How hard is it to violate Pauli's exclusion principle?

Ultracooled rubidium atoms are bosonic, and start to behave as part of a collective fluid, their properties varying together like shown above. Bosons can do this because they don't obey Pauli's principle.
Ultracooled rubidium atoms are bosonic, and start to behave as part of a collective fluid, their properties varying together like shown above. Bosons can do this because they don’t obey Pauli’s principle. Photo: Wikimedia Commons

A well-designed auditorium always has all its seats positioned on an inclined plane. ​Otherwise it wouldn’t be well-designed, would it? Anyway, this arrangement solves an important problem: It lets people sit anywhere they want to irrespective of their heights.

It won’t matter if a taller person sits in front of a shorter one – the inclination will render their height-differences irrelevant.

However, if the plane had been flat, if all the seats were just placed one behind another instead of raising or lowering their distances from the floor, ​then people would have been forced to follow a particular seating order. Like the discs in a game of Tower of Hanoi, the seats must be filled with shorter people coming first if everyone’s view of the stage must be unobstructed.

It’s only logical.​

A similar thing happens inside atoms. ​While protons and neutrons are packed into a tiny nucleus, electrons orbit the nucleus in relatively much larger orbits. For instance, if the nucleus is 2 m across, then electrons would be orbiting it at up to 10 km away. This is because every electron can only be so far away that its negative charge doesn’t pull it into the nucleus.

However, this doesn’t mean all electrons orbit the nucleus at the same distance. They follow an order. Like the seats on the flat floor where taller people must sit behind shorter ones, more energetic electrons must orbit closer to the nucleus than less energetic ones. Similarly, all electrons of the same energy must orbit the nucleus at the same distance.

Over the years, scientists have observed that around every atom of a known element, there are well-defined energy levels, each accommodating a fixed and known number of electrons. These quantities are determined by various properties of electrons, designated by the particle’s four quantum numbers: n, l, m_s, m_l.


1. n is the principle quantum number, and designates the energy-level of the electron.​

​2. l is the azimuthal quantum number, and describes the angular momentum at which the electron is zipping around the nucleus.

​3. m_l is the orbital quantum number and yields the value of l along a specified axis.

4. s is the spin quantum number and describes the “intrinsic” angular momentum, a quantity that doesn’t have a counterpart in Newtonian mechanics.​

So, an electron’s occupation of some energy slot around a nucleus depends on the values of the four quantum numbers. ​And the most significant relation between all of them is the Pauli exclusion principle (PEP): no two electrons with all four same quantum numbers can occupy the same quantum state.

An energy level is an example of a quantum state. This means if two electrons exist at the same level inside an atom, and if their n, l and m_l values are equal, then their m_s value (i.e., spin) must be different: one up, one down. Two electrons with equal n, l, m_l, and m_s values couldn’t occupy the same level in the same atom.

But why?​

The PEP is named for its discoverer, Wolfgang Pauli. Interestingly, Pauli himself couldn’t put a finger why the principle was the way it was. From his Nobel lecture, 1945 (PDF):​

Already in my original paper I stressed the circumstance that I was unable to give a logical reason for the exclusion principle or to deduce it from more general assumptions. I had always the feeling and I still have it today, that this is a deficiency. … The impression that the shadow of some incompleteness [falls] here on the bright light of success of the new quantum mechanics seems to me unavoidable.

It wasn’t that the principle’s ontology was sorted over time. In 1963, Richard Feynman said:

“…. Why is it that particles with half-integral spin are Fermi particles (…) whereas particles with integral spin are Bose particles (…)? We apologize for the fact that we can not give you an elementary explanation. An explanation has been worked out by Pauli from complicated arguments from quantum field theory and relativity. He has shown that the two must necessarily go together, but we have not been able to find a way to reproduce his arguments on an elementary level. It appears to be one of the few places in physics where there is a rule which can be stated very simply, but for which no one has found a simple and easy explanation. (…) This probably means that we do not have a complete understanding of the fundamental principle involved. For the moment, you will just have to take it as one of the rules of the world.

