Category Archive: blog

Apr 01

Universiteit Van Vlaanderen

A bit over 1 month ago, I told you about my adventure at the film studio of “de Universiteit Van Vlaanderen“. Today is the day the movie is officially released. You can find it at the website of de Universiteit Van Vlaanderen: Video. The video is in Dutch as this is a science-communication platform aimed at the local population, presenting the expertise available at our local universities.

 

In addition to this video, I was asked by Knack magazine to write a piece on the topic presented. As computational research is my central business I wrote a piece on the subject introducing the general public to the topic. The piece can be read here (in Dutch).

And of course, before I forget, this weekend there was also the half-yearly daylight saving exercise with our clocks.[and in Dutch]

 

Mar 18

SBDD XXIV: Diamond workshop

The participants to SBDD XXIV of 2019.  (courtesy of Jorne Raymakers, SBDD XXIV secretary) 

 

Last week the 24th edition of the Hasselt diamond workshop took place (this year chaired by Christoph Becher). It’s already the fourth time, since 2016, I have attended this conference, and each year it is a joy to meet up with the familiar faces of the diamond research field. The program was packed, as usual. And this year the NV-center was again predominantly present as the all-purpose quantum defect in diamond. I keep being amazed at how much it is used (although it has a rather low efficiency) and also about how many open question remain with regard to its incorporation during growth. With a little luck, you may read more about this in the future, as it is one of a few dozen ideas and questions I want to investigate.

A very interesting talk was given by Yamaguchi Takahide, who is combining hexagonal-BN and H-terminated diamond for high performance electronic devices. In such a device the h-BN leads to the formation of a 2D hole-gas at the interface (i.e., surface transfer doping), making it interesting for low dimensional applications. (And it of course hints at the opportunities available with other 2D materials.) The most interesting fact, as well as the most mind-boggling to my opinion, was the fact that there was no clear picture of the atomic structure of the interface. But that is probably just me. For experiments, nature tends to make sure everything is alright, while we lowly computational materials artificers need to know where each and every atom belongs. I’ll have to make some time to find out.

A second extremely interesting presentation was given by Anke Krueger (who will be the chair of the 25th edition of SBDD next year), showing of her groups skill at creating fluorine terminated diamond…without getting themselves killed. The surface termination of diamond with fluorine comes with many different hazards, going from mere poisoning, to fire and explosions. The take-home message: “kids don’t try this at home”. Despite all this risky business, a surface coverage of up to 85% was achieved, providing a new surface termination for diamond, with a much stronger trapping of negative charges near the surface, ideal for forming negatively charged NV centers.

On the last day, Rozita Rouzbahani presented our collaboration on the growth of B doped diamond. She studied the impact of growth conditions on the B concentration and growth speed of B doped diamond surfaces. My computational results corroborate her results and presents the atomic scale mechanism resulting in an increased doping concentration upon increased growth speed. I am looking forward to the submission of this nice piece of research.

And now, we wait another year for the next edition of SBDD, the celebratory 25th edition with a focus on diamond surfaces.

Feb 20

Universiteit Van Vlaanderen: Will we be able to design new materials using our smartphone in the future?

Yesterday, I had the pleasure of giving a lecture for the Universiteit van Vlaanderen, a science communication platform where Flemish academics are asked to answer “a question related to their research“. This question is aimed to be highly clickable and very much simplified. The lecture on the other hand is aimed at a general lay public.

I build my lecture around the topic of materials simulations at the atomic scale. This task ended up being rather challenging, as my computational research has very little direct overlap with the everyday life of the average person. I deal with supercomputers (which these days tend to be bench-marked in terms of smartphone power) and the quantum mechanical simulation of materials at the atomic scale, two other topics which may ring a bell…but only as abstract topics people may have heard of.

