that you like. On the order of twenty items, perhaps.

A new algorithm, developed by Evgeny Frolov and Ivan Oseledets of the Skolkovo Institute, provides a much less time consuming, and likely more accurate, way of establishing your preferences. It uses information about items that you do not like, as well as about items that you like. Roughly stated, if you do not like item B, and those who like B also like C, then item C is perhaps not a good suggestion for you.

The details of their algorithm are subtle, and designed for efficient operation. It is not just graph searching. See Arxiv 1607.04228 for their paper — linked below — and a press release also linked below.

Best wishes,
Ken Roberts
01-Aug-2016


http://arxiv.org/abs/1607.04228
Evgeny Frolov and Ivan Oseledets — Fifty Shades of Ratings: How to Benefit from a Negative Feedback in Top-N Recommendations Tasks


Press Release — Skoltech scientists have created an algorithm that improves the quality of recommender systems

Raspberry Pi 3 Multi-Computer

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It now makes sense to use Raspberry Pi 3 boards to build a multi-computer for serious scientific work. In a previous post on 25-Jan-2016 I described my tests using a Raspberry Pi 2 for density functional theory (DFT) calculations using the ELK software. The RP 2 operates at about 1-10th the speed of a Laptop purchased for scientific work, the laptop having an Intel i5-core processor and 4 GB of memory. Details in the previous post. Bottom line: RP version 2 is not economic for building a multi-computer for the ELK DFT tasks.

However, I have now had an opportunity to test the Raspberry Pi version 3, released earlier this year. It is faster, and completed the ELK calculations in 43 minutes elapsed time, vs 79 minutes on the RP version 2, and vs 8 minutes on the i5-core laptop. That means it takes 5.5 RP3 boards to equal the capability of the laptop. Since each RP3 (board only) costs $35 Cdn from a typical supplier (or $45 if one includes a power adaptor, or $75 if one wants a case, cables, etc as well), one can set up a config of about 6 RP3 boards for something like $250. The laptop costs $300 as a refurb system.

There are of course complexities. One has to mount the RP3 boards somewhere — a bit of lumber should suffice for home brew. And six power supplies can probably be replaced by a single supply of larger amperage capacity. Ethernet cables are needed, and a switch, but most tinkerers have that sort of stuff around. Still, all considered, one can probably build a very nice multi-computer out of RP3 boards at about three-quarters the cost of an equivalent multi-computer based upon i5-core refurb laptops.

I’m not planning to actually build this multi-computer. I already have what I need for my calculational tasks. But it’s nice to see the Raspberry Pi become suitable for this serious calculation.

Best wishes,
Ken Roberts
05-May-2016

How to Find Papers

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A site named “ResearchGate” has become one of my tools for locating papers. The site is a bit annoying at times. Sort of like Facebook, for scholarly work. There is a continual stream of “helpful” emails, and popups asking one to consider a job posting, or add a publication, etc etc. Rather like a hyperactive and gossipy personal assistant. Also, there is a lack of confidentiality — if one “follows” a particular paper, to be apprised of future citations of it, all of one’s contacts are advised that one is following that paper. So if one wants to work under the radar, then ResearchGate is not the tool for you. Be advised!

However, on the plus side: ResearchGate is excellent as a way of finding out what a person has published, who has cited it, what that person has published, and so on. I recently was looking at a paper by Neil Turok, “On quantum tunneling in real time”, and wanted to find out if anyone had cited it. Web of Science / Knowledge said not. But Google Scholar reported some related papers, and one of them led me to a paper by Carl Bender and Daniel Hook which cites the Turok paper. And that, in turn, via Hook’s ResearchGate publications list, led me to the very interesting and stimulating paper by Bender and Hook, Arxiv 1011.0121, “Quantum tunneling as a classical anomaly”.

That enables me to return to a subject I’ve been interested in for some time, whether the complex plane tangencies of the Lambert W lines with the strength contours of a quantum well, represent sensitivities which have a physical implication — for example, whether one can devise a sensor which uses that tangency. The QWIP, quantum well infrared photodetector, which is found in night vision apparatus, is an example of such a sensor. In general, a sensor can be made by conditioning a quantum well device at or near a context which changes the number of bound states or a tunneling probability, and then allowing the environment to stress the sensor — changing the energy, changing a dimension, changing temperature, and so on. Finding the Bender and Hook papers, and a couple of Turok papers, offers the possibility of a new look at that topic.