​(R. Feynman, Feynman Lectures of Physics, 3rd Vol., Chap. 4, Addison-Wesley, Reading, Massachusetts, 1963)

The Ramberg-Snow experiment​

In 1990, two scientists, Ramberg and Snow, devised a simple experiment to study the principle. They connected a thin strip of copper to a 50-ampere current source. Then, they placed an X-ray detector over ​the strip. When electric current passed through the strip, X-rays would be emitted, which would then be picked by the detector for analysis.

How did this happen?​

When electrons jump from a higher-energy (i.e., farther) energy-level to a lower-energy (closer) one, they must lose some energy to be permitted their new status. The energy can be lost as light, X-rays, UV- radiation, etc. Because we know how many distinct energy-levels there are in the atoms of each element and how much energy each of those orbitals has, electrons jumping levels for different elements must lose different, but fixed, amounts of energy.

So, when current is passed through copper, extra electrons are introduced into the metal, precipitating the forced occupation of some energy-level, like people sitting in the aisles of a full auditorium.

In this scenario, or in any other one for that matter, an electron jumping from the 2p level to the 1s level in a copper atom ​must lose 8.05 keV as X-rays – no more, no less, no differently.

However, Ramberg and Snow found that, after over two months of data-taking at a basement in Fermilab, Illinois, ​about 1 in 170 trillion trillion X-ray signals didn’t contain 8.05 keV but 7.7 keV.

The 1s orbital usually has space for two electrons going by the PEP. If one slot’s taken and the other’s free, then an electron wanting to jump in from the 2p level must lose 8.05 keV. However, ​if an electron was losing 7.7 keV, where was it going?

After some simple calculations, the scientists made a surprising discovery.​ The electron was squeezing itself in with two other electrons in the 1s level itself – instead of resorting to the aisles, it was sitting on another electron’s lap! This meant that the PEP was being violated with a probability of 1 in 170 trillion trillion.

While this is a laughably minuscule number, it’s nevertheless a positive number even taking into account possibly large errors arising out of the unsophisticated nature of the Ramberg-Snow apparatus. Effectively, where we thought there ought to be no violations, there were.

Just like that, there was a hole in our understanding of the exclusion principle.​

And it was the sort of hole with which we could make lemonade.

Into the kitchen​

So fast forward to 2006, 26 lemonade-hungry physicists, one pretentiously titled experiment, and one problem statement: Could the PEP be violated much more or much less often than once in 170 trillion trillion?

The setup was called the VIP for ‘VIolation of the Pauli Exclusion Principle Experiment’.​ How ingenious. Anyway, the idea was to replicate the Ramberg-Snow experiment in a more sophisticated environment. Instead of a simple circuit that you could build on the table top, they used one in the Gran Sasso National Lab that looked like this.

This is the DEAR (DAΦNE Exotic Atom Research) setup and was slightly modified to make way for the VIP setup. Everything’s self-evident, I suppose. CCD stands for charge-coupled detector, which is basically an X-ray detector.

(The Gran Sasso National Lab, or Laboratori Nazionali del Gran Sasso, is one of the world’s largest underground particle physics laboratories, consisting of around 1,000 scientists working on more than 15 experiments.​ It is located near the Gran Sasso mountain, between the towns of L’Aquila and Teramo in Italy.)​

​After about three years of data-taking, the team of 26 announced that it had bettered the Ramberg-Snow data by three orders of magnitude. According to data made available in 2009, they declared the PEP had been violated only once every 570,000 trillion trillion electronic level-jumps.​

Fewer yet surely

​Hurrah! The principle was being violated 1,000 times less often than thought, but it was being violated still. At this stage, the VIP team seemed to have thought the number could be much lesser, even 100-times lesser. On March 5, 2013, it submitted a paper (PDF) to the arXiv pre-print server containing a proposal for the more-sensitive VIP2.

​You might think that the number is positive, so VIP’s efforts an attempt to figure out how many angels are dancing on the head of a pin.

Well, think about this way. The moment we zero in on one value, one frequency with which anomalous level-jumps take place, then we’ll be in a position to stick the number into a formula and see what that means for the world around us.

