How the Universe Was Made

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

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

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.

Raspberry Pi 3 Multi-Computer

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

Interference Inkjet Printing

Interference Inkjet Printing refers to the production of a color image using colorless ink and an inkjet printer. The color comes from the interference of reflected light, which has passed through thin layers of the ink deposited on the paper.

A recent paper (Jan-2016) by four authors at ITMO University, Saint Petersburg, Russia, describes their technique. Details at links below. Their innovations include the use of an inexpensive office inkjet printer, and an ink formulation which does not require high temperature or other special processing.

It’s impressive work. If this topic interests you, I think there are many opportunities for follow up studies.

Best wishes,
Ken Roberts
18-Feb-2016

Paper:
Aleksandr V. Yakovlev, Valentin A. Milichko, Vladirm V. Vinogradov, and Alexandr V. Vinogradov.
Inkjet Color Printing by Interference Nanostructures.
ACS Nano
http://pubs.acs.org/doi/abs/10.1021/acsnano.5b06074
Paper is unfortunately behind a paywall, but at least abstract has some detail, and the supplementary file with more tech detail is not paywalled:
http://pubs.acs.org/doi/suppl/10.1021/acsnano.5b06074

Paper info at Researchgate, including links to profiles of the four authors:
https://www.researchgate.net/publication/291815114

Press release about the technique:
http://tass.ru/en/science/857145

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How to Find Papers

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

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

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.

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

Snow Crystals 2

Imagine my surprise to see a mainstream media article about snow crystals! Here is the link.

https://www.rt.com/news/326738-designer-snowflakes-lab-california/

It describes the snow crystals grown by Ken Libbrecht in his Caltech lab, and has some great photos.

The text of the article is also worth a read.

Best wishes,
Ken R.
22-Dec-2015

Solar Cells 2

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.

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

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

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.

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

Copper and Bismuth

Today I returned to a book about Density Functional Theory, A Practical Introduction, 2009, by David Sholl and Janice Steckel.  In their first chapter, pp 4-5, they describe the successful use of DFT to determine the reason why copper, ordinarily maleable, becomes brittle when bismuth is added, at a very low level — less than 100 ppm.  Ditto with lead and mercury inclusions.  They describe the work of Schweinfest, Paxton and Finnis (not seen) which determined that the cause was the large bismuth atoms, migrating to grain boundaries within the copper, causing a lack of cohesion between grains.  The original paper by SPF is in Nature, vol 432 (2004), pg 1008 ff.

That reminds me of Goldschmid’s description of the diffusion of copper into bismuth telluride (Bi2Te3), sectrion 6.3, pp 86-87 of his Introduction to Thermoelectricity, 2010.  Copper diffuses very readily into Bi2Te3.  Hence copper contacts soldered to a Bi2Te3 thermoelectric element give poor performance.  Nickel is a better choice of contact material.

There is more to be studied along these lines.  For instance, gold atoms, are almost as large /heavy as lead, etc.  Yet I would expect gold, because of its electronic structure, to be very compatible with copper, and not drastically affect the maleability of copper.  Something to investigate, theoretically and experimentally.

Best wishes,
Ken Roberts,
24-Aug-2015

SciELO Electronic Library

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):

I like the pun in their name!

Best wishes,
Ken Roberts
31-July-2015

Leonardo’s Machines

There is an excellent exhibition of machines constructed from the drawings of Leonardo da Vinci. The exhibit is presently located in Lucca, Italy, at the Chiesa di San Cristoforo, on Via Fillungo, open daily 930 am to 730 pm. I originally saw this exhibit 2.5 years ago in Venice, and recently had another opportunity to view everything carefully. It is a marvelous experience, and if you should have the chance to study Leonardo’s machines, you will surely find it beneficial to your design activities.

The photo is of a self-propelled cart designed by Leonardo. It may have been designed for use in theatrical productions. The motive power is provided by two large spiral springs, wound inversely to one another, which turn the two drive wheels. There is an escapement mechanism which makes the motion smooth. A brake is provided. There is a small wheel for steering the cart. The entire assembly is very ingenious, and makes use of several of the mechanisms found elsewhere in Leonardo’s drawings. Reference folio 812r of the Codex Atlanticus.