Best wishes,
Ken Roberts
31-Jan-2016

ELK Software and Raspberry Pi

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ELK is a software package for density functional theory calculations. Raspberry Pi is an inexpensive computer. I was interested in whether the RasPi might be useful for ELK calculations. This post reports on some timing tests.

ELK is open source software, written in Fortran, available via the SourceForge link given below. ELK runs its calcs on multiple parallel processors. My typical config for using ELK is to run on a two-core or four-core processor such as an Intel i3, i5 or i7 based Laptop, with Linux (Slackware) as the operating system, and OpenMP as the task coordination mechanism. I usually run several jobs, with various parameter choices, exploring some material model. Each job might take from a few hours to a couple days, depending upon the fineness of the grid used for modelling. On a supercomputer cluster, I have in the past run up to 100 jobs concurrently, with a selection of parameter choices, in order to develop an understanding of how the parameters affect the calculation results for the material.

More recently I have been running jobs sequentially, using a more directed exploration of the parameter space vs model results. That latter approach allows for more interaction with the investigation as it progresses, and leads to better intuition. One of my beliefs is that the dynamic tension between improving calculation speed and improving model and math, leads to better understanding. That belief goes back decades, to some success I had in number theory problems by using the tension between calculation and description. To do the work faster, instead of using a faster computer, one can replace parts of the problem description with better math. Eventually, sometimes, the problem collapses into good math and a fairly rapid calculational model. I have a hope in the back of my wish list that certain DFT, QCD, and Molecular Dynamics problems will someday be discovered to have such simplifications.

Anyway, given that approach, I was interested in perhaps using the RasPi as an ELK machine, thereby leaving my main Linux laptop free for other work. The RasPi (couple of links below) has impressive specs, with the model 2B, the current product, having a quad-core processor, 1 GB memory, 8 GB to 32 GB of local disk (depending on the sD photo card installed) with net 4 GB to 28 GB of free storage for user files, an ethernet port, four usb ports for keyboard, mouse, etc, and … best of all … very modest cost. RasPi comes with Linux (Debian based) and a copy of Mathematica. Although I don’t use Mathematica at present, having “paid my dues” already by learning Maple, and not wishing to have to re-learn the quirks of a new tool unless necessary, the low cost access to Mathematica is a nice plus, of possible future benefit.

So, I ran some timing tests. The results in summary: ELK installs on RasPi with no difficulty (details below) and runs its post-install tests successfully. However, RasPi is slow in running ELK, taking 79 elapsed minutes to complete the collection of 18 test calculations. The “top” and “1” command sequence shows that all four of the processor threads were active, so the problem may lie with some other aspect — for example, speed of the sD card or the low (1 GB) memory resulting in less in-memory-buffer file access. Whatever. In contrast, my Intel i5-core based laptop (Lenovo ThinkPad T420 with disk replaced by a 1 TB drive, and Slackware installed instead of Windows) completes the test calculations in 8 minutes. With a 10x speed difference, the ELK approach is not cost effective. It might cost about $600 to get ten stripped down RasPi boards (with board itself, with only power and ethernet connected, and controlled via ssh login sessions), whereas the T420 laptop cost only $300 refurbished. So, I will stick with laptops. However, I look for the next generation of RasPi boards. I think it is an excellent product. There are other applications besides ELK. GROMACS, for molecular dynamics calculations, for instance, might be suitable for the RasPi. For the future.

Some details about the RasPi setup may be helpful to record, for others. The RasPi in Canada is available from canakit.com and one can find other suppliers elsewhere. I was pleased with the speed of delivery from Canakit. I got the full kit, eg with Wifi dongle, but one just needs a stripped down kit for second and subsequent RasPi boards. I replaced the 8 GB sD storage card with a 32 GB sD card, with the latest operating system version (Nov-2015) downloaded from the RasPi support website. The OS boots to an X-Windows screen, and one logs in as user “pi”. To get root access and a command line, I used “sudo passwd root” and set my own password on the root user. Then one can work a bit easier. The aptitude utility was used to install the software needed for ELK, including “aptitude xxx” commands where xxx was these in sequence: “update”, “search fortran”, “install gfortran”, “search openmp”, “install libblacs-openmpi1”, “search lapack”, “install liblapack3”, “search fft”, “install fftw3”. ELK was set up to use the GNU fortran compiler (gfortran), etc.