Also, electrons are only one kind of a class of particles called fermions, all of which are thought to obey the PEP. Perhaps other experiments conducted with other fermions, such as tau leptons and muons, will throw up some other rate of violation. In that case, we’ll be able to say the misbehavior is actually dependent on some property of the particle, like its mass, spin, charge, etc.

Until that day, we’ve got to keep trying.​

(This blog post first appeared at The Copernican on March 11, 2013.)

Where does the Higgs boson come from?

When the Chelyabinsk meteor – dubbed Chebarkul – entered Earth’s atmosphere at around 17 km/s, it started to heat up due to friction. After a point, cracks already present on the chunk of rock weighing 9,000-tonnes became licensed to widen and eventually split off Chebarkul into smaller parts.

While the internal structure of Chebarkul was responsible for where the cracks widened and at what temperature and other conditions, the rock’s heating was the tipping point. Once it got hot enough, its crystalline structure began to disintegrate in some parts.

Spontaneous symmetry-breaking

About 13.75 billion years ago, this is what happened to the universe. At first, there was a sea of energy, a symmetrically uniform block. Suddenly, this block was rapidly exposed to extreme heat. Once it hit about 1015 kelvin – 173 billion times hotter than our Sun’s surface – the block disintegrated into smaller packets called particles. Its symmetry was broken. The Big Bang had happened.

The Big Bang splashed a copious amount of energy across the universe, whose residue is perceivable as the CMBR.

Quickly, the high temperature fell off, but the particles couldn’t return to their original state of perfect togetherness. The block was broken forever, and the particles now had to fend for themselves. There was a disturbance, or perturbations, in the system, and the forces started to act. Physicists today call this the Nambu-Goldstone (NG) mode, named for Jeffrey Goldstone and Yoichiro Nambu.

In the tradition of particle physics treating with everything in terms of particles, the forces in the NG mode were characterised in terms of NG bosons. The exchange of these bosons between two particles meant they were exchanging forces. Since each boson is also a particle, a force can be thought of as the exchange of energy between two particles or bodies.

This is just like the concept of phonons in condensed matter physics: when atoms part of a perfectly arranged array vibrate, physicists know they contain some extra energy that makes them restless. They isolate this surplus in the form of a particle called a phonon, and address the entire array’s surplus in terms of multiple phonons. So, as a series of restlessness moves through the solid, it’ll be like a sound wave moving through it. Simplifies the math.

Anyway, the symmetry-breaking also gave rise to some fundamental forces. They’re called ‘fundamental’ because of their primacy, and because they’re still around. They were born because the disturbances in the energy block, encapsulated as the NG bosons, were interacting with an all-pervading background field called the Higgs field.

The Higgs field has four components, two charged and two uncharged. Another, more common, example of a field is the electric field, which has two components: some strength at a point (charged) and the direction of the strength at that point (neutral). Components of the Higgs field perturbed the NG bosons in a particular way to give rise to four fundamental forces, one for each component.

So, just like in Chebarkul’s case, where its internal structure dictated where the first cracks would appear, in the block’s case, the heating had disturbed the energy block to awaken different “cracks” at different points.

The Call of Cthulhu

The first such “crack” to be born was the electroweak force. As the surroundings of these particles continued to cool, the electroweak force split into two: electromagnetic (eM) and weak forces.

The force-carrier for the eM force is called a photon. Photons can exist at different energies, and at each energy-level, they have a corresponding frequency. If a photon happens to be in the “visible range” of energy-levels, then each frequency shows itself as a colour. And so on…

The force-carriers of the weak forces are the W+, W-, and Z bosons. At the time the first W/Z bosons came to life, they were massless. We know now because of Einstein’s mass-energy equivalence that this means the bosons had no energy. How were they particulate, then?

Imagine an auditorium where an important lecture’s about to be given. You get there early, your friend is late, and you decide to reserve a seat for her. Then, your friend finally arrives 10 minutes after the lecture’s started and takes her seat. In this scenario, after your arrival, the seat was there all along as ‘friend’s seat’, even though your friend took her time to get there.

