S04E17: The Toast Derivation

Tonight Sheldon told us a story.  But by knowing a little physics you can try to adapt it to profit from gold.

 

Can we use physics to make gold more valuable?

Over 2000 years ago, Archimedes was challenged to find out if a crown fabricated for the King of Syracuse was 100% gold, as paid for.    Gold was the densest and most valuable element known to ancients.  So, the story goes,  Archimedes knew that of all known substances, pure gold would displace the least amount of water than any other substance of equal weight known to man.    A given volume of gold, has a mass nearly twenty times (19.3) that of water.

Had the crown been alloyed with silver, a less valuable metal, the same mass would displace more water.   If the goldsmith had replaced only 10% of the crown’s mass with silver, he stood to keep a lot of valuable gold for himself.  But by volume silver only has a mass of ten times (10.5)  that of water.  The crown would displace 10% more water than the same weight of pure gold.

Now in modern times, gold is no longer the densest metal.  We could choose other materials more dense than gold to fool the king, or whoever is buying our gold.   For example, osmium is the densest known element found in the Earth’s crust, but was only discovered in 1802 and was unknown to Archimedes.  It displaces 10% less water than the same mass of gold.

By mixing an alloy of osmium and silver  and gold (or some other metal) you can remake your jewelry with precisely the same density as gold.   Then go down to your local motel where someone is buying gold and cash in.     But there’s a catch.   Osmium is one of the least abundant elements in the Earth’s crust.  It costs almost as much as gold.     Given that Osmium is rarer than gold, maybe you are better off holding onto the osmium anyway.

Artificial elements have an even higher density than osmium so maybe they will work a little better.  Mix in a little plutonium into your gold and you are good to go.    Except that plutonium is one of the  most toxic substance known to man.   Ingest even a small dose and you will die of radiation poisoning.   Try to store more than about 10o ounces for this purpose in one place–if you could even find it– and you have a critical mass that will produce enough neutrons to kill you.

Don't bother alloying plutonium into your gold.

Unfortunately all the dense artificial elements are radioactive.  If they weren’t then they would be found naturally.

Besides all of this would be illegal, or at least fraudulent, so is unsportsmanlike.   If you represent your jewelery as pure gold, or some purity gold, and it is not, you are just stealing.  You may as well not go to so much trouble and just become a burglar.

All of which leads me to a potential honest and legal version of this manipulation.    The atoms of the basic elements are labeled by how many protons they have in their nuclei.  You can add neutrons to their central nucleus and leave their chemical properties essentially unchanged.  Elements with different numbers of neutrons are called isotopes.  For example, iron has many isotopes, but they are all legally, chemically, and  truly iron.

So what about gold?  Why not take the  gold reserves in Fort Knox and the Manhattan Federal Reserve Bank of New York to Oak Ridge National Laboratory and irradiate them with a neutron flux?   As the gold atoms absorb neutrons they would become heavier, and more valuable.   The weight of gold in atomic units is 197 and adding even one neutron would make it 198.   That’s a 0.5% increase in value in the 8000 tons of U.S . gold reserves alone.    A rate of 0.5% is not a bad overnight return 250 billion dollars.  This achieves a version of the dream of the alchemists: turning gold into gold.

Meanwhile the U.S. national debt is 14 trillion dollars.  Our money supply (M2) is 8 trillion. Compared to that,  a quarter trillion dollars of gold is loose change.   The US holdings in gold could never back our currency 100% and would not even fill an Olympic-sized swimming pool.  To those who lament leaving the gold standard, as someone we know might say:

“Your argument is not with the Federal Reserve System, it is with basic mathematics.”

The idea of turning gold into gold could net the US a billion dollars overnight.   I’d be happy with just a 1% commission for the idea.   But again, there’s a catch.  Unfortunately gold only has one stable isotope.  No other isotope of gold has a half-life of longer than an hour, not enough time to sell it.  Meanwhile some of those neutrons would turn gold into lesser valued elements such as bismuth.

And that’s the nightmare of the alchemists: turning  gold into bismuth.

S04E16: The Cohabitation Formulation

There are times when television can be a public service.  On tonight’s whiteboards, the heroes of The Big Bang Theory have derived what to do in case our civilization  is attacked by zombies.

A few weeks ago, during the taping of  the differential equations episode, one of the Warner Brothers executives asked me, “What good are differential equations?”

So my theoretical physicist guest and I  eagerly began a discourse on all the wonderful mathematics of differential equations, but we could tell he was somehow unconvinced.  But then we told him about a recent well-known academic paper:  WHEN ZOMBIES ATTACK!:  MATHEMATICAL MODELLING OF AN OUTBREAK IN ZOMBIE INFECTION and he was on board.

Canadian mathematicians published a mathematical tretise on zombie population growth in the journal Infectious Disease Modelling Research Progress

Differential equations  describe how quickly things change, and how the rates of change affect other processes.  If you want to be really fancy, even the rates of rates of change can be described as well.  If you say, “The world’s population grows by 2% every year”, you have just stated a differential equation.  The solution to that one, unfortunately,  is the same function that describes compound interest.

