It is a common image of electrons “orbiting the nucleus” like they orbit the sun. This picture remains enduring, both in popular images of atoms and in the minds many of us who are better informed. Although it is an appealing picture, the 1913 proposal that the centrifugal power of the revolving electronic just perfectly balances the attraction force of nucleus (an analogy to the force exerted by the moon’s orbit in counteracting the Earth’s gravity) was simply untrue.

Figure 1.1: The most popular science images of the Atom show electrons moving about a nucleus, much like planets orbiting the sun. These images are simply not true. These pictures are based on an old idea about the structure and have held up well partly because it is difficult to draw simple pictures of the modern view of how electrons are arranged in an atom.

This hypothesis is supported by the similarities of gravity and Coulombic interactions. Newton’s Law of gravity is the expression for the force of gravity between mass (Newton’s Law of gravity).

Fgravitym1m2r2(1.1)

With

- M1 and m2 represent the respective masses of objects 1 and 2.
- The distance between the object centers is represented by r

The Coulomb force is the difference between two charged species.

FCoulombq1q2r2(1.2)

With

- q1 and Q2 represent the respective charges of object 1 or 2.
- The distance between the object centers is represented by r

An electron is, however, electrically charged. It has been known that any electric charge that experiences acceleration (changes in velocity and direction) will emit electromagnetic waves, which can cause it to lose energy. Revolving electrons would transform an atom into a mini radio station. However, their energy output would be lower than the electron’s potential energy. Classic mechanics states that the electron would spiral into the nucleus, and then the atom would fall.

Figure 1.2: The classical death spiral of an electron around a nucleus.

Quantum theory comes to your rescue!

It became obvious that an extremely small object like the electron could not be considered a classical particle with a fixed position and velocity by the 1920’s. We can only determine the likelihood of it manifesting at any given point in space. Imagine a magic camera capable of taking a series of images of the electron within the 1s orbital a hydrogen atom and combining the dots into one image. You would get something similar to this. The electron is more likely than not to be found as we get closer to the nucleus.

This is confirmed by this plot which shows the quantity of electron charge per unit volume of space at various distances from the nucleus. This is known as a probability density plot. The per unit volume of space part is very important here; as we consider radii closer to the nucleus, these volumes become very small, so the number of electrons per unit volume increases very rapidly. In this view, it appears as if the electron does fall into the nucleus!

Classic mechanics states that the electron would just spiral into the nucleus, and then the atom would fall. Quantum mechanics tells a completely different story.

The Battle of the Infinities Saves the Electron from Its Death Spiral

The potential energy of an electron is affected by its movement towards the nucleus’ attractive field. In fact, it can approach negative infinity. The total energy of a hydrogen atom is constant, so the potential energy loss is not compensated by an increase in electron’s kinetic (sometimes called “confinement”) energy. This energy determines the electron’s momentum and effective velocity.

So as the electron approaches the tiny volume of space occupied by the nucleus, its potential energy dives down toward minus-infinity, and its kinetic energy (momentum and velocity) shoots up toward positive-infinity. This “battle of the infinities” cannot be won by either side, so a compromise is reached in which theory tells us that the fall in potential energy is just twice the kinetic energy, and the electron dances at an average distance that corresponds to the Bohr radius.

This picture is not perfect according to the Heisenberg uncertainty principle

A particle as small as the electron cannot be considered to have either a definite position or momentum. According to the Heisenberg principle, either the location or momentum of quantum particles such as electrons can be determined as precisely as you wish. However, as one of these quantities becomes more precise, the value of each of them becomes increasingly uncertain. This isn’t a matter of observational difficulty but rather fundamental property of nature.

This means that the electron can’t be considered a “particle” within the confines of an atom. It has no definite energy or location. Therefore, it is misleading to speak about the electron “falling into the nucleus.”

Arthur Eddington, a well-known physicist and astronomer, suggested that “wavicle” would be a better description for the electron, but not in jest!

Probability density vs. Radial probability

However, we can talk about the location where the electron is most likely to manifest itself, that is, the area with the greatest negative charge.

This is just the curve labeled “probability density”; its steep climb as we approach the nucleus shows unambiguously that the electron is most likely to be found in the tiny volume element at the nucleus. But wait! Did we not just say that this does not happen? What we are forgetting here is that as we move out from the nucleus, the number of these small volume elements situated along any radius increases very rapidly with r, going up by a factor of 4πr2. So the probability of finding the electron somewhere on a given radius circle is found by multiplying the probability density by 4πr2. This yields the curve you have probably seen elsewhere, known as the radial probability

This diagram shows the right-hand side. Bohr radius is the peak of the radial probabilities for principal quantum numbers n=1.

The probability density and the radial probability plots, in summary, show two things. The first shows the electron densities at any point in an atom. The second, which is more useful, shows us the relative electron densities for all points within a circle with a given radius.