It is interesting that the ELK test runs produce some warning messages regarding IEEE floating point arithmetic errors: IEEE_INVALID_FLAG, IEEE_OVERFLOW_FLAG, IEEE_UNDERFLOW_FLAG, IEEE_DIVIDE_BY_ZERO. These exceptions are discussed in a post at SourceForge, URL given below. Although I found the specific recommendations of that post to be ineffective, the insight into cause, and why it “does not matter”, is useful — see the third comment in that post. It is something that can guide investigation. Running ELK on Slackware / Intel i5-core with gfortran does not produce those errors, though it produces identical test results. I think that is simply because floating point exception warnings have been turned off in the latter configuration.

That’s it about RasPi and ELK. Each device/program is worth one’s attention if it matches one’s objectives, as these do with my objectives. Neither is perfect, but they are understandable, affordable, open for improvement, and a net benefit for serious work.

URLs below.

Best wishes,
Ken Roberts
25-Jan-2016

For ELK density functional theory calculational software:
http://sourceforge.net/projects/elk/

For Raspbery Pi supply, and for community projects website:
https://www.element14.com/community/welcome
https://www.raspberrypi.org/

Discussion of ELK tests producing IEEE floating point exceptions:
http://sourceforge.net/p/elk/discussion/897820/thread/e87237ad/

[end]

Solar Cells 3

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In this post I’ll talk about the one diode model for solar cells. A solar cell produces a current because, under illumination, exciton pairs — electrons and holes — are produced, and before the electrons and holes can recombine, they flow towards and out through the contact terminals of the solar cell.

A solar cell can be described by a “lumped parameters” model. There is a current source, which represents the current which the solar cell’s material would produce, under standard illumination, in the absence of any losses due to reverse flows or contact resistance. The reverse flow, or recombination of excitons, is represented as a diode in parallel with a shunt resistance. There is also another resistance in the model, due to the contact resistance and other limitations on the flow of electrons to outside the solar cell. That latter resistance is modelled as a resistor in series with the current source / diode / shunt resistor triplet.

Here is a circuit diagram. It’s easier to see an unambiguous circuit than to follow the verbiage in the previous paragraph. The output current and voltage produced by the solar cell are I and V. These two related quantities are measurables. One actually has several values of I and V — for instance Isc the short circuit current (when V is zero), and Voc the open circuit voltage (when I is zero), both under standard illumination. As well one can obtain a value Idk, the dark current when there is no illumination and the solar cell has a bias voltage V applied. However, the other parameters in the circuit diagram are components of the model, to be determined by choosing those parameters to fit the actual I-V curve measurements (see characteristic curve in prior post) against the model. Rs is a series resistance, Rp is a shunt resistance, Iph is the photocurrent, and I0 and n are diode parameters — to be discussed in the next paragraph. Actually, given the current direction convention in the circuit, the output of the solar cell is -I not I. That is just a convention, but it accounts for the shape of the I-V curve in previous post, which looks like a letter J. Solar cells (good ones, at least) are described as having a J-shaped I-V curve, so I have adopted a current direction convention which makes the curve look like a letter J.

fig-1-diode

So, what about the diode? The standard diode model is called the Shockley model, and is described in many places on the net and in books. For an extended discussion focused on the particular context of solar cells see, for instance, “The Physics of Solar Cells”, by Jenny Nelson, chapters 1 and 6. The basic idea of a p-n junction diode is that current flows in one direction based upon the bias voltage across the diode, but it is offset by a thermal counterflow. The current through and voltage across the diode, say Idiode and V1, are related by the equation
Idiode = I0 [exp(q V1 / n k T) – 1]
where T denotes absolute temperature (degrees Kelvin), k denotes Boltzmann’s constant, and q is the magnitude of the electron charge.

There are two parameters in the Shockley diode model: I0 and n. The parameter n is called the “ideality factor”, and is 1 in an “ideal” diode, on the order of 1.05 for a real diode, and somewhere between 1 and 2 for a diode model used for a solar cell. As you can see, the assumption that a solar cell can be modelled using a diode is a bit of a stretch, but it is good enough for most practical purposes. It has the great advantage that there is lots of circuit modelling software for diode circuits. The parameter I0 is chosen to fit the data.