Therefor, I crafted a story taking people on a fast ride down the rabbit hole of my work. Starting from the almost divine power of the computational materials scientist over his theoretical sample, over the reality of nano-scale materials in our day-to-day lives, past the relative size of atoms and through the game nature of simulations and the salvation of computational research by grace of Moore’s Law…to the conclusion that in 25 years, we may be designing the next generation of CPU materials on our smartphone instead of a TIER-1 supercomputer. …did I say we went down the rabbit hole?

The television experience itself was very exhilarating for me. Although my actual lecture took only 15 minutes, the entire event took almost a full day. Starting with preparations and a trial run in the afternoon (for me and my 4 colleagues) followed by make-up (to make me look pretty on television 🙂 … or just to reduce my reflectance). In the evening we had a group diner meeting the people who would be in charge of the technical aspects and entertainment of the public. And then it was 19h30. Tensions started to grow. The public entered the studio, and the show was ready to start. Before each lecture, there was a short interview to test sound and light, and introduce us to the public. As the middle presenter, I had the comfortable position not to be the first, so I could get an idea of how things went for my colleagues, and not to be the last, which can really be destructive on your nerves.

At 21h00, I was up…

and down the rabbit hole we went. 

 

 

Full periodic table, with all elements presented with their relative size (if known)

Full periodic table, with all elements presented with their relative size (if known) created for the Universiteit van Vlaanderen lecture.

 

Feb 06

Roots of Science

Today VLIR (Flemish Inter-university Council) and the Young Academy had a conference on the future of fundamental research in Flanders: Roots of Science. We live in a world where we rely on science more and more to resolve our problems (think climate change, disease control, energy generation, …). In our bizarre world of alternative facts and fake news, science can be utterly ignored in one sentence and proposed as a magical solution in the next.

Although I am happy with the faith some have in the possibilities of science, it is important to remember that it is not magic. This has a very important consequence:

Things do not happen simply because you want them to happen.

 

Many important breakthroughs in science are what one would call serendipity (e.g., the discovery of penicillin by Fleming, development of the WWW as a side-effect of researchers wanting to share their data

at CERN in 1991,…) . In Flanders the Royal Flemish Academy and the Young Academy have written a Standpoint (an evidence-based advisory text)

discussing the need for more researcher-driven research in contrast to agenda-driven research, as they believe this is a conditio sine qua non for a healthy scientific future.

Where government-driven research focuses on resolving questions from society, researcher-driven research allows the researcher to follow his or her personal interest. This not with the primal aim of having short-te

rm return of investment, but with the aim of providing the fundamental knowledge and expertise which some day may be needed for the former. In researcher-driver research, the journey is the goal as this is where scientific progress is made by finding solutions for problems not imagined before.

Do we have to pay for this with our tax-payers money? I think we do. No-one imagined optical drives (CD, DVD, blue-ray) to become a billion euro industry while the laser was being developed in a lab. Who would have thought the transistor would play such an important role in our every-day life? And what about the first computer? Thomas Watson, President of IBM, has allegedly said in 1943: “I think there is a world market for maybe 5 computers.” And, yet, now many of us have more than 5 computers at home (including tables, smartphones,…)! The researchers working on these “inventions” did not do this with your Blue-ray player or smartphone in mind. These high impact applications are “merely” side-products of their fundamental scientific research. No-one at the time could predict this, so why should we be able to do this today? In this sense, you should see funding of fundamental research as a long term investment. Tax-money is being invested in our future, and the future of children and grandchildren. Although we do not know what will be the outcome, we know from the past that it will have an impact on our lives.

Its difficult to make predictions, especially about the future.

Let us therefor support more researcher-driven research.

 

In addition to the Standpoint, there is also a very nice video explaining the situation (with subtitles in English or Dutch, use the cogwheel to select your preference).

