Author Archives: Anshul Kogar

DIY Garage Work

Recently, I heard about a string of YouTube videos where Ben Krasnow of the Applied Sciences YouTube Channel makes a series of scientific instruments in his garage. One of the particularly impressive achievements is his homemade Scanning Electron Microscope, where he constructs a pretty decent instrument with approximately $1500. This is definitely outstanding from an educational viewpoint — $1500 will probably be affordable for many high schools and will enable students to see how to image objects with electrons.

Here are a couple videos showing this and another one of his projects where he uses a laser and a couple optical elements to construct a Raman spectroscopy setup:




Lastly, I’d like to point out that Christina Lee has put together an excellent set of Jupyter code (i.e. IPython Notebook code) to solve various condensed matter physics problems. It’s definitely worth having a look.

Jahn-Teller Distortion and Symmetry Breaking

The Jahn-Teller effect occurs in molecular systems, as well as solid state systems, where a molecular complex distorts, resulting in a lower symmetry. As a consequence, the energy of certain occupied molecular states is reduced. Let me first describe the phenomenon before giving you a little cartoon of the effect.

First, consider, just as an example, a manganese atom with valence 3d^4, surrounded by an octahedral cage of oxygen atoms like so (image taken from this thesis):


The electrons are arranged such that the lower triplet of orbital states each contain a single “up-spin”, while the higher doublet of orbitals only contains a single “up-spin”, as shown on the image to the left. This scenario is ripe for a Jahn-Teller distortion, because the electronic energy can be lowered by splitting both the doublet and the triplet as shown on the image on the right.

There is a very simple, but quite elegant problem one can solve to describe this phenomenon at a cartoon level. This is the problem of a two-dimensional square well with adjustable walls. By solving the Schrodinger equation, it is known that the energy of the two-dimensional infinite well has solutions of the form:

E_{i,j} = \frac{h}{8m}(i^2/a^2 + j^2/b^2)                where i,j are integers.

Here, a and b denote the lengths of the sides of the 2D well. Since it is only the quantity in the brackets that determine the energy levels, let me factor out a factor of \gamma = a/b and write the energy dependence in the following way:

E \sim i^2/\gamma + \gamma j^2

Note that \gamma is effectively an anisotropy parameter, giving a measure of the “squareness of the well”. Now, let’s consider filling up the levels with spinless electrons that obey the Pauli principle. These electrons will fill up in a “one-per-level” fashion in accordance with the fermionic statistics. We can therefore write the total energy of the N-fermion problem as so:

E_{tot} \sim \alpha^2/ \gamma + \gamma \beta^2

where \alpha and \beta parameterize the energy levels of the N electrons.

Now, all of this has been pretty simple so far, and all that’s really been done is to re-write the 2D well problem in a different way. However, let’s just systematically look at what happens when we fill up the levels. At first, we fill up the E_{1,1} level, where \alpha^2 = \beta^2 = 1^2. In this case, if we take the derivative of E_{1,1} with respect to \gamma, we get that \gamma_{min} = 1 and the well is a square.

For two electrons, however, the well is no longer a square! The next electron will fill up the E_{2,1} level and the total energy will therefore be:

E_{tot} \sim 1/\gamma (1+4) + \gamma (1+1),

which gives a \gamma_{min} = \sqrt{5/2}!

Why did this breaking of square symmetry occur? In fact, this is very closely related to the Jahn-Teller effect. Since the level is two-fold degenerate (i.e. E_{2,1} =  E_{1,2}), it is favorable for the 2D well to distort to lower its electronic energy.

Notice that when we add the third electron, we get that:

E_{tot} \sim 1/\gamma (1+4+1) + \gamma (1+1+4)

and \gamma_{min} = 1 again, and we return to the system with square symmetry! This is also quite similar to the Jahn-Teller problem, where, when all the states of the degenerate levels are filled up, there is no longer an energy to be gained from the symmetry-broken geometry.

This analogy is made more complete when looking at the following level scheme for different d-electron valence configurations, shown below (image taken from here).


The black configurations are Jahn-Teller active (i.e. prone to distortions of the oxygen octahedra), while the red are not.

In condensed matter physics, we usually think about spontaneous symmetry breaking in the context of the thermodynamic limit. What saves us here, though, is that the well will actually oscillate between the two rectangular configurations (i.e. horizontal vs. vertical), preserving the original symmetry! This is analogous to the case of the ammonia (NH_3) molecule I discussed in this post.

The Physicist’s Proof II: Limits and the Monty Hall Problem

As an undergraduate, I was taught the concept of the “physicist’s proof”, a sort of silly concept that was a professor’s attempt to get us students to think a little harder about some problems. Here, I give you a “physicist’s proof” of the famous Monty Hall problem, which (to me!) is easier to think about than the typical Bayesian approach.

