# Monthly Archives: May 2016

One of the recurring struggles of being a physicist, especially for those early in their career, is how to balance depth and breadth of topics. In pursuing a PhD, it is necessary to study a particular topic in great detail, read the previous literature on the subject, and in some sense, become an expert in a very narrow area. One then needs to solve a problem in this area. In reality, this is all that is needed to obtain a PhD.

To become a good physicist, though, requires that one has a broad and general overview of, in our case, condensed matter physics and even topics beyond. Obviously, this is not the only trait one must have in order to become a good physicist, but it is indeed one of them.

Colloquially, there is therefore a balance that needs to be struck between “knowing a little bit about everything and a lot about nothing” vs. “knowing everything about something that is almost nothing and nothing about anything“.

Becoming a good physicist therefore requires both a broad physical knowledge and a depth of knowledge in a few specialized topics. It requires one to “zoom in” and focus on a narrow field, and solve a problem. It then requires one to “zoom out” to understand its implications on the grander scale for condensed matter physics or physics in general.

The thing about striking this balance between depth and breadth is that it is extremely difficult to do! There are questions that arise like:

• For us in condensed matter physics, is learning particle physics “too broad”?
• What about learning topics like computer science, electronics or economics?

I think that these questions are challenging to answer, partly because the answers will vary from person to person. There are numerous examples of physicists pursuing subjects like economics, biology, neuroscience, philosophy and computer science with great success.

During graduate school, the strategy I employed was to spend the day doing research, remaining narrow, while spending the evening reading widely in attempt to broaden my knowledge and understand why my research was of any importance at all. This was a decent strategy for me, but I can see others pursuing different schemes.

I still struggle with this dichotomy relatively often, and it is not one I see vanishing any time soon. I’m curious to know how others approach this problem, so please feel free to comment.

## European League Underdogs

I’m taking a little detour away from physics blogging today to dip my toe into the world of football (soccer). With Leicester City’s success in the Premier League this season, I feel compelled to dedicate one post to this most unlikely of events. To those not familiar with English football, prior to the beginning of the season, Leicester City FC was famously a whopping 5000/1 bet to win the Premier League. There is an amusing BBC video demonstrating how unlikely this victory was by comparing to some other 5000/1 bets. Teaser: bookies are giving 5000/1 odds on Elvis being alive.

The last decade of world football has been dominated by a style of football colloquially referred to as “tiki-taka”. This type of football has been most often associated with Barcelona FC (and more recently with Bayern Munich) at the club level and by Spain on the international front. Tiki-taka is built on a heavily possession-based game. This high percentage of possession has two effects:

1. It denies the opposition chances to score by denying them the ball
2. Players of extremely high technical ability are required so that:
• They can retain possession in tight spaces
• They are able to find openings in the opposing team’s defense

Tiki-taka has probably been the most successful strategy in modern football, but it requires the best players in the world to execute the strategy. Therefore, this is a luxury that only teams with a great deal of money can employ.

The question then naturally arises: how is a team with fewer resources supposed to compete against the teams with better players?

To answer this question, I would like to concentrate on three teams, one from the Premier League (Leicester City), one from the German Bundesliga (Borussia Dortmund) and one from La Liga in Spain (Athletico Madrid). In the past seven years, these three teams, despite being at a huge financial disadvantage compared to other teams in their leagues, were still able to win their respective domestic leagues.

For those who are unfamiliar with football, in the Premier League, the teams with the most money are Manchester City, Manchester United, Chelsea and (despite what Arsene Wenger will tell you) Arsenal. In the German league, it is without a doubt Bayern Munich. In the Spanish league, Real Madrid and Barcelona are by far the richest teams in the league.

So how did the aforementioned underdogs overcome the odds and win their respective leagues on a shoestring budget? I contend that the answer is with a three-pronged strategy:

#### 1. Tackle Hard

Take a look at the plot below. It ranks the season’s champions in terms of tackles per game. For example, in the Bundesliga season ending in 2013, the champions were Bayern Munich and they ranked 18th (dead last) in their league in terms of tackles/game. It is important to take into consideration when reading the plots that the English and Spanish leagues have 20 teams, while the German league has 18 teams.

This plot shows that on the way to their league titles, Athletico Madrid and Leicester ranked 1st in the league in the number of tackles per game. Borussia Dortmund ranked 2nd in tackles/game for both their 2011 and 2012 trophies.

