Monthly Archives: June 2014

Log Slice Table

Here’s a log slice table that’s pretty easy to make. What you need is a thick slice of some interesting wood — elm is a good choice as it has curious lines within it. And you need a router and a portable belt sander. Let the wood sit for a long time — years perhaps — to ensure it is thoroughly dried out inside. The next time you have a tree cut down, ask the workmen to save you 3-4 slices of the trunk, each slice about 15-20 cm thickness.

The second photo shows the log slice being flattened using the router. The wood box provides a sturdy frame with two flat rails, the slider for the router moves at right angles to that, and you draw the router back and forth to mill the log slice flat. The little wooden strips on top, alongside the router, keep the router from drifting so far to one side that you might gouge out the slider boards.

The router bit is one of the square-bottom profiles; I like to use a 1.2-cm bit, as a 1.8-cm bit chatters too much. Very small depth cuts, maybe 4-6 mm, are made on each traverse. After one level has been taken off, a thin board (foamboard or plywood) can be put under the log, to raise it up for the next round. You are essentially being your own milling machine, without the cost or complexity of a real machine. The tradeoff is your time — it will take several hours to get the log slice just right. I like to flip the slice over a couple of times, to ensure that each face is being milled parallel to the other, with no unwanted bevels at the edges.

Once the log slice is the right thickness with two parallel faces, I like to remove the bark around the edge, so it does not flake off later. I use a small tool with about a 1-mm diameter rounded blunt tip — originally intended for rug hooking, I think. Then sand the faces, and finish them with urethane. Both top and bottom, and the edges, so the log slice does not absorb moisture unevenly.

To attach the legs (which can be anything you find convenient and attractive), I make a plywood flat which can be screwed underneath the log slice, with its profile about 2-3 cm inside the profile of the base of the log slice. Screws are run down through that flat, into the legs. Then the flat is screwed to the base of the log slice. One final coat of urethane, including the legs, and the table is complete.

I’ve made three of these log slice tables in the past, for personal use and gifts, and the one I’m making presently will be the fourth, another gift. It’s been a really enjoyable activity. If you make one of these tables, I hope you have fun.

Perhaps I should mention cautions. Protective goggles and ear muffs, of course. More importantly, there are lots of wood chips flying about — you might want to consider a dust mask. And some woods (eg, black cherry) are poisonous when breathed in as sawdust. Check the woodworking websites if you’re in doubt about the wood which you’re working with. As with any power tool, know where each of your hands is positioned at all times when you are going to use the tool.

Best wishes,
Ken Roberts
29-Jun-2014

Thermoelectric Cooler 2

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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

Thermoelectric Cooler

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I recently had an opportunity to test a thermoelectric cooler, and am very happy with its operation. It maintains the contents of the refrigeration compartment at 7.0 degrees C when the room temperature is 20.1 degrees C. A difference of 13.1 degrees C. That is better than the stated capability of the cooler; the brochure says it can cool its contents 10 degrees C.

For the past couple of years, I’ve been involved with a thermoelectric materials working group, colleagues who are trying to develop improved thermoelectric materials. My particular role has been to work on certain mathematical and computational tasks which arise during our activities. Mostly, it’s quite abstract. I don’t usually encounter actual materials or thermoelectric modules. So, when I do get a chance to fiddle with some practical device, even in an “amateur” way, it’s rather an enjoyable diversion.

This particular cooler is intended to plug into a car’s 12 Volt DC socket (aka cigarette lighter). There is also an adapter, so that it can be run from the AC wall outlet when the car is not in use. For instance, when travelling, the cooler can be brought into the motel room overnight and run from the wall outlet.

One of the advantages of a thermoelectric cooler is that it is essentially vibrationless. There is a fan for this cooler, but it does not rely upon a compressor. Vibrationless coolers are particularly desirable for instance as wine cabinets, because vibration can disturb the sediments in the wine.

