# Do electrons have some volume, area or shape?

I am in 8th grade now and when I was in 6th grade, my science book had diagrams of the electronic configuration of atoms. The electrons were round like spherical balls. Is it true that the electrons have some volume, area or shape and they revolve around the nucleus?

• Pictures are simplifications and should not be taken as real representations of the properties of subatomic properties that are best described by quantum mathematics. Mar 23 '18 at 13:16
• Not too say that even point particle are meaningless out of the exact domain where they are modelled in such a mathematical way. Within a molecule or atom, perhaps an electron itself has no size no shape but is surely better modeled by its wave function or the square of the latter. These latter are quantum mechanical as well , have shapes and volumes, and are of relevance when chemistry is involved. Mar 23 '18 at 13:58
• Related on Physics: Do electrons have shape?
– rob
Mar 23 '18 at 20:30

Quoting from the Nobel lecture of Hans G. Dehmelt (1989):

With the rise of Dirac’s theory of the electron in the late twenties their size shrunk to mathematically zero. Everybody “knew” then that electron and proton were indivisible Dirac point particles with radius R = 0 and gyromagnetic ratio g = 2.00. The first hint of cuttability or at least compositeness of the proton came from Stern’s 1933 measurement of proton magnetism in a Stern-Gerlach molecular beam apparatus. However this was not realized at the time. He found for its normalized dimensionless gyromagnetic ratio not g = 2

...

Similar later work at still higher energies found 3 quarks inside the “indivisible” proton. Today everybody “knows” the electron is an indivisible atomon, a Dirac point particle with radius R = 0 and g = 2.00.... But is it? Like the proton, it could be a composite object. History may well repeat itself.

So there is no evidence that the electron is anything but a point particle, yet a non-zero size has not been ruled out either.

New Measurement of the Electron Magnetic Moment Using a One-Electron Quantum Cyclotron Phys. Rev. Lett. 97, 030801 (2006) (more-official link) shows that any radius of the electron is less than $10^{-18}$ meters.

• But from a QFT perspective, the electron isn't a point particle because it isn't in any one place. It's a wave that's necessarily spread out over some volume depending on its state. Mar 23 '18 at 18:22
• @Shufflepants Strictly speaking there is a theoretical question about the effective volume of an idealized free electron, truly unaffected by interaction with the universe. But this is probably only an asymptotic scenario in the real universe.
– Ian
Mar 23 '18 at 18:42
• @Ian I believe the wave function of a free electron truly unaffected by interaction with the universe slowly spreads out over time. That is as time goes on, the size of the spherical region that contains >50% of the distribution "mass" grows without bound over time. My point is that from what we know now about QFT, all indications point to them being spread out excitations in a quantum field, not little balls or little points. Mar 23 '18 at 19:01
• @Shufflepants Well, particle/wave dualism hasn't gone anywhere. You can't say it's not a particle at all. Mar 24 '18 at 16:39

Atoms are composed of a positively charged nucleus and an outer shell of negatively charged electrons. When two atoms come into close proximity, their electron shells repel, preventing the atoms from sharing the same space. The "volume" of an object can generally be understood as the total measure of space that is unavailable for other objects to occupy, as the electron shells of the atoms of the two objects would repel, preventing the intruding object from entering that space.

This macroscopic conception does not apply to an electron. However, there are other measures of an electron that can be considered analogous to the dimensions of macroscopic objects. For instance, quantum mechanics states that all objects exhibit both wave-like and particle-like behavior (although, for larger objects, the wave-like behavior is insignificant). When an electron is analyzed as a wave, that wave has a wavelength, which is called the electron's "de Broglie wavelength". There is also a phenomenon called Compton scattering for which there is a characteristic length that is used in modeling the interaction between a photon and an electron. One could also ask "For what distance would putting two electrons at that distance require more energy than the mass-energy of the electrons?"

All of these, as well as other quantities, can in various sense be considered the "size" of an electron. To what extent these quantities "actually are" the electron's size is a matter of opinion. Some people consider one or more of these quantities to be the size, without any qualification, some consider them to be merely analogous to the concept of size, and others consider them to simply be quantities that are completely distinct from the macroscopic concept of size.

One thing to realize about science is that we use various words in everyday language without bothering to come up with a precise definition of what those words mean. We just have an intuitive sense of the general meaning of words. So when we start doing science, we build off of everyday language, taking everyday words as inspiration for precise terms, but we often have definitions that can be more or less precise than everyday meanings, or only loosely connected. For instance, "energy" in everyday meaning can refer to a wide array of concepts, such as "optimism", "enthusiasm", "friendliness", etc., but in physics it's much more precise. "Salt", on the other hand, in everyday meaning refers specifically to table salt, but in chemistry it is used more generally for ionic compounds. Words can have different definitions that are equivalent in everyday use, but they are distinct when we try to apply them to more exotic situations.

So while "Do electron have volume" may sound like an objective question, if we really look at it, it has to be understood as "Do electrons have a property that corresponds to the 'normal' concept of 'volume'?", and that question depends on how exactly one understand the "normal" definition of volume, how one defines that concept precisely, how one extends that concept to the microscopic realm, and what one considers to be a valid "correspondence".

It is generally accepted that physical laws are the same regardless of direction, and other than spin and some narrow symmetry breaking, electrons haven't been found to act different depending on direction. Which implies that electrons don't have a "shape" in the sense of "looking different" from different direction. So, if they are modeled as having a shape, it makes sense to have that shape be a sphere, possibly pictured with a vector representing spin.

Also, saying that they "revolve" around the nucleus is a simplification. Position and velocity are, somewhat like volume, concepts for which there are rough analogues in quantum mechanics, but they aren't quite the same. In a spherical orbital, an electron has a certain probability of being observed at each location, and this probability is the same for all points at the same distance from the nucleus. So even though the electron has a quantity that is referred to as "velocity", it is some sense not moving (in the sense that the probability of it being found in a particular location is not changing over time). Even though the electron's wavefunction is "moving", the magnitude of the wavefunction is not changing, and so it is both moving in one sense, and not moving in another. Actually explaining this clearly requires getting into quantum mechanics.

• In fact in the spherical shells, the most likely place for the electron to be found is in the middle of the nucleus. Where the Bohr radius for that shell (the place where the simplified models pretend that the electron is orbiting at) is just the spherical shell that contains 50% of the distribution mass. That is, the chance of interaction with that electron inside or outside that shell is 50/50. Mar 23 '18 at 19:06
• Reading this, I don't see an actual, clear, literal answer to the question, for an 8th grader. Would you mind adding "no, by our current knowledge, eletrons do not have a volume, area or shape in the sense you are probably thinking about" at the top, or something like this?
– AnoE
Mar 24 '18 at 16:38
• @Shufflepants : Er... The maximum likelihood location of the electron is in the shell, slightly inside the Bohr radius. The average location of the electron is at the center. The average need not be particularly likely. Mar 24 '18 at 19:14
• @EricTowers Hmmm, yeah, I guess I don't understand the difference between the distribution function and the density function. umich.edu/~chem461/QMChap7.pdf Mar 25 '18 at 22:56

Observation of a single electron in a Penning trap suggests the upper limit of the particle's radius to be $10^{−22}$ meters.