Note that it is rare for a team to win the league with the highest number of tackles per game. This is especially true in Germany where from 2013-2016 Bayern Munich, who won the Bundesliga in those four seasons, ranked dead last in tackles/game.

#### 2. Tackle Smart

Just because an underdog team tackles harder, however, doesn’t mean that they will miraculously start winning games. Take a look now at the plot below. It shows the end-of-season league position of the team with the highest number of tackles/game.

One can see that there is no pattern at all! In fact, if we take the average league position over the past seven years of the top-tackling teams we get:

1. Premier League: 9.4
2. Bundesliga: 8.4
3. La Liga: 9.4

Keeping in mind that the Bundesliga only has 18 teams, this says that the tackles/game statistic is basically irrelevant! However, I showed in the previous section that Leicester, Athletico and Dortmund all had a high tackles/game trait in common. What gives?!

The data shows that Leicester, Athletico and Dortmund do something different. They don’t just tackle hard, they tackle hard and smart. This means three things:

1. They seek to win the ball back in dangerous positions
2. After recognizing a dangerous position, they win the ball back by pressing with more than one player
3. They take advantage of the turnovers by producing goal opportunities, especially on the counter attack

This means that what these three teams do is defend well as a team. It’s no use having many great tacklers on the team if they do not collectively pressurize the ball when they see an opponent in a weak position. If one watches as much football as I do, one will realize the extent to which Athletico Madrid has turned this collective defending almost into an art form.

#### 3. Don’t Worry about Possession and Take Advantage of Counter Attacks

The last important ingredient is to take advantage of this collective tackling and use it to benefit on the counter attack. Many teams that tackle hard are too slow to get the ball forward and therefore lose their tackling advantage.

Leicester (8 counter attack goals) and Dortmund (11 counter attack goals) ranked 1st in the number of counter attack goals scored in the 2016 and 2012 seasons, when they respectively won their domestic titles. Athletico Madrid (8 counter attack goals), on the other hand, ranked 3rd, behind (you guessed it!) Barcelona (10 counter attack goals) and Real Madrid (13 counter attack goals). It should be kept in mind, however, that Real Madrid and Barcelona outscored Athletico over the season by 27 and 23 goals respectively. Therefore, it can be said that these three underdogs put a huge emphasis on effective counter-attacking football.

Needless to say, this means that these three teams do not put an emphasis on possession-based football unlike their tiki-taka counterparts. In their championship winning seasons, the highest possession any of these teams had over the course of the season was Dortmund in 2012 with a measly 53.3%. In fact, in the 2016 season, Leicester had 44.3% possession over the course of the season, third last in the league! Athletico also routinely finishes outside the top 5 in La Liga in possession stats, including their title-winning season.

#### Putting All of it Together

Okay, so that was probably a lot to process, so let me summarize and paint a picture of what the data tells us.

It turns out that the underdog champions have quite a bit in common with each other. One significant revelation from these statistics is that it is not only important to tackle hard but to also tackle smart. This intelligence in tackling requires the entire team to be on the same wavelength. Teammates need to help each other to win the ball back and the team’s forwards need to make themselves available for a pass immediately after dispossessing the opposition. Take a look at this video (taken from here) to see how fast Leicester can turn defense into attack. Athletico and Dortmund also employ similar lightning-paced counterattacks.

There is an important lesson here for teams that do not have the wealth of the Barcelonas and Real Madrids of the world. There is a way by which one can beat these teams. This requires an extreme dedication to a collective defensive strategy coupled with an emphasis on direct counterattacking football on the offensive end. To play with this strategy requires the entire team to buy into this ethos. These teams cannot have “luxury players” that don’t graft.

Lastly, let me say that following Dortmund and Athletico’s domestic success, both teams went on to play in the UEFA Champions League (UCL) final. In Athletico’s case, they are in the European Cup final again this year (to be played May 28th). This begs the question as to how far Leicester can go next year in the UCL. Unfortunately, bookies have wised up and are only giving Leicester 100/1 odds to win the entire thing.