Dec 26

Defensive Programming and Debugging

Last few months, I finally was able to remove something which had been lingering on my to-do list for a very long time: studying debugging in Fortran. Although I have been programming in Fortran for over a decade, and getting quite good at it, especially in the more exotic aspects such as OO-programming, I never got around to learning how to use decent debugging tools. The fact that I am using Fortran was the main contributing factor. Unlike other languages, everything you want to do in Fortran beyond number-crunching in procedural code has very little documentation (e.g., easy dll’s for objects), is not natively supported (e.g., find a good IDE for fortran, which also supports modern aspects like OO, there are only very few who attempt), or you are just the first to try it (e.g., fortran programs for android :o, definitely on my to-do list). In a long bygone past I did some debugging in Delphi (for my STM program) as the debugger was nicely integrated in the IDE. However, for Fortran I started programming without an IDE and as such did my initial debugging with well placed write statements. And I am a bit ashamed to say, I’m still doing it this way, because it can be rather efficient for a large code spread over dozens of files with hundreds of procedures.

However, I am trying to repent for my sins. A central point in this penance was enlisting for the online MOOC  “Defensive Programming and Debugging“. Five weeks of intense study followed, in which I was forced to use command line gdb and valgrind. During these five weeks I also sharpened my skills at identifying possible sources of bugs (and found some unintentional bugs in the course…but that is just me). Five weeks of hard study, and taking tests, I successfully finished the course, earning my certificate as defensive programmer and debugger. (In contrast to my sometimes offensive programming and debugging skills before 😉 .)

Dec 24

Merry Christmas & Happy New Year

Having fun with my xmgrace-fortran library and fractal code!

Oct 31

Daylight saving and solar time

For many people around the world, last weekend was highlighted by a half-yearly recurring ritual: switching to/from daylight saving time. In Belgium, this goes hand-in-hand with another half-yearly ritual; The discussion about the possible benefits of abolishing of daylight saving time. Throughout the last century, daylight saving time has been introduced on several occasions. The most recent introduction in Belgium and the Netherlands was in 1977. At that time it was intended as a measure for conserving energy, due to the oil-crises of the 70’s. (In Belgium, this makes it painfully modern due to the current state of our energy supplies: the impending doom of energy shortages and the accompanying disconnection plans which will put entire regions without power in case of shortages.)

The basic idea behind daylight saving time is to align the daylight hours with our working hours. A vision quite different from that of for example ancient Rome, where the daily routine was adjusted to the time between sunrise and sunset. This period was by definition set to be 12 hours, making 1h in summer significantly longer than 1h in winter. As children of our time, with our modern vision on time, it is very hard to imagine living like this without being overwhelmed by images of of impending doom and absolute chaos. In this day and age, we want to know exactly, to the second, how much time we are spending on everything (which seems to be social media mostly 😉 ). But also for more important aspects of life, a more accurate picture of time is needed. Think for example of your GPS, which will put you of your mark by hundreds of meters if your uncertainty in time is a mere 0.000001 seconds. Also, police radar will not be able to measure the speed of your car with the same uncertainty on its timing.

Turing back to the Roman vision of time, have you ever wondered why “the day” is longer during summer than during winter? Or, if this difference is the same everywhere on earth? Or, if the variation in day length is the same during the entire year?

Our place on earth

To answer these questions, we need a good model of the world around us. And as is usual in science, the more accurate the model, the more detailed the answer.

Let us start very simple. We know the earth is spherical, and revolves around it’s axis in 24h. The side receiving sunlight we call day, while the shaded side is called night. If we assume the earth rotates at a constant speed, then any point on its surface will move around the earths rotational axis at a constant angular speed. This point will spend 50% of its time at the light side, and 50% at the dark side. Here we have also silently assumed, the rotational axis of the earth is “straight up” with regard to the sun.