There is rapid advance in thermoelectric materials. But we are still at the infancy of this field. New materials and material preparation improvments are discovered frequently. The design configuation of thermoelectric modules in applications is still very rudimentary. Some of the subtleties, such as focusing of thermal radiation, have not been explored very much at all. I contrast the field of thermoelectricity, to the field of radio electronics. We are at the equivalent of the “crystal receiver” stage.

There is a tremendous opportunity to do interesting and useful work. There are only perhaps some 10,000 people working in the field, and the potential for energy efficiency improvements is such that the world will probably have 100,000 to 1 million practical engineers and other skills involved a decade or two hence. If one is looking for interesting work, over a long duration of increasingly sophisticated applications, consider thermoelectricity!

Best wishes,
Ken Roberts
27-Jun-2014

Zebras White with Black Stripes

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An old question is whether zebras are white with black stripes, or black with white stripes. Here is some evidence, that zebras are white with black stripes. A photo of a parent and baby zebra. You can see that the young zebra has brown stripes, changing to black as it matures, changing first on the front portion of its body.

My thanks to the bright youngster who, upon seeing this photo, immediately realized that it solved the old riddle.

Best wishes,
Ken Roberts
24-Jun-2014

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.

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

Sparrow Feathers

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This sparrow has a problem with his feathers — they are not hooked together properly. The flight feathers of a bird — other than down feathers which are fluffy and for warmth — have barbs which provide cross-attachments. Particularly in the tail of this bird, if you enlarge the photo, you can see what looks like raggedness.

This sparrow is one of several orphaned birds that was re-released near our house. Most thrive. This fellow can fly, but he is not as strong as others. Rather, he is strong, but he cannot fly long distances or with as much ease as other sparrows. We noticed him soaking his tail in the birdbath, which is perhaps a conscious behaviour to try to fix up his tail problem.

We are hoping that this little fellow will outgrow his feather problems and find a suitable niche hereabouts.

Here’s a link to the Wikipedia page about Feathers.
http://en.wikipedia.org/wiki/Feather

Best wishes,
Ken Roberts
22-Jun-2014

Luna Moth

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Today we found a Luna Moth outside, resting in the shade underneath the arm of a wooden chair. Here is a photo. The wingspan is about 10 cm.

The photo has been left fairly high-resolution so you can zoom in, if you wish, to examine the “eye spots”.

This moth is really neat to see. Here is a link to the Wikipedia page with details about luna moths.

http://en.wikipedia.org/wiki/Actias_luna

Best wishes,
Ken Roberts
18-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

Voting Day in Ontario

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Today June 12th is Voting Day in Ontario. Just a reminder that it is important to vote!

I have in the past been involved, in a minor way, in some of my party’s discussions about policies. In those talks I tend to emphasize the importance of listening to young people. Old people (like myself) are very outspoken, have the time to participate, have self-confidence and experience, and also… are often self-centered and even greedy. As well, we turn out to vote. The result can be policies over-weighted towards the interests of older people.

That is not good. There must be balance, in policies, in order to maintain balance in our society.

The offsetting factor, though, is that old people tend to be grandparents, or have some sort of feelings of nurture towards younger people. By the time one is a grandparent, one realizes that nurturing the young cannot be done just via one’s own descendents. One must think more widely. So, the way I have found to appeal to oldsters, to think of the needs of youngsters, is via the grandparent or uncle/aunt perspective.

I was most impressed by the response from one young person, when I asked her what was needed for young people in our society. She said that she was ok, being a student, etc, but that the real need was to help young people who did not have her advantages. That generosity is perhaps characteristic of young people, and it is different from what I might hear from most oldsters. So, young people will not necessarily advocate for their personal interest.

We must do it for them, our our roles as grandparents, uncles and aunts, in the broader social context.

Can you provide work? Guidance? Listen to someone? Please consider it.

Will young people turn out to vote in large numbers? If they are asked what they think, and we listen to them. Then they will speak.

