Atomic orbitals: from the probabilistic cloud to images of the infinitely small

Structure of the Atom

It is by looking at the smallest constituents of matter that scientists would explain, in the 20th century, the functioning of the entire universe.

The Atomic Nucleus and Its Constituents

An atom consists of a nucleus around which one or more electrons move. What characterises the nucleus is its number of protons (Z) which ranges from 1 to 118 – it is this that determines the element. For example, iron (Fe26) has 26 protons, and 26 is its atomic number (Z). The 118 elements of the periodic table, from the lightest (hydrogen) to the heaviest observed to date (oganesson, Z=118), are now all known. The number of neutrons (N) ranges from 0 to about 177 for experimentally observed nuclei (as is the case for the superheavy isotope oganesson-294, with 118 protons and 176 neutrons), and characterises the isotopes of the element. For example, hydrogen (H1) has one proton and no neutrons, deuterium (H2) has one proton and one neutron, tritium (H3) has one proton and two neutrons.

Isotopes and Their Particularities

These three forms of hydrogen each have only one electron, since there is only one positive charge – the single proton.

Note that only in the case of hydrogen do we give different names to the isotopes of the element; in all other cases, we only indicate the number of nucleons, which allows us to find the number of neutrons. For example, iron (Fe26) has several isotopes, including Fe56 – we understand that Fe56 has 30 neutrons, Fe57 has 31, Fe58 has 32; the number of neutrons indeed distinguishes the isotopes.

The Size of the Atom and the Nucleus

Since 1811, we have also known the approximate size of an atom. Amedeo Avogadro (1776-1856) estimated the size of atoms to be 10⁻¹⁰ metre (Ångström), i.e. a little more than 10 millionths of a millimetre. In 1911, Ernest Rutherford (1871-1937) discovered the atomic nucleus.

Rutherford refined the structure of the atom by bombarding a gold leaf with particles from the radioactive decay of uranium. He gave the atomic nucleus a size on the order of 10⁻¹⁴ metre.

Rutherford's Model and the Quantum Revolution

Rutherford represented each atom as a miniature solar system, with the nucleus at the centre and electrons orbiting like planets.

Since the advent of quantum mechanics (1920s), the picture of the nucleus has become disturbing: the nucleus is no longer a system of corpuscles bound together. The nucleus, governed by quantum mechanics, cannot be observed directly because to probe the nucleons inside the nucleus, one would have to illuminate the particles with light so energetic that the nucleus would disintegrate instantly.

The same is true for the electron: we no longer picture the electron as a corpuscle moving in a well‑defined orbit around the nucleus. The electron is neither a wave nor a corpuscle, yet we speak of wave‑particle duality, which is the foundation of quantum mechanics. The electron does not follow a unique trajectory; it is probably located in a region around the nucleus called the atomic orbital.

Electron Cloud or Atomic Orbital

Wave‑Particle Duality and de Broglie's Hypothesis

Since 1924, all matter is associated with a wave – this is the hypothesis of Louis de Broglie (1892-1987). With this hypothesis, he generalised to all particles of matter the "wave‑particle duality" introduced for light by Max Planck (1858-1947) at the beginning of the 20th century.

All subatomic particles therefore have a wavelength. The wavelength λ of a subatomic particle and its momentum p are related by the equation: λ = h/p, where h is Planck's constant, and p is the momentum, i.e. the product of mass and velocity (p = mv). In the semi‑classical Bohr model, the electron takes about 150 attoseconds (1.5×10⁻¹⁶ seconds) to "go around" the nucleus of the hydrogen atom on its first orbit; this duration defines the typical time scale of attosecond physics experiments, which now allow us to probe electron dynamics in real time.

Thanks to Einstein's formula (E = mc²), all matter is endowed with an associated energy. In other words, the shorter the wavelength, the higher the energy (E = h/λ). This energy will modify the shape of atoms. The foundations of quantum mechanics were thus laid.

What Do the Particles of the Infinitely Small Look Like?

Properties of Subatomic Particles

Matter is composed of very small particles (electrons, neutrinos, quarks) that have mass, charge, energy, size, a wave, and spin.

The Electron Cloud, Much Larger Than the Nucleus

The particles inside the atomic nucleus still remain inaccessible to direct observation today. The electron cloud, on the other hand, occupies the entire spatial extent of the atom, since it is about 100,000 times larger than its nucleus: if the nucleus were represented by an orange 3 cm in radius, the electron cloud would extend over about 3 kilometres.

The Wave Function and Its Interpretation

In wave quantum mechanics, a particle is represented by a wave function (state vector), but it is very difficult to picture the fundamental concept of quantum mechanics or the quantum state of a system. In 1926, Erwin Schrödinger (1887-1961) published the equation that now bears his name, allowing one to calculate this wave function for a given system.

That same year, 1926, Max Born (1882-1970) gave an interpretation of the wave function in which the square of its modulus represents the probability of finding the particle at a precise location when a measurement is made. This probabilistic interpretation, for which he would receive the Nobel Prize in Physics in 1954, became a pillar of the so‑called "Copenhagen" interpretation of quantum mechanics.