Now, one can write an equation to relate I and V for the one diode model of a solar cell. The voltage drop across the entire solar cell is V = I*Rs plus V1; that is, V1 = V – I*Rs. The current through the shunt resistance is V1/Rp. The current through the diode is Idiode as per the formula above. Putting everything together, and taking account of the sign conventions, one ends up with the equation
I = I0 [exp(q (V – I Rs) / n k T) + (V – I Rs)/Rp – Iph.
This equation is exact, at least insofar as the Shockley diode model and the other components in the one diode model are adequate as a description of actual solar cells. If one chooses a value of V, one can solve for the corresponding value of I. And vice versa. So the I-V curve corresponding to this equation can be drawn, and one can adjust the five parameters I0, Rs, Rp, n and Iph to fit the solar cell model to the actual I-V curve for a solar cell. (The parameters q and k are constants, and the parameter T reflects the working context of the solar cell, so they do not have to be adjusted to fit the model to the data.)

However, choosing a value of V and solving for I means solving an implicit equation. Likewise, choosing a value of I and solving for V also means solving an implicit equation. It would be nice to have an explicit equation, I = f(V) or V = f(I), either one. The difficulty is that I and V each appears both in linear terms and in exponential terms in the model equation above.

Those sorts of equations, involving exponentials and linear terms in a variable, often can be solved using the Lambert W function, and this equation is one which can be solved. The basic solutions were found by Jain and Kapoor in 2004 (see refs in working paper for the journal reference), and I will not bother to transcribe them here. J&K give explicit equations for both alternatives, I = f(V) and V = f(I). Because it is the V = f(I) form which turns out to be best for extending to the two diode model to be considered later, I will show the V = f(I) form here:
V = f(I) = I Rs + (I + Iph + I0) Rp
– (n k T / q) LamW((q / (n k T)) I0 Rp exp[(q / n k T) Rp (I + Iph + I0) ] )
where LamW() denotes the principal branch of the Lambert W function.

That’s a mess, I know. But at least it is an explicit expression. Given a value of I, one can calculate V. That calculation process is much less time-consuming than iterative refinement to solve an implicit equation of the form function(V,I) = 0.

However, there’s a problem. That particular formula V = f(I) can experience arithmetic overflow. And that’s where our working paper comes in. I’ll discuss some computational considerations, for the one-diode model, in the next post.

By the way … Merry Christmas !

Best wishes,
Ken Roberts
25-Dec-2015

Solar Cells 2

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A solar cell produces an electric current when its material surface is illuminated. In order to standardize the methods of measuring solar cell performance, and comparing different cells, certain conventions are adopted. For instance, standard illumination, which is the amount and wavelengths of the light falling on the solar cell’s surface.

A typical solar cell might have an area of 2.2 cm^2. Sometimes the current produced by a solar cell is quoted as so-many amps (or milliamps or microamps) under standard illumination. That statistic is appropriate when considering a particular solar cell or comparing two models of solar cells. Other times the current produced by a solar cell is quoted as a “current areal density”, that is, so many amps per square centimeter of solar cell material. That statistic is appropriate when considering a proposed solar cell material, or comparing two types of material or two choices of processing options for preparing solar cells using a particular material.

Solar cells are assembled into modules, which are groups of individual solar cells connected together (eg, in series) pre-packaged for easy handling. Modules are built into arrays, which make up solar panels; the panel includes the structural framework. Solar panels are the structures one notices on roofs and in fields.

The context of our recent work has been a mathematical model which is used for describing individual solar cells. As you can see there are many other aspects of solar cells and solar panels which are worth investigation (and have been investigated). Our work is just one part of the efforts of thousands.

I’m no expert on solar power. But I want to share the bits of information (or misunderstanding!) which I’ve gathered during my work. In the rest of this post I’ll describe some general aspects of solar cell performance. The details of the math model I’ll discuss in other posts — or you can go directly to the working paper at Researchgate if you want.

There are two measurements on the performance of a solar cell which are relatively easy to make: Isc = the short circuit current, and Voc = the open circuit voltage. Isc is determined by the amount of current the solar cell will push when it is under standard illumination and its contacts are shorted (no voltage difference); that current is largely determined by the series resistance within the solar cell. Voc is determined as the voltage difference between the contacts of the solar cell, again under standard illumination, when the load is missing — ie, an open circuit (no current flowing), or the voltage into an infinite load. Under other loads, there is a current less than Isc, and a voltage difference less than Voc.