Best wishes,
Ken Roberts
12-Jun-2014

Motion Mountain

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A new edition of Motion Mountain – The Adventure of Physics, the e-book by Christoph Schiller, is available for downloading. I first encountered this book about four years ago, and tremendously enjoyed browsing in it — until the pressure of coursework drew me away. This new edition has reminded me, and I’m enjoying it as much as before. Perhaps more, as I now bring a bit of background with physics concepts to the reading. The subtleties of Schiller’s discussion of can motion exist, is motion just a matter of perception, are enhanced by the close encounter I had four years ago with a philosophy of space-time course.

It is possible to get a good physics education online. The facts are all available. The only things missing are a framework for discussion with others (though the physics stack exchange and other websites can help) and access to the prior literature (though arxiv.org and archive.org and other websites help there also). Schiller’s open writing style, roughly described as speaking with one rather than at one, provides some of the feeling of having a discussion. And his e-book has numerous active links to throught-provoking resources.

Motion Mountain is free. The pdf files (one per volume, 6 volumes in all) can be downloaded directly from Schiller’s site, which is at the URL http://www.motionmountain.net/ This is a truly wonderful learning resource.

Best wishes,
Ken Roberts
10-Jun-2014

Spoon Chime

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Here’s a nice design for a wind chime, made from a fork and five spoons.

To start, use a small hand sledge to flatten the fork and the spoons. Drill a hole in the fork handle, and in the flat part of the fork before the tines branch out. Photos below. Use pliers to twist the tines out, horizontally, with a small bend up at the end of each tine. Drill holes in the spoon handles. Tie strings from the spoon handles to the fork, and from the fork handle to an S-hook or loop which the chime hangs from.

This string is ordinary cotton twine, stained by soaking it in tea. Tougher string is desirable, but I don’t have any right now. The knots have been secured against weather by painting them with urethane.

We’ve had one of the these chimes for years, a gift from a friend, and have made three others to give to friends.

Best wishes,
Ken Roberts
09-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

Conformal Mapping Website

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There’s a very nice conformal mapping book and collection of mappings online at
http://harald-dalichau.de — provided by Prof. Harald Dalichau. I’m finding it a useful resource for some conformal mapping explorations I’m presently doing. He has written a book, the complete text of which (in English or in German) is online. He also provides a comprehensive dictionary of mappings, with illustrations, and also Basic programs which do the relevant calculations. It’s quite helpful.

To get the book, etc, go to the home page above, click on Bucher, then click on one of the “Conf Mapping – Chapters” links on the left of the page, or the “Conf Mapping – Mappings” link for the mappings themselves.

Enjoy!

Best wishes,
Ken Roberts
06-Jun-2014

Graphene Ammonia Sensor

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There’s an interesting article recently published about using graphene, or carbon nanotubes, in ammonia sensors. The basic idea is that the presence of NH3 molecules changes the conductance of the graphene sheet or the carbon nanotube. The article develops an analytical model for the conductance, and compares it with experimental data for carbon nanotubes.

Here is the reference: Elnaz Akbari, Vijay K. Arora, et al (6 others), “An analytical approach to evaluate the performance of graphene and carbon nanotubes for NH3 gas sensor applications”, Beilstein Journal of Nanotechnology, 2014, vol 5, pp 726-734. URL is http://www.beilstein-journals.org/bjnano/content/pdf/2190-4286-5-85.pdf (free access).

The first two figures of that article are understandable for schoolchildren, and I plan to show them to some youngsters when we meet for our next science discussion. It will be a good visual introduction to the topic of how sensors work.

The article thereafter gets quite mathematical and detailed — too much math for the kids — but has a development of equations for conductance, using Fermi-Dirac integrals, that I wish to think about further. So this article is a pretty good find — it has something which will benefit and interest each of us.

Best wishes,
Ken Roberts
03-Jun-2014

Transcendental Equation Solution

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The equation $a\,e^{bx} + bx = c$ looks hard to solve. It’s said to be transcendental, ie not algebraic, like a polynomial equation. But the Lambert W function is the right tool for solving that equation exactly. Many transcendental equations are not hard to solve. This one is easy, if you have the right tools for the purpose.