Most of the statistics were obtained at http://www.whoscored.com

## Diffraction, Babinet and Optical Transforms

In an elementary wave mechanics course, the subject of Fraunhofer diffraction is usually addressed within the context of single-slit and double-slit interference. This is usually followed up with a discussion of diffraction from a grating. In these discussions, one usually has the picture that light is “coming through the slits” like in the image below:

Now, if you take a look at Ashcroft and Mermin or a book like Elements of Modern X-ray Physics by Als-Nielsen and McMorrow, one gets a somewhat different picture. These books make it seem like X-ray diffraction occurs when the “scattered radiation from the atoms add in phase”, as in the image below (from Ashcroft and Mermin):

So in one case it seems like the light is emanating from the free space between obstacles, whereas in the other case it seems like the obstacles are scattering the radiation. I remember being quite confused about this point when first learning X-ray diffraction in a solid-state physics class, because I had already learned Fraunhofer diffraction in a wave mechanics course. The two phenomena seemed different somehow. In their mathematical treatments, it almost seemed as if for optics, light “goes through the holes” but for X-rays “light bounces off the atoms”.

Of course, these two phenomena are indeed the same, so the question arises: which picture is correct? Well, they both give correct answers, so actually they are both correct. The answer as to why they are both correct has to do with Babinet’s principle. Wikipedia summarizes Babinet’s principle, colloquially, as so:

the diffraction pattern from an opaque body is identical to that from a hole of the same size and shape except for the overall forward beam intensity.

To get an idea of what this means, let’s look at an example. In the images below, consider the white space as openings (or slits) and the black space as obstacles in the following optical masks:

What would the diffraction pattern from these masks look like? Well, below are the results (taken from here):

Apart from minute differences close to the center, the two patterns are basically the same! If one looks closely enough at the two images, there are some other small differences, most of which are explained in this paper.

Hold on a second, you say. They can’t be the exact same thing! If I take the open space in the optical mask on the left and add it to the open space on the mask to the right, I just have “free space”. And in this case there is no diffraction! You don’t get the diffraction pattern with twice the intensity. This is of course correct. I have glossed over one small discrepancy. First, one needs to realize that intensity is related to amplitude as so:

$I \propto |A|^2$

This implies that the optical mask on the left and the one on the right give the same diffraction intensity, but that the amplitudes are 180 degrees out of phase. This phase doesn’t affect the intensity, though, as in the formula above intensity is only related to the magnitude of the amplitude. Therefore the masks, while giving the same intensity, are actually slightly different. The diffraction pattern will then cancel when the optically transparent parts of the two masks are added together. It’s strange to think that “free space” is just a bunch of diffraction patterns cancelling each other out!

With this all in mind, the main message is pretty clear though: optical diffraction through slits and the Ashcroft and Mermin picture of “bouncing off atoms” are complementary pictures of basically the same diffraction phenomenon. The diffraction pattern obtained will be the same in both cases because of Babinet’s principle.

This idea has been exploited to generate the excellent Atlas of Optical Transforms, where subtleties in crystal structures can be manipulated at the optical scale. Below is an example of such an exercise (taken from here). The two images in the first row are the optical masks, while the bottom row gives the respective diffraction patterns. In the first row, the white dots were obtained by poking holes in the optical masks.

Basically, what they are doing here is using Babinet’s principle to image the diffraction from a crystal with stacking faults along the vertical direction. The positions of the atoms are replaced with holes. One can clearly see that the effect of these stacking faults is to smear out and broaden some of the peaks in the diffraction pattern along the vertical direction. This actually turns out to gives one a good intuition of how stacking faults in a crystal can distort a diffraction pattern.

In summary, the Ashcroft and Mermin picture and the Fraunhofer diffraction picture are really two ways to describe the same phenomenon. The link between the two explanations is Babinet’s principle.

## With Great Beam Time Comes Great Humor

There is a well-written and amusing article on PBS, that focuses on the role that humor has played in the history of physics. I regurgitate one of the jokes from the article below:

James Clerk Maxwell and Michael Faraday were traveling together on the Caledonian Express. Outside it was pouring rain. Maxwell pointed to a fellow in a red cap and uniform standing near an open window.

“That trainman is incompetent,” Maxwell said. “He collects nary a ticket and calls out the wrong names for all the stations. Mostly all he does it is lean against the window with his eyes shut. What a disgrace to the railway!”

Suddenly a flash of lightning entered the carriage through the window and hit the trainman directly. The current bounced right off of him and leapt to the floor. Miraculously, the man was completely unscathed.

“Aye, that’s simple,” said Maxwell. “He’s a bad conductor.”

I agree with the article that humor serves an important daily function in our lives, one that is often under-appreciated.