The “I-V characteristic” for a solar cell is the curve which relates the current and the voltage. The graph shows a typical example, for a silicon solar cell manufactured about 3 decades ago. The current is shown as a negative number because of the convention used in the test circuit. What you can see in this curve is that this solar cell has Isc about 102 milliamps and Voc about 520 millivolts. The solar cell puts out a fairly constant current, between 102 milliamps and 95 milliamps, regardless of the load resistance, as long as the voltage drop does not exceed say 420 millivolts. That is, the load resistance can be up to say 420/95 = 4.3 ohms. As the load resistance goes higher, the solar cell’s current output drops rapidly towards zero. A load resistance of about 4.3 ohms, for this solar cell, is the “sweet spot” when the I*V product is a maximum. Power equals I*V (for direct current circuits like a solar cell), and hence that sweet spot is known as the “maximum power point” or MPP of the solar cell. The “fill factor” or FF of the solar cell is the ratio between the I*V product at the MPP, and the Isc*Voc product.

fig-jk1-blue-png

The maximum power point can shift if the illumination on the solar cell changes. For example, if the solar cell falls into shadow. The calculations to determine the MPP for an array of solar cells, for load balancing, can represent a lot of work. The MPP can be estimated by searching, varying the load slightly, but sometimes the power curve will have multiple peaks, and automated searching may not find the optimal operating conditions. It is important to have a good mathematical model for the I-V characteristic, in order to rapidly find the MPP under changed illumination. Further, that model should be suitable for implementation on very inexpensive microprocessors, such as may be used in a field installation. That’s where the method which S. R. Valluri and I recently described in our working paper is likely to be most useful.

That’s it for this post. More later!

Best wishes,
Ken R.
20-Dec-2015

Solar Cells 1

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This is the first of a series of posts I wish to make about solar cells. I am particularly interested in the diode models which are used to calculate the I-V characteristic curve of a solar cell. These models have an implicit equation relating current I and voltage V, and that implicit equation can be solved to give voltage as an explicit function of current, for instance. The solution uses the Lambert W function, which is what first led me to the topic of solar cell models.

The calculation of V = f(I) for a solar cell, using the exact formula, can be difficult using computer hardware arithmetic, such as in Fortran or C. Overflow of the arithmetic hardware may occur, because some intermediate numbers in the calculation are extraordinarily large. That poses a problem, since for many applications, such as load balancing of solar cell array panels, it is desirable to be able to perform solar cell calculations on inexpensive computer hardware, such as micro-controllers.

That situation led me in Spring 2015 to write a brief note (Arxiv 1504.01964) suggesting that a variant function y = g(x) = log(W(exp(x))), where W() is the principal branch of the Lambert W function, might be a better way to perform solar cell calculations. That article, though correct, was perhaps a bit terse. Recently S. R. Valluri and I have prepared a working paper which sets out the solar cell application of the y = g(x) function in detail, with example calculations for two actual silicon solar cells (one-diode model) and one actual organic solar cell (two-diode model).

Our working paper is now available on Researchgate at
https://www.researchgate.net/publication/287195509

My intent is, via a sequence of posts in this blog, to walk through that working paper. It may be enjoyable (for me, at least) to go through the material gradually, perhaps explaining some of the details which had to be condensed for the working paper itself.

Best wishes,
Ken Roberts
16-Dec-2015

Snow Crystals

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A snow crystal is a single crystal of snow, whereas a snow flake is a clump of snow crystals. Snow crystals show a six-sided symmetry, and often lie in a single plane. There is a book by W. A. Bentley, who photographed snow crystals for about five decades; his 2,400-some photos are in a 1931 book which has been republished by Dover. See the figure for an example, and see the end of this post for pointers to the book and some online resources.

snow-crystal

Why are snow crystals symmetrical? There are a couple of hypotheses. (A) One, from Ken Libbrecht at Caltech Physics, is this: “Branches begin to sprout from the six corners of the hexagon… Since the atmospheric conditions (eg, temperature and humidity) are nearly constant across the small crystal, the six budding arms all grow out at roughly the same rate.” Secondly, Libbrecht notes, symmetrical crystals are rare — irregular crystals are much more common.

Another hypothesis (B) is that snow crystals grow upon a charged core, and the charge promotes growth which fills in gaps in the structure. That is, symmetrical growth is a lower energy state, hence encouraged.