The book “Leonardo’s Machines: Secrets and Inventions in the Da Vinci Codices”, edited by Mario Taddei and Edoardo Zanon, text by Domenico Laurenza, translated into English by Joan M. Reifsnyder, published in 2005 by Giunti Editore S.p.A., is closely related to the exhibition. If you cannot attend, perhaps you will have an opportunity to view this or another book.

The exhibition has a website: http://www.leonardoavenezia.com with some information. However, I noticeed only a brief mention of the Lucca exhibit at this website. I understand that the exhibit in Lucca is expected to remain there for several months. I am not certain that the Venice exhibition is continuing concurrently, but I believe that may be the case.

Best wishes,
Ken Roberts
04-May-2015

Transparent Metals 2

Recently I’ve been reading more about anharmonic properties of crystals, for instance metals. There are phenomena, such as thermal expansion, or electrical conductivity (resistivity), that appear to primarily be related to imperfections in the crystal structure, or to non-harmonic behaviour of the material. A material with a purely harmonic response should have no resistance, and should have no dimensional change when heated or cooled. Instead of thinking of the flow of electrons through a material, one might think of a wave function for the distributed gas of electrons.

This is somewhat prompted by a reading of Blakemore’s book on solid state, mentioned in my post about a year ago (April 20th, 2014) on the topic of transparent metals. I had occasion to return to Blakemore’s book again because of a new investigation. And he is as stimulating as ever!

So here’s the basic idea: Instead of thinking about the flow of electrons (as particles), think about propagation of a wave. That leads one to thinking about optics, and indexes of refraction, and the possibility of focusing etc. For instance, it is standard to talk about the speed with which electrons flow through a metal, and compare different models in terms of the wildly different speeds they predict. Do the three noble metals — copper, silver and gold — have the same speeds of electron flow? Otherwise said, would they have the same index of refraction? Can we make a copper-silver-gold alloy that can act as a lens or a waveguide?

Conversely, one can think of traditional wave models, and consider them in terms of particle flow models, eg the resistance (or conductivity) of glass.

Just some late-night idle musing. Maybe something will intrigue you.

Best wishes,
Ken Roberts
14-Mar-2015

Belgacem Paper — Lambert W Function

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

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

Bismuth-Telluride Antimony-Telluride Alloys

The past few days I attended a summer course on thermoelectric materials at the Univ of Toronto. A good course, and an opportunity to hear the thoughts of some of the best researchers in the field. A friend pointed out that there are two styles of summer course, one truly an introduction, the other more of a symposium for fellow researchers. I have to say that this course was the latter variety, and I missed most of the details, as I am not expert on making of or properties of specific materials. I did gain some understanding of a method of calculation of properties, which perhaps I will discuss in another post. Right now I want to talk about a specific material.

One of the presentations reminded me of what I believe is an open question regarding bismuth-telluride Bi2Te3 and antimony-telluride Sb2Te3 alloys. I mentioned the topic to a colleague at the course, and also had a chance to ask one of the lecturers, an expert, whether this was still an open question. I’m not an expert, but will set out my thoughts for possible exploration by others with suitable background and equipment.

Here’s the question. Consider the following graph, figure 1 from a paper by Mac Smith, et al, in 1962 (reference given below). It exhibits a difference between the two materials which can be used, for instance, to purify a mix by zone melting. Where the two curves are separated, if a bar of solid mix is moved thru a heater, one material will tend to migrate to one end of the bar, the other material to the other end of the bar. Zone melting is a well known technique for removing impurities from the middle of a bar of material by migrating the impurities to the ends.

However, in this case, the two curves come together when the molar ratio of Bi:Sb is either 1:2 or 2:1. Such a mix cannot be purified by zone melting. Moreover, if one has a bar whose gross contents of Bi:Sb is 1:2 or 2:1 (with Te in appropriate proportions so one is actually dealing with bismuth-telluride or antimony-telluride), the zone melting process should tend to average out the contents, producing a uniformly distributed alloy which might be described as Bi(6-x)Sb(x)Te9, where x is either 2 or 4.