In reality, this is actually not the case. The earths rotational axis is tilted by about 23° from an axis perpendicular to the orbital plane. If we now consider a fixed point on the earths surface, we’ll note that such a point at the equator still spends 50% of its time in the light, and 50% of its time in the dark. In contrast, a point on the northern hemisphere will spend less than 50% of its time on the daylight side, while a point on the southern hemisphere spends more than 50% of its time on the daylight side. You also note that the latitude plays an important role. The more you go north, the smaller the daylight section of the latitude circle becomes, until it vanishes at the polar circle. On the other hand, on the southern hemisphere, if you move below the polar circle, the point spend all its time at the daylight side. So if the earths axis was fixed with regard to the sun, as shown in the picture, we would have a region on earth living an eternal night (north pole) or day (south pole). Luckily this is not the case. If we look at the evolution of the earths axis, we see that it is “fixed with regard to the fixed stars”, but makes a full circle during one orbit around the sun.* When the earth axis points away from the sun, it is winter on the northern hemisphere, while during summer it points towards the sun. In between, during the equinox, the earth axis points parallel to the sun, and day and night have exactly the same length: 12h.

So, now that we know the length of our daytime varies with the latitude and the time of the year, we can move one step further.

How does the length of a day vary, during the year?

The length of the day varies over the year, with the longest and shortest days indicated by the summer and winter solstice. The periodic nature of this variation may give you the inclination to consider it as a sine wave, a sine-in-the-wild so to speak. Now let us compare a sine wave fitted to actual day-time data for Brussels. As you can see, the fit is performing quite well, but there is a clear discrepancy. So we can, and should do better than this.

Instead of looking at the length of each day, let us have a look at the difference in length between sequential days.** If we calculate this difference for the fitted sine wave, we again get a sine wave as we are taking a finite difference version of the derivative. In contrast, the actual data shows not a sine wave, but a broadened sine wave with flat maximum and minimum. You may think this is an error, or an artifact of our averaging, but in reality, this trend even depends on the latitude, becoming more extreme the closer you get to the poles.

This additional information, provides us with the extra hint that in addition to the axial tilt of the earth axis, we also need to consider the latitude of our position. What we need to calculate is the fraction of our latitude circle (e.g. for Brussels this is 50.85°) that is illuminated by the sun, each day of the year. With some perseverance and our high school trigonometric equations, we can derive an analytic solution, which can then be calculated by, for example, excel.

Some calculations

The figure above shows a 3D sketch of the situation on the left, and a 2D representation of the latitude circle on the right. α is related to the latitude, and β is the angle between the earth axis and the ‘shadow-plane’ (the plane between the day and night sides of earth). As such, β will be maximal during solstice (±23°26’12.6″) and exactly equal to zero at the equinox—when the earth axis lies entirely in the shadow-plane. This way, the length of the day is equal to the illuminated fraction of the latitude circle: 24h(360°-2γ). γ can be calculated as cos(γ)=adjacent side/hypotenuse in the right hand side part of the figure above. If we indicate the earth radius as R, then the hypotenuse is given by Rsin(α). The adjacent side, on the other hand, is found to be equal to R’sin(β), where R’=B/cos(β), and B is the perpendicular distance between the center of the earth and the plane of the latitude circle, or B=Rcos(α).

Combining all these results, we find that the number of daylight hours is:

24h*{360°-2arccos[cotg(α)tg(β)]}

 

How accurate is this model?

All our work is done, the actual calculation with numbers is a computer’s job, so we put excel to work. For Brussels we see that our model curve very nicely and smoothly follows the data (There is no fitting performed beyond setting the phase of the model curve to align with the data). We see that the broadening is perfectly shown, as well as the perfect estimate of the maximum and minimum variation in daytime (note that this is not a fitting parameter, in contrast to the fit with the sine wave). If you want to play with this model yourself, you can download the excel sheet here. While we are on it, I also drew some curves for different latitudes. Note that beyond the polar circles this model can not work, as we enter regions with periods of eternal day/night.

 

After all these calculations, be honest:

You are happy you only need to change the clock twice a year, don’t you. 🙂

 

 

* OK, in reality the earths axis isn’t really fixed, it shows a small periodic precession with a period of about 41000 years. For the sake of argument we will ignore this.