Similarly, the W/Z bosons, which became quite massive later on, were initially massless. They had to have existed when the weak force came to life, if only to account for a new force that had been born. The debut of massiveness happened when they “ate” the NG bosons – the disturbed block’s surplus energy – and became very heavy.

Unfortunately for them, their snacking was irreversible. The W/Z bosons couldn’t regurgitate the NG bosons, so they were doomed to be forever heavy and, consequently, short-ranged. That’s why the force that they mediate is called the weak force: because it acts over very small distances.

You’ll notice that the W+, W-, and Z bosons make up for only three components of the Higgs field. What about the fourth component?

Enter: Higgs boson

That’s the Higgs boson. And now, getting closer to pinning down the Higgs boson means we’re also getting closer to pinning down the Higgs mechanism as valid, a quantum mechanical formulation within which we understand the behaviours of these particles and forces. This formulation is called the Standard Model.

(This blog post first appeared at The Copernican on March 8, 2013.)

Higgs boson closer than ever

The article, as written by me, appeared in The Hindu on March 7, 2013.

Ever since CERN announced that it had spotted a Higgs boson-like particle on July 4, 2012, their flagship Large Hadron Collider (LHC), apart from similar colliders around the world, has continued running experiments to gather more data on the elusive particle.

The latest analysis of the results from these runs was presented at a conference now underway in Italy.

While it is still too soon to tell if the one spotted in July 2012 was the Higgs boson as predicted in 1964, the data is convergent toward the conclusion that the long-sought particle does exist and with the expected properties. More results will be presented over the upcoming weeks.

In time, particle physicists hope that it will once and for all close an important chapter in physics called the Standard Model (SM).

The announcements were made by more than 15 scientists from CERN on March 6 via a live webcast from the Rencontres de Moriond, an annual particle physics forum that has been held in La Thuile, Italy, since 1966.

“Since the properties of the new particle appear to be very close to the ones predicted for the SM Higgs, I have personally no further doubts,” Dr. Guido Tonelli, former spokesperson of the CMS detector at CERN, told The Hindu.

Interesting results from searches for other particles, as well as the speculated nature of fundamental physics beyond the SM, were also presented at the forum, which runs from March 2-16.

Physicists exploit the properties of the Higgs to study its behaviour in a variety of environments and see if it matches with the theoretical predictions. A key goal of the latest results has been to predict the strength with which the Higgs couples to other elementary particles, in the process giving them mass.

This is done by analysing the data to infer the rates at which the Higgs-like particle decays into known lighter particles: W and Z bosons, photons, bottom quarks, tau leptons, electrons, and muons. These particles’ signatures are then picked up by detectors to infer that a Higgs-like boson decayed into them.

The SM predicts these rates with good precision.

Thus, any deviation from the expected values could be the first evidence of new, unknown particles. By extension, it would also be the first sighting of ‘new physics’.

Bad news for new physics, good news for old

After analysis, the results were found to be consistent with a Higgs boson of mass near 125-126 GeV, measured at both 7- and 8-TeV collision energies through 2011 and 2012.

The CMS detector observed that there was fairly strong agreement between how often the particle decayed into W bosons and how often it ought to happen according to theory. The ratio between the two was pinned at 0.76 +/- 0.21.

Dr. Tonelli said, “For the moment, we have been able to see that the signal is getting stronger and even the difficult-to-measure decays into bottom quarks and tau-leptons are beginning to appear at about the expected frequency.”

The ATLAS detector, parallely, was able to observe with 99.73 per cent confidence-level that the analysed particle had zero-spin, which is another property that brings it closer to the predicted SM Higgs boson.

At the same time, the detector also observed that the particle’s decay to two photons was 2.3 standard-deviations higher than the SM prediction.

Dr. Pauline Gagnon, a scientist with the ATLAS collaboration, told this Correspondent via email, “We need to asses all its properties in great detail and extreme rigour,” adding that for some aspects they would need more data.

Even so, the developments rule out signs of any new physics around the corner until 2015, when the LHC will reopen after a two-year shutdown and multiple upgrades to smash protons at doubled energy.