A wave function is therefore a probability amplitude, or a probability density of presence of the system at a given position and time. This function has a complex value. If a real number represents, for example, the length of a segment on a line, a complex value is represented by a vector in a plane. This vector has not only a length in space but also a phase corresponding to the direction of the vector.

Probability of Presence and the Electron Cloud

If we no longer represent the electron as a point particle on a regular orbit around the nucleus, how can we form a mental image of it?

The electron does not follow a single trajectory around the nucleus; it is located somewhere in a "vast region" called the electron cloud or atomic orbital. The state of an electron is therefore represented by this volume of space around the nucleus in which it is delocalised. The ground state of a hydrogen atom measures about 1 Ångström, i.e. 10⁻¹⁰ metre.

To picture the electron in this region, imagine a grain of rice about 5 mm in size moving in a sphere about 50 metres in diameter. But it is not always a sphere; the shape of this region of atomic space depends on the energy of the electron and its angular momentum.

The Varied Shapes of Orbitals

A Geometry Dictated by Quantum Numbers

The orbitals of the electron can take different characteristic shapes depending on the nature of the atom. For example, the orbital of the hydrogen atom in the top row of the image has a spherical shape; the orbital in the second row has the shape of two water droplets; the orbital in the third row has the shape of four water droplets.

The Effect of Excitation on Orbital Shape

In summary, the orbital corresponds to the region of space where the electron is delocalised. The state of the electron is a superposition of all possible positions inside the atomic orbital, whose shape varies. The shape of the orbital changes when the atom is excited, as in the first row. If the atom is excited further, the shape changes again, as in the second row or second electron shell.

Why Doesn't the Electron Collapse onto the Nucleus?

The answer comes from quantum mechanics. According to Heisenberg's uncertainty principle, one cannot know precisely both the position and the velocity of a particle like the electron. If the electron were stationary on the nucleus, its position would be perfectly known, but its velocity uncertainty would become infinite. However, infinite energy would be required to achieve that state, which is impossible.

The electron can therefore only exist in well‑defined energy levels. Even at the lowest level, it retains a minimum energy, called "zero‑point energy". It is this energy that prevents it from collapsing onto the nucleus: it remains in equilibrium in an orbital, where the attraction of the nucleus is balanced by this minimum energy.

Analogy: Imagine a very large spring that you try to compress. The more you push, the more it resists. Zero‑point energy acts like this resistance: it prevents the electron from sticking to the nucleus.

Seeing the Invisible: Direct Imaging of Atomic Orbitals

Spectacular Observations Since 2016

For a long time, the atomic orbital remained a purely mathematical object – a solution to the Schrödinger equation that could only be imagined or calculated, with no hope of observing it directly. Things began to change from the mid‑2010s. In 2016, a research team filmed, using femtosecond scanning tunnelling microscopy, the ultrafast motion of a single molecular orbital. In 2023, two independent teams went further: one used atomic force microscopy (AFM) to directly observe the signature of electronic orbitals of single metal atoms within phthalocyanine molecules; the other developed a technique based on scanning transmission electron microscopy (STEM) capable of mapping electron density at the atomic scale in a two‑dimensional material.

An Indirect Measurement but Confirmed by Theory

These feats do not mean that we "see" an electron as we would observe a pebble: these techniques remain indirect measurements of the charge distribution or of the current induced by the microscope tip, later interpreted in the light of quantum theory. But they spectacularly confirm, at the scale of a single atom, the shapes predicted for a century by Schrödinger's equation: spheres, dumbbells, and multiple lobes. As for the nucleons inside the nucleus itself, they remain beyond the reach of any direct observation, for the reason given above: probing them would require an energy so high that the nucleus would disintegrate.

Rydberg Atoms: When the Orbital Becomes Giant

Orbitals with Micrometric Dimensions

In a highly excited state, called the "Rydberg state" (named after the Swedish physicist Johannes Rydberg), an electron can occupy an orbital whose principal quantum number n (shell number) is very high, typically between 50 and 100 in laboratory experiments. Since the radius of an electron orbit grows as the square of n, the size of these atoms explodes: it becomes on the order of a thousand to ten thousand times that of an atom in its ground state. Concretely, for n≈50, the orbital radius already exceeds one micrometre, a size approaching that of certain bacteria.

Valuable Tools for Quantum Physics

These "giant" atoms are extremely fragile: their peripheral electron, very far from the nucleus, is weakly bound and sensitive to the slightest electric field or collision. But this fragility itself makes them valuable tools for quantum physics. The team of Serge Haroche, Nobel laureate in physics 2012, used Rydberg atoms as non‑destructive probes to detect the presence of a single photon trapped in a cavity, paving the way for experimental studies of quantum decoherence. Today, arrays of individual Rydberg atoms, trapped by lasers, are also used as a promising platform for quantum computing and simulation, because their giant orbitals allow strong and controllable interactions between neighbouring atoms.