** Unfortunately, the data available for sunrises and sunsets has only an accuracy of 1 minute. By taking averages over a period of 7 years, we are able to reduce the noise from ±1 minute to a more reasonable value, allowing us to get a better picture of the general trend.

External links

Oct 25

Newsflash: Materials of the Future

This summer, I had the pleasure of being interviewed by Kim Verhaeghe, a journalist of the EOS magazine, on the topic of “materials of the future“. Materials which are currently being investigated in the lab and which in the near or distant future may have an enormous impact on our lives. While brushing up on my materials (since materials with length scales of importance beyond 1 nm are generally outside my world of accessibility), I discovered that to cover this field you would need at least an entire book just to list the “materials of the future”. Many materials deserve to be called materials of the future, because of their potential. Also depending on your background other materials may get your primary attention.

In the resulting article, Kim Verhaeghe succeeded in presenting a nice selection, and I am very happy I could contribute to the story. Introducing “the computational materials scientist” making use of supercomputers such as BrENIAC, but also new materials such as Metal-Organic Frameworks (MOF) and shedding some light on “old” materials such as diamond, graphene and carbon nanotubes.

Aug 12

Dangerous travel physics

Tossing coins into a fountain brings luck, tossing them of a building causes death and destruction?

 

We have probably all done it at one point when traveling: thrown a coin into a wishing well or a fountain. There are numerous wishing wells with legends describing how the deity living in the well will bring good fortune in return for this gift. The myths and legends often originate from Celtic, German or Nordic traditions.

In case of the Trevi fountain, there is the belief that if you throw a coin over your left shoulder using your right hand, you will return to Rome…someday. As this fountain and legend are iconic parts of our western movie history, many, many coins get tossed into it (more than 1 Million € worth each year, which is collected an donated to charity).

In addition to these holiday legends, there also exist more recent “coin-myths”: Death by falling penny. These myths are always linked to tall buildings, and claim that a penny dropped from the top of such a building could kill someone if they hit him.

Traveling with Newton

In both kinds of coin legends, the trajectory of the coin can be predicted quite well using Newton’s Laws. Their speed is low compared to the speed of light, and the coins are sufficiently large to keep the world of quantum mechanics hidden from sight.

The second Law of Newton states that the speed of an object changes if there is a force acting on it. Here on earth, gravity is a major player (especially for Physics exercises). In case of a coin tossed into a fountain, gravity will cause the coin to follow a roughly parabolic path before disappearing into the water. The speed at which the coin will hit the water will be comparable to the speed with which it was thrown…at least if there isn’t to much of a difference in height between the surface of the water and the hand of the one throwing the coin.

But, what if this difference is large? Such as in case of the penny being dropped from a tall building. In such a case, the initial velocity is zero, and the penny is accelerated toward the ground by gravity. Using the equations of motion for a uniform accelerated system, we can calculate easily the speed at which the coin hits the ground:

x = x0 + v0*t + ½ * g * t²

v=v0+g*t

If we drop a penny from the 3rd floor of the Eiffel Tower (x0=276.13m, x=0m, v0=0 m/s, g=-9.81m/s²) then the first equation teaches us that after 7.5 seconds, the penny will hit the ground with a final speed (second equation) of -73.6 m/s (or -265 km/h)*. With such a velocity, the penny definitely will leave an impression. More interestingly, we will get the exact same result for a pea (cooked or frozen), a bowling ball, a piano or an anvil…but also a feather. At this point, your intuition must be screaming at you that you are missing something important.

All models are wrong…but they can be very useful

The power of models in physics, originates from keeping only the most important and relevant aspects. Such approximations provide a simplified picture and allow us to understand the driving forces behind nature itself. However, in this context, models in physics are approximations of reality, and thus by definition wrong, in the sense that they do not provide an “exact” representation of reality. This is also true for Newton’s Laws, and our application above. With these simple rules, it is possible to describe the motion of the planets as well as a coin tossed into the Trevi fountain.