The equation a*exp(bx) + bx = c above shows up in models of diodes or photovoltaic cells, in the relationship between current I and voltage V. If you solve for either I in terms of V, or for V in terms of I, you will end up with an equation of the above form. See the website http://pvpmc.org for some background about photocells and diodes, a circuit diagram for a diode equivalent of a photocell, and the Shockley model of the diode. Or there are many other resources.

You might think that equation above is a bit special, because the constant b appears in two places. But if one has some other coefficient in front of x, simply multiply the equation by a constant factor so you get bx both in the exponential and as the linear x term. And put the constant c on the right hand side. That is a very tidy way to write the equation relating a linear variable and its exponential, and that is the form which I will solve here.

Write y = exp(bx). Thus bx is log(y). Rewrite the equation as ay + log(y) = c.

We want log(ay) on the left, to match the ay term. So add log(a) to each side:

ay + log(y) + log(a) = c + log(a)

Since the sum of two logs is the log of the product, that becomes

ay + log(ay) = c + log(a)

Now take exponentials. We get

exp(ay)*ay = exp(c + log(a)) = a*exp(c)

The left hand side is in the form that defines the Lambert W function, namely w*exp(w) = z, which is the definition of w = LamW(z). So we have

ay = LamW(a*exp(c))

Divide by a, and recall that y is exp(bx), to get

exp(bx) = y = (1/a)*LamW(a*exp(c))

Now take logs, and divide by b, to get the solution for x:

x = (1/b)*log((1/a)*LamW(a*exp(c))

This solution, in its diode application, is used in modeling of diode and photocell arrays. It is very fast compared to having to find solutions by manual searching, because computer software is widely available to calculate the LamW function. (Just as there are sine, tangent, exponential, log, square root, and many other functions available in the software that we can use without having to write our own subroutines.)

Now, let’s think a bit more widely. If two diodes are in parallel, their currents add. That is like adding two Lambert W functions. If two diodes are in series, their currents satisfy a harmonic mean relationship, of the form 1/x1 + 1/x2 = 1/x. Will it surprise you that there is a LamW interpretation of that? Basically, one can make an analogue computer out of diodes, that will do arithmetic using LamW functions as a basis.

Recall the delay differential equation solution mentioned in a prior post? Also LamW. Actually, a sum of LamW evaluations, one per branch, if one is working in the complex plane. Like a Fourier series, but better because it incorporates delay. Delay of course happens in electronic circuits — more LamW interaction with analog computer ideas.

A tremendous amount of money is spent running particular programs on digital computers. Runs of models in density functional theory (DFT) or lattice quantum chromodynamics (QCD) can take days or weeks of supercomputer time. And the job is never done. Once a solution is obtained, one decides to use more points in the grid or lattice, and to use more digits of precision to deal with noise in the result arising from discrete approximations to nature.

Meanwhile, nature manages to compute quite well physically. Want to know the lattice constant for the Gold FCC crystal? Measure a chunk of the metal, determine its specific mass, and do a little arithmetic. Or use some sort of diffraction technique. Lots of ways to get the job done. Want the most cumbersome, slowest method? Use DFT. You will be lucky to choose an algorithm and discretization that gives you a value within 3 percent of reality. It is acceptable in some circles to get a rough idea, and even to use that rough idea to confirm a hypothesis about crystal structure. Maybe not for Gold — that is pretty well known — but for more complicated materials.

Nature, however, is not so tolerant. If you take a chunk of gold metal, and increase its lattice constant by 2 percent, it will melt! To get the lattice constant up by a couple of percent, you will have to add energy, and bring the metal up to about 1350-1400 Kelvin. Whereupon it melts. It’s not melting because of the temperature, not directly. Rather, at a lattice constant about 1.02 times room temperature lattice constant, gold can no longer hold its FCC crystal structure, and the atoms start to become mobile — what we call melting or liquid.

If we cannot answer all questions about nature using digital computers, maybe we should be giving nature a chance to answer some of those questions for us. Thus my fondness for analog computation, geometric solutions and such.

Best wishes,
Ken Roberts
02-Jun-2014