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## Electron configurations

In this blog, I discussed the “electron configuration” nomenclature which is so loved by chemists (strange people that they are….). Just to remind you, the noble gas neon, which is at number 10 in the periodic table, may be written as $1s^{2} \; 2s^{2} \; 2p^{6}$. If you add together the superscripts you get $2+2+6=10$, the number of electrons in neutral Helium. Titanium, which is at number 22 in the periodic table may be written as $1s^{2} \; 2s^{2} \; 2p^{6} \; 3s^{2} \; 3p^{6} \; 3d^{2} \; 4s^{2}$. Again, if you add together the superscripts you get $2+2+6+2+6+2+2=22$, the number of electrons in neutral Titanium. I explained in the blog that the letters s,p,d and f refer to “sharp, principal, diffuse” and “fine“, as this was how the spectral lines appeared in the 1870s when spectroscopists first started identifying them.

But, what I didn’t address in that blog on the electron configuration nomenclature is why do electrons occupy different shells in atoms? In hydrogen, the simplest atom, the 1 electron orbits the nucleus in the ground state, the n=1 energy level. If it is excited it will go into a higher energy level, n=2 or 3 etc. But, with a more complicated atom like neon, which has 10 electrons, the 10 do not all sit in the n=1 level. The n=1 level can only contain up to 2 electrons, and the n=2 level can only contain up to 8 electrons, the n=3 level can only contain up to 18 electrons, and so on. This leads to neon having a “filled” n=1 level (2 electrons), and a filled n=2 level (8 electrons), which means it does not seek additional electrons. This is why it is a noble gas.

Titanium on the other hand, with 22 electrons, has a filled n=1 level (2 electrons), a filled n=2 level (8 electrons), a partially filled n=3 level (8 electrons out of a possible 18), and a partially filled n=4 level (2 electrons out of a possible 32). Because it has partially filled n=3 and n=4 levels, and it wants them to be full, it will seek additional electrons by chemically combining with other elements.

What is the reason each energy level has a maximum number of allowed electrons?

It is all due to something called the Pauli exclusion principle.

Wolfgang Pauli, after whom the Pauli exclusion principle is named. He came up with the idea in 1925. In addition to this principle, he also came up with the idea of the neutrino.

## The energy level n

Niels Bohr suggested in 1913 that electrons could only occupy certain orbits. I go into the details of his argument in this blog, but to summarise it briefly here, he suggested that something called the orbital angular momentum of the electron had to be divisible by $\hbar \text{ where } \hbar = h/2\pi, \text{ } h$ being Planck’s constant. We now call these the energy levels of an atom, and we use the letter n to denote the energy level. So, an electron in the second energy level will have $n=2$, in the third energy level it will have $n=3$ etc.

As quantum mechanics developed over the next 15-20 years it was realised that an electron is fully described by a total of four (4) quantum numbers, not just its energy level. The energy level $n$ came to be known as the princpical quantum number. The other three quantum numbers needed to fully describe the state of an electron are

• its orbital angular momentum, $l$
• its magnetic moment, $m_{l}$ and
• its spin, $m_{s}$

## The orbital angular momentum $l$ quantum number

As I mentioned above, spectroscopists noticed that atomic lines could be visually categorised into “sharp”, “principal”, “diffuse” and “fine“, or $s,p,d \text{ and } f$. It was found that the following correspondence existed between these visual classifications and the orbital angular momentum $l$. This is the second quantum number. $l$ can only take on certain values from $0 \text{ to } (n-1)$. So, for example, if $n=3, \; l \text{ can be } 0,1 \text{ or } 2$.

spectroscopic name and orbital angular momentum
Spectroscopic Name letter orbital angular momentum $l$
sharp s $l=0$
principal p $l=1$
diffuse d $l=2$
fine f $l=3$

As this table shows, the reason a line appears as a “sharp” (s) line is because its orbital angular momentum $l=0$. If it appears as a “principal” (p) line then its orbital angular momentum must be $l=1$, etc.

## The magnetic moment quantum number $m_{l}$

The third quantum number is the magnetic moment $m_{l}$, which can only take on certain values. The magnetic moment only shows up if the electron is in a magnetic field, and is what causes the Zeeman effect, which is the splitting of an atom’s spectral lines when an atom is in a magnetic field. The rule is that the magnetic moment quantum number can take on any value from $-l \text{ to } +l$, so e.g. when $l=2, \text{ } m_{l}$ can take the values $-2, -1, 0, 1 \text{ and } 2$ (5 possible values in all). If $l=3 \text{ then } m_{l} \text{ can be } -3, -2, -1, 0, 1, 2, 3$ (7 possible values).

## The spin quantum number $m_{s}$

The final quantum number is something called the spin. Although it is only an analogy (and not to be taken literally), one can think of this as the electron spinning on its axis as it orbits the nucleus, in the same way that the Earth spins on its axis as it orbits the Sun. The spin can, for an electron, take on two possible values, either $+1/2 \text{ or } -1/2$.

## Putting all of this together

Let us first of all consider the $n=1$ energy level. The only allowed orbital angular momentum allowed in this level is $l=0$, which means the only allowed values of $m_{l}$ is also 0 and the allowed values of the spin are $+1/2 \text{ and } -1/2$. So, in the $n=1$ level, the only allowed state is $1s$, and this can have two configurations, with the electron spin up or down (+1/2 or -1/2), meaning the $n=1$ level is full when there are 2 electrons in it. That is why we see $1s^{2}$ for Helium and any element beyond it in the Periodic Table. But, what about the $n=2, n=3$ etc. levels?