So what’s the difference between the coin tossed into a fountain and planetary motion on the one hand, and our assorted objects being dropped from the Eiffel Tower on the other hand?

Friction as it presents itself in aerodynamic drag!

Aerodynamic drag gives rise to a force in the direction opposite to the movement, and it is defined as:

FD= ½ *Rho*v²*CD*A

This force depends on the density Rho of the medium (hence water gives a larger drag than air), the velocity and surface area A in the direction of movement of the object, and CD the drag coefficient, which depends on the shape of the object.

If we take a look at the planets and the coin tosses, we notice that, due to the absence of air between the planets, no aerodynamic drag needs to be considered for planetary motion. In case of a coin being tossed into the Trevi fountain, there is aerodynamic drag, however, the speeds are very low as well as the distance traversed. As such the effect of aerodynamic drag will be rather small, if not negligible. In case of objects being dropped from a tall building, the aerodynamic drag will not be negligible, and it will be the factors CD and A which will make sure the anvil arrives at the ground level before the feather.

Because this force also depends on the velocity, you can no longer make direct use of the first two equations to calculate the time of impact and velocity at each point of the path. You will need a numerical approach for this (which is also the reason this is not (regularly) taught in introductory physics classes at high school). However, using excel, you can get a long way in creating a numerical solution for this problem.[Excel example]

As we know the density of air is about 1.2kg/m³, CD for a thin cylinder (think coin) is 1.17, the radius of a penny is 9.5 mm and its mass is 2.5g, then we can find the terminal velocity of the penny to be 11.1 m/s (40 km/h). The penny will land on the ground after about 25.6 seconds. This is quite a bit slower than what we found before, and also quite a bit more safe. The penny will reach its terminal velocity after having fallen about 60 m, which means that dropping a penny from taller buildings (the Atomium [102 m], the Eiffel Tower [276.13 m, 3rd floor, 324 m top], the Empire State Building [381 m] or even the Burj Khalifa [829.8 m]) will have no impact on the velocity it will have when hitting the ground: 40km/h.

This is a collision you will most probably survive, but which will definitely leave a small bruise on impact.

 

*The minus sign indicates the coin is falling downward.

Jul 17

Building bridges towards experiments.

Quantum Holy Grail: The Ground-State

Quantum mechanical calculations provide a powerful tool to investigate the world around us. Unfortunately it is also a computationally very expensive tool to use, which puts a boundary on what is possible in terms of computational materials research. For example, when investigating a solid at the quantum mechanical level, you are limited in the number of atoms that you can consider. Even with a powerful supercomputer at hand, a hundred to a thousand atoms are currently accessible for “routine” investigations. The computational cost also limits the number of configurations/combinations you can calculate.

However, in the end— and often with some blood sweat and tears—these calculations do provide you the ground-state structure and energy of your system. From this point forward you can continue characterizing its properties, life is beautiful and happy times are just beyond the horizon. At this horizon your experimental colleague awaits you. And he/she tells you:

Sorry, I don’t find that structure in my sample.

After recovering from the initial shock, you soon realize that in (materials science) experiments one seldom encounters a sample in “the ground-state”. Experiments are performed at temperatures above 0K and pressures above 0 Pa (even in vacuum :p ). Furthermore, synthesis methods often involve elevated temperatures, increased pressure, mechanical forces, chemical reactions,… which give rise to meta-stable configurations. In such an environment, your nicely deduced ground-state may be an exception to the rule. It is only one point within the phase-space of the possible.

So how can you deal with this? You somehow need to sample the phase-space available to the experiment.

Sampling Phase-Space for Ball-milling synthesis.

For a few years now, I have a very fruitful collaboration with Prof. Rounaghi. His interest goes toward the cheap fabrication of metal-nitrides. Our first collaboration focused on AlN, while later work included Ti, V and Cr-nitrides. Although this initial work had a strong focus on simple corroboration through the energies calculated at the quantum mechanical level, the collaboration also allowed me to look at my data in a different way. I wanted to “simulate” the reactions of ball-milling experiments more closely.