## Broken Symmetry and Degeneracy

Often times, when I understand a basic concept I had struggled to understand for a long time, I wonder, “Why in the world couldn’t someone have just said that?!” A while later, I will then return to a textbook or paper that actually says precisely what I wanted to hear. I will then realize that the concept just wouldn’t “stick” in my head and required some time of personal and thoughtful deliberation. It reminds me of a humorous quote by E. Rutherford:

All of physics is either impossible or trivial.  It is impossible until you understand it and then it becomes trivial.

I definitely experienced this when first studying the relationship between broken symmetry and degeneracy. As I usually do on this blog, I hope to illustrate the central points within a pretty rigorous, but mostly picture-based framework.

For the rest of this post, I’m going to follow P. W. Anderson’s article More is Different, where I think these ideas are best addressed without any equations. However, I’ll be adding a few details which I wished I had understood upon my first reading.

If you Google “ammonia molecule” and look at the images, you’ll get something that looks like this:

With the constraint that the nitrogen atom must sit on a line through the center formed by the triangular network of hydrogen atoms, we can approximate the potential to be one-dimensional. The potential along the line going through the center of the hydrogen triangle will look, in some crude approximation, something like this:

Notice that the molecule has inversion (or parity) symmetry about the triangular hydrogen atom network. For non-degenerate wavefunctions, the quantum stationary states must also be parity eigenstates. We expect, therefore, that the stationary states will look something like this for the ground state and first excited state respectively:

Ground State

First Excited State

The tetrahedral (pyramid-shaped) ammonia molecule in the image above is clearly not inversion symmetric, though. What does this mean? Well, it implies that the ammonia molecule in the image above cannot be an energy eigenstate. What has to happen, therefore, is that the ammonia molecule has to oscillate between the two configurations pictured below:

The oscillation between the two states can be thought of as the nitrogen atom tunneling from one valley to the other in the potential energy diagram above. The oscillation occurs about 24 billion times per second or with a frequency of 24 GHz.

To those familiar with quantum mechanics, this is a classic two-state problem and there’s nothing particularly new here. Indeed, the tetrahedral structures can be written as linear combinations of the symmetric and anti-symmetric states as so:

$| 1 \rangle = \frac{1}{\sqrt{2}} (e^{i \omega_S t}|S\rangle +e^{i \omega_A t}|A\rangle)$

$| 2 \rangle = \frac{1}{\sqrt{2}} (e^{i \omega_S t}|S\rangle -e^{i \omega_A t}|A\rangle)$

One can see that an oscillation frequency of $\omega_S-\omega_A$ will result from the interference between the symmetric and anti-symmetric states.

The interest in this problem, though, comes from examining a certain limit. First, consider what happens when one replaces the nitrogen atom with a phosphorus atom (PH3): the oscillation frequency decreases to about 0.14 MHz, about 200,000 times slower than NH3. If one were to do the same replacement with an arsenic atom instead (AsH3), the oscillation frequency slows down to 160 microHz, which is equivalent to about an oscillation every two years!

This slowing down can be simply modeled in the picture above by imagining the raising of the barrier height between the two valleys like so:

In the case of an amino acid or a sugar, which are both known to be chiral, the period of oscillation is thought to be greater than the age of the universe. Basically, the molecules never invert!

So what is happening here? Don’t worry, we aren’t violating any laws of quantum mechanics.

As the barrier height reaches infinity, the states in the well become degenerate. This degeneracy is key, because for degenerate states, the stationary states no longer have to be inversion-symmetric. Graphically, we can illustrate this as so:

Symmetric state, $E=E_0$

Anti-symmetric state, $E=E_0$

We can now write for the symmetric and anti-symmetric states:

$| 1 \rangle = e^{i\omega t} \frac{1}{\sqrt{2}} (|S\rangle + |A\rangle)$

$| 2 \rangle = e^{i\omega t} \frac{1}{\sqrt{2}} (|S\rangle - |A\rangle)$

These are now bona-fide stationary states. There is therefore a deep connection between degeneracy and the broken symmetry of a ground state, as this example so elegantly demonstrates.

When there is a degeneracy, the ground state no longer has to obey the symmetry of the Hamiltonian.

Technically, the barrier height never reaches infinity and there is never true degeneracy unless the number of particles in the problem (or the mass of the nitrogen atom) approaches infinity, but let’s leave that story for another day.

## John Oliver on Science

John Oliver on Last Week Tonight did a bit about how science is represented in the media. It is sad, funny and most of it true. You can watch it here:

Amusingly, he shows a clip of an interview with Brian Nosek, whose work I have discussed in a similar context previously.