In the case of zombies, the paper approaches the problem with all the necessary variables:  “zombies” (Z);  humans who are not yet zombies  (H); and zombies who have been neutralized — by some mechanism detailed in “Night of the Living Dead” — whom they called “removed” (R).  (Humans that die of natural causes, are also considered “removed”. How lucky is that during a zombie apocalypse?).

The role of differential equations comes into play when you consider the critical outcome: the rate of zombie increase or decrease.   For example, in the authors’ model, the number of zombies can increase by resurrection of “removed” humans or zombies.  It is just a rate proportional to the population of removed humans and zombies, so  call it +constant*R where R is the number of dead bodies around.  The “+” sign denotes an increase and the constant depends on the movie.   Because this term affects a rate, we are constructing a differential equation.

Lucky for us, the number of zombies can also decrease by removing their heads or other unpleasantness.  The probability that this happens depends now on two populations though, not one.   For example, if there were zero humans (H)  to neutralize the zombies, this  rate would be zero.  Likewise if there were no zombies (Z) to neutralize, the rate would be zero too.  The answer, the  product of H and Z, behaves just this way.  The authors describe this mathematically as –constant*H*Z, where the “-” sign corresponds to a decrease in zombies.

Unfortunately a zombie meeting a susceptible human doesn’t always work out so well.  A zombie might win and turn the human  into a zombie.  So the authors add another term like the one above but now with a plus sign: +constant*H*Z.  The “+” sign indicates an increase in zombies.

In "Zombie College" (a series of 12 animated shorts written by Big Bang Theory writer, Eric Kaplan) the constant c is explored.

(Watch Zombie College)

So you are ready to put these three parts together to form  the basic equation.   The net result of the three processes above is a rate of change of Zombies, Z’ which is just a sum of the last three paragraphs.   Calling the constants a, b and c, we have the rate of change of zombies described by

Z’= +a*R  – b*H*Z + c*H*Z.

(The variables on the board follow the paper: using S, for “susceptibles” instead of H for “human”.  But I thought H was clearer and it’s my blog.)

The authors add a few other details, such as birth rate, to find the formulas for the rate of change of humans, H’, and removed people R’.   The three equations’ variables depend on each other so these become  especially nasty kinds of differential equations, called coupled differential equations.  Undaunted, the mathematicians sharpen their pencils and follow the time honored tools for solving such coupled differential equations.  The net result is…

Everyone becomes a zombie.

(Technically, if humans  try to neutralize all the zombies everywhere at the same time, we have a fighting chance, but the authors consider the necessary cooperation unlikely.)

Full disclosure:  There is one thing I don’t understand about their model.  The authors allow zombies destroyed by humans to be counted in R.   I am no zombie movie expert, but I’d figure once you remove the head of a zombie, or even the head of  a non un-dead dead human, they cannot turn into a zombie.   (Please, someone with zombie cred pipe up in the comments.)   It would be easy enough to change the model to account for this though.  That is the beauty of differential equations.    Maybe their grim conclusion would be averted.

Despite the many scenarios envisioned, the outlook is always bleak, as summarized in the authors’ conclusions:

…an outbreak of zombies will result in the collapse of civilisation, with every human infected, or dead. This is because human births and deaths will provide the undead with a limitless supply of new bodies to infect, resurrect and convert. Thus, if zombies arrive, we must act quickly and decisively to eradicate them before they eradicate us.

While the example may seem frivolous, the mathematics of the differential equations are real.  Conclusions drawn from this study have impact on other infectious diseases with latent manifestation, presumably helping us address infections such as HIV.   For more about how mathematics can help you survive the zombie apocalypse,  or battle persistent disease in underdeveloped countries, see  my friend Jennifer Ouellette’s handy new book.

Want to solve some differential equations yourself?   I’m still waiting for someone to program the iPhone differential equation handwriting recognition and equation solver app.  You’ve had a few weeks now people.

S04E15: The Benefactor Factor

Pssst.   Do you want to buy a cryogenic centrifugal pump?     Tonight we find out that Leonard and the physics department want one.  And clearly it is big and expensive.

A cryogenic centrifugal pump. Although the one Leonard needs is even bigger.

It isn’t surprising that Leonard wants one. Modern physics experiments are often looking for extremely rare events.  Maybe just once per year a dark matter particle might bump into a cold, pure liquid detector.  Or perhaps once per year we might see an extremely rare radioactive decay that means something important to us.   The problem is that physicists need to look at a lot of material for an extremely long time without being fooled.

Physicists look for rare decays and events often by the small amount of light they emit.   To do the job, we use what is basically a television camera.  A small amount of light knocks an electron out of a metal called a “photocathode”.  A careful array of voltages are set up so that the electron gains energy and hits another metal producing several more electrons.  The process, called multiplication, is repeated until a detectable signal is present.  In order to not absorb the electron the entire structure is put into a vacuum tube.  The net result is called a “photomultiplier tube”.

A "photomultiplier tube" is the workhorse of physicists. It turns light into an electrical signal. It works better when cold.