These two alternative hypotheses might be tested, by a statistical examination of snow crystal photographs. Consider two arms separated by 60 degrees; call that type 60 symmetry. Or consider two arms separated by 120 degrees; that is type 120 symmetry. Or, opposite arms, type 180 symmetry. If hypothesis A (external conditions) holds, we would expect the context to be roughly constant across the crystal; hence type 60, vs 120, vs 180 symmetry should be about the same across a population of crystals. If hypothesis B holds (charge migration, energy minimum) we would expect type 60 symmetry to be stronger than type 120 symmetry, and type 120 to be stronger than type 180.

So there is interesting work to be done. Opportunity beckons.

Here are some pointers to get you started…

Book: Snow Crystals, by W. A. Bentley and W. J. Humphreys, 1931, Dover reprint 1962. A beautiful book to browse.

Website: http://snowcrystals.com — this website was written by Ken Libbrecht of Caltech Physics, and links to his academic pages at Caltech.

Website: http://www.its.caltech.edu/~atomic/snowcrystals/faqs/faqs.htm — this web page has the hypothesis A which was quoted above, as well as other interesting info, and links to other comparable pages.

Best wishes,
Ken Roberts
01-Dec-2015

SciELO Electronic Library

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SciELO is a very good library of scientific publications, mostly from researchers in South American countries. It has articles in Spanish, Portuguese and English, and also has search engine interfaces in each of those languages. I’ve found it an excellent alternative resource for finding articles that do not show up in the customary search engines such as provided by my university’s libary catalogue. It contains about half a million articles.

The SciELO search engine is quite flexible. For instance, if one wants to find articles in Spanish or English, about diode (juntura) models which use the Lambert W function, one can use this search term: ((diode) OR (juntura)) AND (Lambert)

Here is a link to the English language search interface:
http://www.scielo.org/php/index.php

And a link to a Wikipedia article with some background about the project:
http://en.wikipedia.org/wiki/SciELO

And, so we have a nice picture to head up this post, here is the SciELO logo (of course, copyright owned by them):
scielo-logo

I like the pun in their name!

Best wishes,
Ken Roberts
31-July-2015

Belgacem Paper — Lambert W Function

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A recent paper by C. H. Belgacem deserves mention, as a good illustration of how to use the Lambert W function to explicitly solve a problem in semiconductor design.  Previous methods required iterative solution of a design equation.  That can be time-consuming and also has the disadvantage that there is no analytic formula for the solution of the design equation.

The paper is “Explicit Solution for Critical Thickness of Semicircular Misfit Dislocation Loops in Strained Semiconductor Heterostructures”, by Chokri Hadj Belgacem, published online 01-March-2015 in the journal “Silicon”.

I am particularly appreciative of Belgacem’s paper because of his reference 15, to a paper by Willams in 2005.  Williams paper was not previously familiar to me.  It has some really stimulating ideas.  I cannot usefully contribute to the discussions in Belgacem’s field of semiconductor design, but acknowledge his help in finding the discussions around the Williams paper.  That relates to the task of finding exact solutions of the Schrodinger equation.

Best wishes,
Ken Roberts
11-Mar-2015

Lane-Emden Equation

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The Lane-Emden equation is a second order differential equation which is used to model a spherically symmetric gas cloud, eg a stellar interior. Although it is a considerably simplified model (eg, no rotation), it still provides a good starting point. Traditionally the LE equation is written in terms of a linear independent variable, say x, which represents the relative radius of the gas cloud. The variable x goes from 0 (center of cloud) to 1 (outer boundary of cloud). With the assumption of a particular type of model of the gas cloud’s thermodynamics — called a polytropic model — one ends up with the LE equation. The model is characterized by a parameter n, called the polytropic index (n is a non-negative real number, not just an integer) which determines a function f (which depends upon the choice of polytropic index n), which in turn determines the density profile of the cloud, and thereby its other profiles, eg its mass profile.

It turns out, however, that the function f is an even function, and its power series expansion, for instance, involves only even powers of x. For instance, f(x) equals 1 – (1/6)*x^2 + (n/120)*x^4 + … and so on, with terms for x^6, x^8 etc. So one asks, whether it might be useful to write the LE equation (a second order differential equation) in terms not of independent variable x, but in terms of independent variable s = x^2. The variable s is the relative surface area of a spherical shell of relative radius x. At x=1 (outer boundary of the gas cloud), the value of s=1, ie relative area of that outer shell, is also 1.