The Monty Hall problem, which was developed on a TV game show, goes something like this (if you already know the Monty Hall problem, you can skip the paragraphs in italics):

Suppose you are a contestant on a game show where there are three doors and a car behind one of them. You must select the correct door to win the car.

Image result for monty hall problem

You therefore select one of the three doors. Now, the host of the show, who knows where the car is, opens a different door for you and shows you that there is no car behind that door.

There are two remaining unopened doors — the one you have chosen and one other. Now, before you find out whether or not you have guessed correctly, the host gives you the option to change your selection from the door you initially chose to the other remaining unopened door.

Should you switch or should you remain with you initial selection?

When I first heard this problem, I remember thinking, like most people, that there was a 50/50 chance of the car being behind either door. However, there is a way to convince yourself that this is not so. This is by taking the limit of a large number of doors. I’ll explain what I mean in a second, but let me just emphasize that taking limits is a common and important technique that physicists must master to think about problems in general.

In Einstein’s book, Relativity, he describes using this kind of thinking to point out absurd consequences of Galilean relativity. Einstein imagined himself running away from a clock at the speed of light: in this scenario, the light from the clock would be matching his pace and he would therefore observe the hands of the clock remaining stationary and time standing still. Were he able to run just a little bit faster than the light emanating from the clock, he would see the hands of the clock start to wind backwards. This would enable him to violate causality!  However, Einstein held causality to be a dearer physical truth than Newton’s laws. Special relativity was Einstein’s answer to this contradiction, a conclusion he reached by considering a physical limit.

Now, let us return to the Monty Hall problem. And this time, instead of three doors, let’s think about the limit of, say, a million doors. In this scenario, suppose that you have to choose one door from one million doors instead of just three. For the sake of argument, suppose you select door number 999,983. The host, who knows where the car is, opens all the other doors, apart from door number 17. Should you stick to door 999,983 or should you switch to door 17?

Let’s think about this for a second — there are two scenarios. Either you were correct on your first guess and the car is behind door 999,983 or you were incorrect on your first guess and the car is behind door 17. When you initially made your selection, the chance of you having made the right guess was 1/1,000,000! The probability of you having chosen the right door is almost zero! If you had chosen any other door apart from door 17, you would have been faced with the same option: the door you chose vs. door 17. And there are 999,999 doors for you to select and not win the car. In some sense, by opening all the other doors, the host is basically telling you that the car is behind door 17 (there is a 99.9999% chance!).

To me, at least, the million door scenario demonstrates quite emphatically that switching from your initial choice is more logical. For some reason, the three door case appears to be more psychologically challenging, and the probabilities are not as obvious. Taking the appropriate limit of the Monty Hall problem is therefore (at least to me) much more intuitive!

Especially for those who are soon to take the physics GRE — remember to take appropriate limits, this will often eliminate at least a couple answers!

For completeness, I show below the more rigorous Bayesian method for the three-door case:

Bayes theorem says that:

P(A|B) = \frac{P(B|A) P(A)}{P(B)}

For the sake of argument, suppose that you select door 3. The host then shows you that there is no car behind door 2. The calculation goes something like this. Below, “car3” translates to “the car was behind door 3” and “opened2” translates to “the host opened door 2”

P(car3|opened2) = \frac{P(opened2 | car3) P(car3)}{P(opened2)}

The probabilities in the numerator are easy to obtain P(opened2 | car3) = 1/2 and P(car3) = 1/3. However, the P(opened2) is a little harder to calculate. It helps to enumerate all the scenarios. Given that you have chosen door three, if the car is behind door 1, then the probability that the host opens door two is 1. Given that you have chosen door three and are correct, the probability of the host choosing door 2 is 1/2. Obviously, the probability of the car being behind door 2 is zero. Therefore, considering that all doors have a 1/3 possibility of having the car behind them at the outset, the denominator becomes:

P(opened2) = 1/3*(1/2 + 1 + 0) = 1/2

and hence:

P(car3|opened2) = \frac{1/2*1/3}{1/2} = 1/3.

Likewise, the probability that the car is behind door 1 is:

P(car1|opened2) = \frac{P(opened2 | car1) P(car1)}{P(opened2)}

which can similarly be calculated:

P(car1|opened2) = \frac{1*1/3}{1/2} = 2/3.

It is a bizarre answer, but Bayesian results often are.

An Undergraduate Optics Problem – The Brewster Angle

Recently, a lab-mate of mine asked me if there was an intuitive way to understand Brewster’s angle. After trying to remember how Brewster’s angle was explained to me from Griffiths’ E&M book, I realized that I did not have a simple picture in my mind at all! Griffiths’ E&M book uses the rather opaque Fresnel equations to obtain the Brewster angle. So I did a little bit of thinking and came up with a picture I think is quite easy to grasp.