The number of electrons in each electron shell
State Principal quantum number $n$ Orbital quantum number $l$ Magnetic quantum number $m_{l}$ Spin quantum number $m_{s}$ Maximum number of electrons
1s 1 0 0 +1/2, -1/2 2
n=1 level Total = 2
2s 2 0 0 +1/2, -1/2 2
2p 2 1 -1,0,1 +1/2, -1/2 6
n=2 level Total = 8
3s 3 0 0 +1/2, -1/2 2
3p 3 1 -1,0,1 +1/2, -1/2 6
3d 3 2 -2,-1,0,1,2 +1/2, -1/2 10
n=3 level Total = 18
4s 4 0 0 +1/2, -1/2 2
4p 4 1 -1,0,1 +1/2, -1/2 6
4d 4 2 -2,-1,0,1,2 +1/2, -1/2 10
4f 4 3 -3,-2,-1,0,1,2,3 +1/2, -1/2 14
n=4 level Total = 32
5s 5 0 0 +1/2, -1/2 2
etc.

The astute readers amongst you may have noticed that the electron configuration for Titanium, which was $1s^{2} \; 2s^{2} \; 2p^{6} \; 3s^{2} \; 3p^{6} \; 3d^{2} \; 4s^{2}$, suggests that the $n=4$ level starts being occupied before the $n=3$ level is full. After all, the $n=3$ level can have up to 18 electrons in it, with up to 10 electrons in the $n=3, l=2$ (d) state. In the $n=3$ level the (s) and (p) states are full, but not the (d) state. With only 2 electrons in the $n=3, l=2$ (d) state, the $4s$ state starts being populated, and has 2 electrons in it. Why is this?

I will explain the reason in a future blog, but it has to do with the “shape” of the orbits of the different states. They are different for different values of orbital angular momentum $l$.

## Cosmic neutrinos detected by IceCube

Several months ago, I blogged about the experiment being done by a neutrino detector called IceCube at the South Pole to try to determine the nature of cosmic rays. A couple of weeks ago it was announced by the IceCube team that they had detected, for only the second time ever, neutrinos coming from beyond our Solar System.

## What are neutrinos?

Neutrinos are amongst the most mysterious and elusive particles in nature. They were first proposed back in 1930 by Wolgang Pauli to solve a problem to do with radioactive beta decay. In radioactive beta decay, a neutron will turn into a proton, spitting out a high energy beta particle (which is actually an electron) from the nucleus. Experiments showed that the energy of these electrons varied, which seemed to violate the principle of the conservation of energy.

Pauli suggested that the energy was actually being shared between two particles, the electron and a new particle which he dubbed the neutrino, which means “little neutral one” in Italian. However, it was not until 1956 that they were first actually detected. The reason they took so long to detect is that they do not interact very much with matter. They have no electrical charge, so do not feel the electromagnetic force. They have next to no mass so do not feel the gravitational force, and they do not feel the strong nuclear force which keeps atomic nuclei together.

The only force they feel is the weak nuclear force. As a consequence of how little neutrinos interact with matter, they can pass through the Earth essentially unimpeded. Every seconds, billions pass through your body without interacting at all with any of the atoms in your body. However, very rarely, a neutrino will directly strike an atomic nucleus, and this collision enables us to detect them. IceCube uses huge columns of very pure water buried below the ice-sheet in Antarctica to shield the neutrino detectors from the background radiation and cosmic rays.

## Neutrinos from the Sun

The Sun converts Hydrogen to Helium in its core, in a process known as the proton-proton chain. During this process, in addition to large amounts of energy being produced, neutrinos are generated.

The proton-proton chain in the core of the Sun, which converts Hydrogen to Helium, producing energy and neutrinos in the process.

The Sun is the strongest source of neutrinos beyond our terrestrial laboratories, but when physicists first started detecting neutrinos from the Sun in the 1960s they discovered a problem. It seemed that the Sun was only producing one third of the neutrinos that calculations predicted, or at least we were only detecting one third. This became known as the solar neutrino problem, and was not solved until the last 15 years. As this is quite a fascinating and involved story, I will talk about the solar neutrino problem and its resolution in more detail in a future blog.

## Supernova 1987A

In February 1987 a star was seen to explode in the nearby Large Magellanic Cloud, a satellite galaxy of the Milky Way. It was seen independently by Ian Shelton and Oscar Duhalde on the same evening whilst both were observing at the Las Campanas Observatory in Chile.

This was the first naked-eye supernova since the early 17th Century, and of course allowed astronomers to study supernovae in detail for the first time. But, 3 hours before anyone had seen the supernova, a burst of neutrinos was detected by 3 separate neutrino detectors, the Kamiokande II detector in Japan, the Irive-Michigan-Brooklyn detector in the USA and the Baksan detector in Russia. These neutrinos (strictly speaking, anti-neutrinos) were produced when the core of the dying star collapsed to form a neutron star. In this process, protons and electrons combine to produce neutrons and anti-neutrinos, in a process known as reverse beta decay. The detection of this burst of anti-neutrinos from supernova 1987A was the first time neutrinos were detected from beyond our Solar System.

## The IceCube detections

In the announcement from the IceCube team, they have stated that IceCube has detected 28 cosmic neutrinos to date, but as of yet they do not know from which objects these neutrinos have come. This does, however, bring us a step closer to realising the promise of using neutrinos to better understand the nature of astrophysical objects. In particular, as I described in my previous blog, neutrinos hold the promise of enabling us to understand the origin of high energy cosmic rays. Because the rays themselves are bent by interstellar magnetic fields, tracing their origin is night-on impossible. But, neutrinos are not affected by the magnetic fields, and so should travel to us from their cosmic source in a straight line.

To date, IceCube is the only neutrino detector in the World which is capable of detecting cosmic neutrinos, but with other neutrino detectors being planned and built, we may indeed soon be entering a new era of astronomy.