Due to the size-limitations of quantum mechanical calculations I played with the following idea:

  • Assume there exists a general master reaction which describes what happens during ball-milling.

X Al + Y Melamine → x1 Al + x2 Melamine + x3 AlN + …

where all the xi represent the fractions of the reaction products present.

  • With the boundary condition that the number of particles needs to be conserved, you end up with a large set of (x1,x2,x3,…) configurations which each have a certain energy. This energy is calculated using the quantum mechanical energies of each product. The configuration with the lowest energy is the ground state configuration. However, investigating the entire accessible phase-space showed that the energies of the other possible configurations are generally not that much higher.
  • What if we used the energy available due to ball-milling in the same fashion as we use kBT? And sample the phase-space using Boltzmann statistics.
  • The resulting Boltzmann distribution of the configurations available in the phase-space can then be used to calculate the mass/atomic fraction of each of the products and allow us to represent an experimental sample as a collection of small units with slightly different configurations, weighted according to their Boltzmann distribution.

This setup allowed me to see the evolution in end-products as function of the initial ratio in case of AlN, and in our current project to indicate the preferred Iron-nitride present.

Grid-sampling vs Monte-Carlo-sampling

Whereas the AlN system was relatively easy to investigate—the phase space was only 3 dimensional— the recent iron based system ended up being 4 dimensional when considering only host materials, and 10 dimensional when including defects. For a small 3-4D phase-space, it is possible to create an equally spaced grid and get converged results using a few million to a billion grid-points. For a 10D phase-space this is no longer possible. As you can no longer keep all data-points (easily) in storage during your calculation (imagine 1 Billion points, requiring you to store 11 double precision floats or about 82Gb) you need a method that does not rely on large arrays of data. For our Boltzmann statistics this gives us a bit of a pickle, as we need to have the global minimum of our phase space. A grid is too course to find it, while a simple Monte-Carlo just keeps hopping around.

Using Metropolis’s improvement of the Monte-Carlo approach was an interesting exercise, as it clearly shows the beauty and simplicity of the approach. This becomes even more awesome the moment you imagine the resources available in those days. I noted 82Gb being a lot, but I do have access to machines with those resources; its just not available on my laptop. In those days MANIAC supercomputers had less than 100 kilobyte of memory.

Although I theoretically no longer need the minimum energy configuration, having access to that information is rather useful. Therefore, I first search the phase-space for this minimum. This is rather tricky using Metropolis Monte Carlo (of course better techniques exist, but I wanted to be a bit lazy), and I found that in the limit of T→0 the algorithm will move toward the minimum. This, however, may require nearly 100 million steps of which >99.9% are rejected. As it only takes about 20 second on a modern laptop…this isn’t a big issue.

Finding a minimum using Metropolis Monte Carlo.

Finding a minimum using Metropolis Monte Carlo.

Next, a similar Metropolis Monte Carlo algorithm can be used to sample the entire phase space. Using 109 sample points was already sufficient to have a nicely converged sampling of the phase space for the problem at hand. Running the calculation for 20 different “ball-milling” energies took less than 2 hours, which is insignificant, when compared to the resources required to calculate the quantum mechanical ground state energies (several years). The figure below shows the distribution of the mass fraction of one of the reaction products as well as the distribution of the energies of the sampled configurations.

Metropolis Monte Carlo distribution of mass fraction and configuration energies for 3 sets of sample points.

Metropolis Monte Carlo distribution of mass fraction and configuration energies for 3 sets of sample points.

This clearly shows us how unique and small the quantum mechanical ground state configuration and its contribution is compared to the remainder of the phase space. So of course the ground state is not found in the experimental sample but that doesn’t mean the calculations are wrong either. Both are right, they just look at reality from a different perspective. The gap between the two can luckily be bridged, if one looks at both sides of the story. 

 

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