This raises the question, in my mind, if what we are dealing with is not simply an alloy, ie a mixture of crystals of one material with another, each crystal being pure Bi2Te3 or pure Sb2Te3, but rather there is a crystal which has 2 or 4 Bismuth, 4 or 2 Antimony, and 9 Tellurium atoms. The “a” lattice constants of Bi2Te3 and Sb2Te3 are different, though there are subtleties — see the paper by Mac Smith, et al for details.

I’ve not read a large fraction of the literature on (Bi,Sb)-(te,Se) materials, but the few that I’ve seen that use a mixture of Bi and Sb with Te, have seemed to focus on mix ratios 1:3 or 3:1, not 1:2 or 2:1. Maybe there is an opportunity being missed? H. J. Goldsmid, in his 2009 book “Introduction to Thermoelectricity”, reproduces the Smith, et al, figure 1 on page 104, as figure 7.6, with the following remark: “The meeting of the liquidus and solidus curves at Sb2Te3 concentrations of one-third and two-thirds suggests that there may be some measure of ordering at these compositions.” When I asked the expert who had lectured at the summer course, he said that he believed that concentrations 1:2 and 2:1 had been determined to be optimal for thermoelectric figure of merit, but he was not aware of any work on the “measure of ordering” remark of Goldsmid’s. That was of course an off the cuff response, to a casual question, and there may have been additional work since 1962, or since 2009 — but at least it suggests that there is something to investigate. H. J. Goldsmid is the expert on bismuth-telluride and its relatives, and in 2009 had some 55 years experience investigating its properties. If he says something is a topic worth investigation, that is pretty good advice!

I would expect that, if there is a novel crystal structure, that its increased complexity of structure might be useful for improving its thermoelectric figure of merit — with the usual considerations of orientation, magnetic field etc.

My purpose with this post is simply to point out the opportunity. I’m sure that those with suitable training and experimental apparatus can, using more modern techniques, revisit the 1962 work of Mac Smith, et al, and obtain a better understanding. I’ve not done my own literature search. This is definitely a side-observation for me, not on my main path of exploration. If this topic interests you, I hope you have fun and perhaps find something useful.

Best wishes,
Ken Roberts
13-July-2014

The paper referenced is: Mac J. Smith, R. J. Knight, C. W. Spencer, “Properties of Bi2Te3-Sb2Te3 Alloys”, Journal of Applied Physics, vol 33, pp 2186-2190, 1962.

Thermoelectric Cooler 2

I’ve been running the thermoelectric cooler (prior post) steadily for a significant fraction of a day, and measuring its power consumption and the temperature difference. It uses about 1.0 kilowatt-hour per day of power (from the wall outlet, via convertor to 12V DC). That costs about 25 cents US. The temperature difference maintained is about 13.3-13.5 degrees C.

Perhaps I should elaborate on that 25 cents per kwh figure. When I take my electrical bill, and look at the power actually used (before markup of quantity by about 9 percent for transmission losses), and divide that into the total charge (inlcuding various service charges, etc), it is about 22 cents Canadian, or 20 cents US. However, a couple of decades ago when I was reading a lot about solar power alternatives, especially for off-grid situations, a figure was offered that if you were on-grid, local solar generation would be an economic break-even if the grid charge was about 25 cents per kwh. Considering capital costs, etc. Thus I’ve taken to using 25 cents/kwh as my guideline for evaluating the economic cost or value of a kilowatt hour of electricity, however produced.

There are always other considerations. But it’s nice to have a rule of thumb, so that one does not have to get into some complicated calculations when a simple Yes/Maybe/No question is being asked, and if the answer is Yes or No then the action to take is simple. It’s mostly the Maybe answers that require a further detailed investigation.

Best wishes,
Ken Roberts
28-Jun-2014