There is a possibility for getting some physical insight from such a rewrite of the LE equation. The physics which is being modeled involves transport of energy and force between nested shells, and such transport may be conceptually more meaningful if it considered as a function of relative area rather than relative radius.

I have prepared a short pdf (2 pages) which describes such a rewrite of the Lane-Emden equation. Nothing new there; I’m sure such a rewrite has been done by others, as the LE equation has been a subject of study for over a century, and there is a considerable literature. Still, it may be of interest to others who are working their way into the details of the Lane-Emden equation and the stellar interior models for which it is a starting place. I give a few references, including to Chandrasekhar’s classic book on Stellar Interiors, which is still a better explanation than some of the more recent books, as it illustrates some motivations. And also a reference to a very interesting little paper by Klaus Rohe, who used Python to calculate rational expressions for the first 15 coefficients of a power series representation of the LE equation (ie, up to the x^28 term), as expressions in the polytropic index n. His paper is at Arxiv 1409.2008 if you want to go there directly.

Best wishes,
Ken Roberts
07-Mar-2015

Link to pdf file mentioned above, with my rewrite of Lane-Emden equation using relative area…
https://lasi2.wordpress.com/wp-content/uploads/2015/03/lane-emden-rewrite.pdf

Grey County Waterfalls

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Grey County, Ontario, Canada has many waterfalls. The terrain is rocky, trees changing from deciduous to cedars depending upon elevation, and water flows from the highlands, north or south. As a result, there are many pleasant places to walk among the woods, and waterfalls to encounter. Here is a photograph of (part of) one waterfall that I like especially. Unfortunately, I cannot remember whether this is Jones Falls or Walter’s Falls.

jones-falls-IGP9208

Here is a link to the county’s tourism info about waterfalls.
http://www.visitgrey.ca/travel-experiences/waterfalls-and-waterways/waterfall-tour/
The brochure linked from there is large, when printed, and the pdf file made available online does not have enough resolution to read all the descriptions and directions. The solution: if you visit Grey county, stop by one of the tourism booths and obtain a paper copy of the Waterfalls of Grey County brochure.

Water motion is very appealing to me. I like to stand along the edge of a placid shallow lake, and observe the formation of sand ripples. I consider the ability to model water motion one of the best tests of the capabilities of physics models. We are not very advanced at present. That is putting it politely. Turbulence, cavitation, foaming, and other dynamic behaviour of as “simple” a molecule as H2O, are still beyond our models. There is much opportunity for interesting work in future.

Best wishes,
Ken Roberts
23-Jun-2014

Look-Alikes

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I have been thinking about the quantum mechanics 1-dimensional potential barrier. The model supposes an incident particle beam (waveforms of some energy or momentum) encountering a square potential barrier, of certain height and thickness, and associated with that there is a reflected component and a transmited component. The entire system, as seen via a monoenergetic beam of particles of unit flux, can be described using two parameters, let’s say B and C, which are complex numbers, depending upon the energy (or momentum) and phase of the incident particle. That’s it … two complex numbers would describe the model situation entirely.

Suppose there were another potential barrier, not a square potential, that gave the same two complex numbers for the relected and transmitted waves. What then … are those two systems the “same” in nature? Do they look the same when tested with particles of only that certain energy, and will look different if we utilize a spread of various energies for incident particles? How many tests are desirable to distinguish one potential barrier from another?

There are situations in which nature forces us to extend our descriptions, because otherwise we reason towards inconsistencies in the mathematics. I wonder if we can have 1-dimensional models in nature? We might suppose that a flat layer of material, with the incident beam normal to that layer, would be close to a 1-dimensional model. But edge effects, and other realities of physical matter, must be considered as well. Perhaps all our models should utilize 3 spatial dimensions, or we get “non-sense” via the mathematics.

Wigner and Dirac 2

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Following the thread of Wigner and Dirac leads to some interesting places. This one, is a paper by Wigner in a book published as a tribute to honour Dirac. The book is Aspects of Quantum Theory, edited by Abdus Salam and E. P. Wigner, published in 1972. The paper, by Wigner, is “On the time-energy uncertainty relation”, chapter 14, pp 237-247.

Wigner presents some very interesting thoughts about the (delta-time)*(delta-Energy) variant of the Heisenberg uncertainty principle. This variant, in contrast to the more familiar (delta-space)*(delta-momentum) description, is not as crisp a concept. Some writers claim that (delta-time)*(delta-energy) is something entirely different. Others that (delta-this)*(delta-that) relationships having a minimum value is simply an artifact of Fourier transformations, with no inherent physical significance, aside from our way of perceiving the world. So Wigner’s perspective is significant as a contribution to that discussion.