First, let me briefly remind you what Brewster’s angle is, since many of you have probably not thought of the concept for a long time! Suppose my incident light beam has both components, s– and p-polarization. (In case you don’t remember, p-polarization is parallel to the plane of incidence, while s-polarization is perpendicular to the plane of incidence, as shown below.) If unpolarized light is incident on a medium, say water or glass, there is an angle, the Brewster angle, at which the light comes out perfectly s-polarized.

An addendum to this statement is that if the incident beam was perfectly p-polarized to begin with, there is no reflection at the Brewster angle at all! A quick example of this is shown in this YouTube video:

So after that little introduction, let me give you the “intuitive explanation” as to why these weird polarization effects happen at the Brewster angle. First of all, it is important to note one important fact: at the Brewster angle, the refracted beam and the reflected beam are at 90 degrees with respect to each other. This is shown in the image below:

Why is this important? Well, you can think of the reflected beam as light arising from the electrons jiggling in the medium (i.e. the incident light comes in, strikes the electrons in the medium and these electrons re-radiate the light).

However, radiation from an oscillating charge only gets emitted in directions perpendicular to the axis of motion. Therefore, when the light is purely p-polarized, there is no light to reflect when the reflected and refracted rays are orthogonal — the reflected beam can’t have the polarization in the same direction as the light ray! This is shown in the right image above and is what gives rise to the reflectionless beam in the YouTube video.

This visual aid enables one to use Snell’s law to obtain the celebrated Brewster angle equation:

n_1 \textrm{sin}(\theta_B) = n_2 \textrm{sin}(\theta_2)


\theta_B + \theta_2 = 90^o

to obtain:

\textrm{tan}(\theta_B) = n_2/n_1.

The equations also suggest one more thing: when the incident light has an s-polarization component, the reflected beam must come out perfectly polarized at the Brewster angle. This is because only the s-polarized light jiggles the electrons in a way that they can re-radiate in the direction of the outgoing beam. The image below shows the effect a polarizing filter can therefore have when looking at water near the Brewster angle, which is around 53 degrees for water.

To me, this is a much simpler way to think about the Brewster angle than dealing with the Fresnel equations.

Spot the Difference

A little while ago, I wrote a blog post concerning autostereograms, more commonly referred to as Magic Eye images. These are images that, at first sight, seem to possess nothing but a random-seeming pattern. However, looked at in a certain way, a three-dimensional image can actually be made visible. Below is an example of a such an image (taken from Wikipedia):

Autostereogram of a shark

In my previous post about these stereograms, I pointed out that the best way to understand what is going on is to look at a two-image stereogram (see below). Here, the left eye looks at the left image while the right eye looks at the right image, and the brain is tricked into triangulating a distance because the two images are almost the same. The only difference is that part of the image has been displaced horizontally, which makes that part appear like it is at a different depth. This is explained at the bottom of this page, and an example is shown below:

Random Dot Stereogram

Boring old square

In this post, however, I would like to point out that this visual technique can be used to solve a different kind of puzzle. When I was in middle school, one of the most popular games to play was called Photo-Hunt, essentially a spot-the-difference puzzle. You probably know what I’m referring to, but here is an example just in case you don’t:

The bizarre thing about these images is that if you look at them like you would a Magic Eye image, the differences between the two images essentially “pop out” (or rather they flicker noticeably). Because each of your eyes is looking at each image separately, your brain is tricked into thinking there is a single image at a certain depth. Therefore, the differences reveal themselves, because while the parts of the image that are identical are viewed with a particular depth of view, the differences don’t have the same effect. Your eyes cannot triangulate the differences, and they appear to flicker. I wish I had learned this trick in middle school, when this game was all the rage.

While this may all seem a little silly, I noticed recently while zoning out during a rather dry seminar, that I could notice very minute defects in TEM images using this technique. Here is an example of an image of a bubble raft (there are some really cool videos of bubble rafts online — see here for instance), where the defects immediately emerge when viewed stereoscopically (i.e. like a Magic-Eye):


Bubble raft image taken from here

I won’t tell you where the defects are, but just to let you know that there are three quite major ones, which are the ones I’m referring to in the image. They’re quite obvious even if not viewed stereoscopically.

Because so many concepts in solid state physics depend on crystal symmetries and periodicity, I can foresee entertaining myself during many more dry seminars in the future, be it a seminar with tons of TEM images or a wealth of diffraction data. I have even started viewing my own data this way to see if anything immediately jumps out, without any luck so far, but I suspect it is only a matter of time before I see something useful.

