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.)