As for the search for Supersymmetry, a favoured theoretical concept among physicists to accommodate phenomena that haven’t yet found definition in the Standard Model: Dr. Pierluigi Campana, LHCb detector spokesperson, told The Hindu that there have been only “negative searches so far”.

Ironing out an X-ray wrinkle

A version of this post, as written by me, originally appeared in The Copernican science blog on March 1, 2013.

One of the techniques to look for and measure the properties of a black hole is to spot X-rays of specific energies coming from a seemingly localized source. The radiation emanates from heated objects like gas molecules and dust particles in the accretion disc around the black hole that have been compressed and heated to very high temperatures as the black hole prepares to gorge on them.

However, the X-rays are often obscured by gas clouds surrounding the black hole, even at farther distances, and then other objects on the path of its long journey to Earth. And this is a planet whose closest black hole is 246 quadrillion km away. This is why the better telescopes that study X-rays are often in orbit around Earth instead of on the ground to minimize further distortions due to our atmosphere.


One of the most powerful such X-ray telescopes is NASA’s NuSTAR (Nuclear Spectroscopic Telescope Array), and on February 28, the government body released data from the orbiting eye almost a year after it was launched in June 2012. NuSTAR studies higher-energy the sources and properties of higher-energy X-rays in space. In this task, it is also complemented by the ESA’s XMM-Newton space-telescope, which studies lower-energy X-rays.

The latest data concerns the black hole at the centre of the galaxy NGC 1365, which is two million times the mass of our Sun, earning it the title of Supermassive Black Hole (SMBH). Around this black hole is an accretion disc, a swirling vortex of gases, metals, molecules, basically anything unfortunate enough to have come close and subsequently been ripped apart. Out of this, NuSTAR and XMM-Newton zeroed in on the X-ray emissions characteristic of iron.

What the data revealed, as has been the wont of technology these days, is a surprise.

What are signatures for?

Emission signatures are used because we know everything about them and we know what makes each one unique. For instance, knowing the rise and dip of X-ray brightness coming from an object at different temperatures allows us to tell whether the source is iron or something else.

By extension, knowing that the source is iron lets us attribute the signature’s distortions to different causes.

And the NuSTAR data has provided the first experimental proof that iron’s signature’s distortion is not due to gas-obscuration, but due to another model called prograde rotation which attributes the distortion to the black hole’s gravitational pull.

A clearer picture

As scientists undertake a detailed analysis of the NASA data, they will also resolve iron’s signature. This means the plot of its emissions at different temperatures and times will be split up into different “colours” (i.e., frequencies) to see how much each colour has been distorted.

With NuSTAR in the picture, this analysis will assume its most precise avatar yet because the telescope’s ability to track higher-energy X-rays lets it penetrate well enough into the gas clouds around black holes, something that optical telescopes like the one at the Keck Observatory can’t. What’s more, the data will also be complete at the higher-energy levels, earlier left blank because XMM-Newton or NASA’s Chandra couldn’t see in that part of the spectrum.

If the prograde rotation model is conclusively proven after continued analysis and more measurements, then for the first time, scientists will have a precision-measurement tool on their hands to calculated black hole spin.

How? If the X-ray distortions are due to the black hole’s gravitational pull and nothing else, then the rate of its spin should be showing itself as the amount of distortion in the emission signature. The finer details for this can be picked out from the resolved data, and a black hole’s exact spin for the first time be pinned down.

The singularity

NGC 1365 is a spiral galaxy about 60 million light-years in the direction of the constellation Fornax and a prominent member of the much-studied Fornax galaxy cluster. Apart from the black hole, the galaxy hosts other interesting features such as a central bar of gas and dust adjacent to prominent star-forming regions and a type-1a supernova discovered as recently as October 27, 2012.

As we sit here, I can’t help but imagine us peering into our tiny telescopes, picking up on feebly small bits of information, and adding an extra line in our textbooks, but in reality being thrust into an entirely new realm of knowledge, understanding, and awareness.

Now, we know there’s a black hole out there – a veritable freak of nature – spinning as fast as the general theory of relativity will allow!