Book Review – The Gene

Following the March Meeting, I took a vacation for a couple weeks, returning home to Bangkok, Thailand. During my holiday, I was able to get a hold of and read Siddhartha Mukherjee’s new book entitled The Gene: An Intimate History.

I have to preface any commentary by saying that prior to reading the book, my knowledge of biology embarrassingly languished at the middle-school level. With that confession aside, The Gene was probably one of the best (and for me, most enlightening) popular science books I have ever read. This is definitely aided by Mukherjee’s fluid and beautiful writing style from which scientists in all fields can learn a few lessons about scientific communication. The Gene is also touched with a humanity that is not usually associated with the popular science genre, which is usually rather dry in recounting scientific and intellectual endeavors. This humanity is the book’s most powerful feature.

Since there are many glowing reviews of the book published elsewhere, I will just list here a few nuggets I took away from The Gene, which hopefully will serve to entice rather than spoil the book for you:

  • Mukherjee compares the gene to an atom or a bit, evolution’s “indivisible” particle. Obviously, the gene is physically divisible in the sense that it is made of atoms, but what he means here is that the lower levels can be abstracted away and the gene is the relevant level at which geneticists work.
    • It is worth thinking of what the parallel carriers of information are in condensed matter problems — my hunch is that most condensed matter physicists would contend that these are the quasiparticles in the relevant phase of matter.
  • Gregor Mendel, whose work nowadays is recognized as giving birth to the entire field of genetics, was not recognized for his work while he was alive. It took another 40-50 years for scientists to rediscover his experiments and to see that he had localized, in those pea plants, the indivisible gene. One gets the feeling that his work was not celebrated while he was alive because his work was far ahead of its time.
  • The history of genetics is harrowing and ugly. While the second World War was probably the pinnacle of obscene crimes committed in the name of genetics, humans seem unable to shake off ideas associated with eugenics even into the modern day.
  • Through a large part of its history, the field of genetics has had to deal with a range of ethical questions. There is no sign of this trend abating in light of the recent discovery of CRISPR/Cas-9 technology. If you’re interested in learning more about this, RadioLab has a pretty good podcast about it.
  • Schrodinger’s book What is Life? has inspired so much follow-up work that it is hard to overestimate the influence it has had on a generation of physicists that transitioned to studying biology in the middle of the twentieth century, including both Watson and Crick.

While I could go on and on with this list, I’ll stop ruining the book for you. I would just like to say that at the end of the book I got the feeling that humans are still just starting to scratch the surface of understanding what’s going on in a cell. There is much more to learn, and that’s an exciting feeling in any field of science.

Aside: In case you missed March Meeting, the APS has posted the lectures from the Kavli Symposium on YouTube, which includes lectures from Duncan Haldane and Michael Kosterlitz among others.

Do you ever get the feeling that…

…when you look at science today that things seem blown way out of proportion?

I get the feeling that many press releases make a big deal out of experiments/theoretical work that are not groundbreaking, are not going to cause an upheaval in anyone’s way of thinking and frankly, are humdrum science (not to diminish the importance of humdrum science!).

In all honesty, really great scientific works are rare and sometimes it takes a long time to recognize the importance of a great leap in understanding. There are many examples of this, but here’s one: Gregor Mendel, who I would refer to as the discoverer of the gene, died before his work was recognized as truly path-breaking, which took about 50 years.

A lot of good science happens all the time, but let’s not kid ourselves — the science is not as revolutionary as a lot of press releases make it seem. Of course, most professional scientists are aware of this, but to the young graduate student and to the public at large, press releases can easily be mistaken for groundbreaking science and often are. How many times have you come across someone from outside of science excited about an article they read online that you know is either extremely speculative or actually pretty mundane? It is hard to respond to reactions like this because you don’t want to dampen someone’s excitement about a subject you care about!

I don’t know what is driving all of this — the media, funding agencies, university rankings or some other metric, but to be perfectly honest, I find much of the coverage on sites like Phys.Org ugly, cynical and detrimental.

While it can be argued that this media coverage does serve some important purpose, it seems to me that this drive to “sell one’s work” may have the adverse effect of exacerbating impostor syndrome (especially among younger colleagues), which is already rampant in physics departments as well as in other academic fields (i.e. you feel like because you need to “sell your work”, and because it gets blown way out of proportion, that you have manipulated people into thinking your work is more important than you really know it to be).

If you just went about your business, trying to do science you think is worthy (without the citation-counting and the excessive media coverage), my guess is science (and more importantly scientists!) would probably be healthier.

I know this viewpoint is pretty one-dimensional and lacks some nuance, so I would like to encourage comments and especially opposing opinions.