The problem in looking for something rare, is that other processes don’t stop for you.  The sensitive light detectors are so sensitive that they often emit electrons without being struck by light.  These give false currents, even without any light, in a dark room.  The unwanted signal is  called a “dark current”.    Making the detectors cold greatly removes this effect.  But now you need to move a large amount, even tons, of cold liquids around.  That’s the job of a cryogenic centrifugal pump.

And while you are at it, you can remove the second source of noise: radioactivity.  A centrifugal pump can push noble elements, like xenon, through small holes but other molecules are larger and can’t fit.   But this is like the sieve in your kitchen lets the water through, but not the pasta.  The key here is also adsorption.   If a small molecule fits into one of the pores, then it is absorbed.  A trick is to find a material that catches what you don’t want.

The material in a molecular sieve blocks all but the smallest molecules and atoms.

Small versions of cryogenic centrifugal pumps are not terribly expensive.  But university budgets are tight. Physicists like Leonard still want to find rare events and are now dreaming of detectors with a ton or even ten tons of pure cold liquid, such as liquid xenon and argon.  For that they will need a large, expensive one.

Besides, my comedy friends tell me, words with the “hard C”  or “K” sound are funny.

S04E14: The Thespian Catalyst

Graphene is so yesterday.   This decade’s material-of-the-century are the tellurides.

At his official 2010 Nobel Prize acceptance lecture in Stockholm, Dr. Novoselov shows graphene, and Sheldon.

In his lecture, Sheldon told  his class (and about 15 million onlookers) about the strange behavior seen recently in certain compounds of bismuth, tellurium and tin. These strange new substances are insulators conductors insulators insulators and conductors simultaneously.  These tellurides and their cousins are part of a new class of recently discovered materials called, as Sheldon said,  topological insulators.

In materials such as typical plastics, electrons are pinned to the underlying structure and don’t move.    Because they can be used to keep conductors from shorting out, they are called insulators.  Relative to the best conductors, the electrical conductivity of the best insulators is 10²⁶ smaller, that’s a factor of 10,000,0000,000,000,000,000,000,000.   Few quantities in physics vary by so much.

On the whiteboards tonight, viewers saw bismuth telluride, cadmium telluride, and mercury telluride  making cameo appearances.    In these materials, the bulk volume is insulating–while the surfaces conduct.  At the same time.  How can that be?

Some clever wag may point out we could do this by just electroplating some plastic.  That was one of Richard Feynman’s first jobs and would be conductive on the outside but insulating in the middle.  But the difference here is that would be two materials.  Physicists never imagined this could be done with a single material at once.

The key difference from normal insulators is the reason they are called “topological”.  The description stems from the branch of mathematics called topology that characterizes the fundamental shapes of objects.   You can stretch a doughnut to form a coffee cup (one hole), but cannot make it into an object with two holes.   In the same way, the underlying structure of electron orbitals in an ordinary insulator can be represented by a simple loop.  A loop topologically distinct from the simplest possible knot:  a trefoil knot.

The structure of electron orbits in topological insulators are akin to the trefoil knot.

It turns out the interactions of the electrons’ spin with their orbital angular momentum creates a mathematical structure described by the trefoil knot.   For reasons beyond the ability of your science consultant to understand, the difference becomes apparent on the surface of the material: becoming metallic and conductive for the topological insulators but remaining non-conductive for normal insulators.  If you have a good explanation, please leave a comment.

Such effects have been seen before, but only with difficult-to-create flat structures.  But now, just like Hollywood, physicists have gone 3D with the advent  bismuth telluride.  Topological insulators can be created with standard semiconductor fabrication technology.  The simultaneous insulating and conducting nature of  the topological insulators is not an effect that can only be produced in expensive labs with high vacuums or extreme cooling.   These materials behave this way even at room temperature on the lab bench, or even held in your hand.

Work has heated up over the last five years and  many other compounds have been found to display not only the dual properties of topological insulators, crystals made of bismuth, selenium and copper have been made superconducting,moving electrons with no dissipation at all.

Topological insulators hold promise for new types of computing and materials whose applications we have not even thought of yet.  Their behavior is interesting in and of itself to physicists.  Sad to say, some popular articles have fallen prey yet again to the monopole falacy. This is the same annoying error that Sheldon complained about to Ira Flatow on NPR’s Science Friday.   Now in this latest article (and others) it says one of the interesting features of topological insulators is to make quasi-particle versions of axions, analogues of what are being sought in elementary particle physics.  However, just as with the magnetic monopole claims, that article misses the point completely:   Particle physicists don’t look for new particles  just to see their mathematical behavior.  We look for them because their existence  means something about the Universe.  In the case of the axion, it would validate certain explanations about why deep symmetries exist in nature. Axions could even be the dark matter in the galaxy.  But an axion-like-thing observed in a condensed matter system is not an axion.  It has none of that meaning.    Materials are topological insulators are still interesting in their own right. Such popular articles mislead readers at a deep level and do a disservice to these new materials by compromising the description of why they actually are interesting.

In fact topological insulators are so promising, we can only hope Sheldon’s boards will some day make a second appearance in Stockholm.  (And if, like Sheldon’s students, you want to tweet how boring this post is…hit the button below.)