Wigner’s thoughts on this topic are a bit aside from my present pursuit, which is related to some other ideas of Wigner’s, but this little paper is worth noting for future followup.

Every time I read Wigner’s writings, something fresh and interesting is discovered.

Best wishes,
Ken Roberts
15-Jun-2014

Wigner and Dirac

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I’ve been reading a brief sketch of Eugene Wigner’s life, written by Jagdish Mehra, in vol 1 of Wigner’s collected papers. Well written, and enjoyable. Great scientists come from various origins with various career intents. Wigner’s original plan was to be a chemical engineer. He took courses in physics and inorganic chemistry, and worked on the crystal structure of rhombic sulphur. His doctorate was in chemical engineering. All this study up to about 1925. By 1928, three joint papers with his friend John von Neumann, and in 1931, Wigner’s book on group theory and atomic spectra.

These years were a time of great intellectual ferment. Recently I ran across the 1928 volume of Zeitschrift for Physik which begins with four papers on the (new) Fermi-statistics (nowadays Fermi-Dirac statistics) version of the gas theory of electrons in metals, two papers by Arnold Sommerfeld and two by colleagues working with him. These years were also a time of great political turmoil; it is interesting how much scientific work was done in those times, despite the political disruptions. “Carry On and Keep Thinking”.

One other bit … Dirac married Wigner’s younger sister, Margit, in 1934. Many of the other marriages noted in Mehra’s biographical sketch of Wigner were between people with physics connections. It was a close-knit community of people with shared interests and outlooks upon the world.

My university library has only volume 1 of Wigner’s collected works (which seems to run to 5 volumes of scientific plus general articles), but even this single volume has much useful scientific information.

Best wishes,
Ken Roberts
14-Jun-2014

Nice Integral for Zeta

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Reading Feynman’s Statistical Mechanics book is a pleasure. There are gems to be found everywhere. Here is one little item, from page 37. You probably know this already, but it was nice to see the following explicit link between Fermi-Dirac integrals and the Riemann zeta function.

Consider the integral I = int(from 0 to +infinity) of (x / (exp(x) + 1)) dx. Here it is in Latex, which will give a nice picture at the top of this post: I=\int_0^\infty{\frac{x\,dx}{e^x+1}} However, I will stick with plain text for the math in the remainder of this post. It is simpler to write plaintext math fast, and you can make the necessary translations to math symbolisms.

Write x/(exp(x)+1)=(x*exp(-x))/(1+exp(-x)) and expand that as a power series in w=exp(-x) as x*w-x*w^2+x*w^3-… which is, because w=exp(-x), simply an alternating sum of terms of the form x*exp(-n*x) for some n=1,2,3,… Those terms are to be integrated from 0 to +infinity to determine the value of I.

The integral of x*exp(-n*x)*dx over [0,+infinity] can be determined by integration by parts, differentiating u=x and integrating dv=exp(-n*x)*dx, to obtain 1/n^2. Thus I equals the alternating sum 1-1/2^2+1/3^2-1/4^2+…

You probably already recognize that. But if not, then let J=1+1/2^2+1/3^2+… which is the Riemann zeta function zeta(2) and (proof we owe to Euler in the first instance) equals pi^2/6. J/4=J/2^s equals 1/2^2+1/4^2+1/6^2+… so I=J-J/2. Hence I equals pi^2/12.

Very tidy. Familiar to you very likely. I suppose in some sense, previously familiar to me. But it was nice to encounter the Riemann zeta function in a discussion of a physical topic, ie the statistical mechanics of a Fermi gas. There are interesting connections between number theory and physics. This is only one indication of links.

I have read that Euler worked for ten years on the proof that zeta(2) equals pi^2/6. It was a famous problem, and there was numerical evidence, but a proof was not found — though some of the best minds (eg, the Bernoullis) were working on the challenge. Euler made his attempt, and probably revisited the problem from time to time, and ten years after he first encountered it, came up with his method of solution. Moral of this anecdote: keep working away at a problem, revisit it occasionally, and don’t be discouraged if you are not getting a solution rapidly.

Best wishes,
Ken Roberts
07-Jun-2014