What is the physical size of an electron?

[rorix_bw]

Hi guys. Sorry if this is in the wrong topic, but what is the physical size of an electron? I understand it can sometimes behave as a wave. But when it is a particle, just how [STRIKE]big[/STRIKE] small is that particle?

 

[Naty1]

read here:
https://www.physicsforums.com/showthread.php?t=536195&highlight=size+of+electron

Also, I would point out an electron can have almost any size depending on environmental conditions...like a cloud around an atom, electron degeneracy in other cases, and virtually infinite when not confined...

 

[Bill_K]

The size of an electron is zero. Period. Quantum field theory describes particles that are pointlike. All experimental evidence confirms this up to the highest collider energy, an order of magnitude 1 TeV, corresponding to a distance of 10^-17 cm. All elementary particles, to the best of our knowledge, are pointlike. Nucleons, of course have internal structure, and have a size of the order of a fermi.

When we say a particle "behaves like a wave," we are talking about a wavefunction that gives the probability of finding the pointlike particle at a particular location.

 

[chrisbaird]

Electron wavefunctions tend to take the size of their container. If you place an electron in a quantum well, it will tend to spread out to fill the well. If you bind it to an atom, it takes generally ends up the size of the atom (I know there's a lot more to it than this, but I'm trying to keep it simple). An electron is always partially wave-like and partially particle-like at the same time. It is sometimes more particle-like than at other times. So if you ask what is the radius of the electron when it is acting most like a particle, I would say zero. It acts like a point particle. If you want to pretend an electron is a classical ball of charge, you can come up with the radius 3x10^-15 m.

 

[cmb]

The size of an electron is zero. Period.

I think you should un-period that.

Sure, you can work with models in which the electron is defined without dimension, but this is not the whole story.

One line of research looking at testing the standard model is to determine if the electron is spherical. Clearly, it needs to have a size to be spherical!

http://www3.imperial.ac.uk/newsandeventspggrp/imperialcollege/newssummary/news_26-5-2011-8-58-6

 

[humanino]

I think you should un-period that.

Sure, you can work with models in which the electron is defined without dimension, but this is not the whole story.

One line of research looking at testing the standard model is to determine if the electron is spherical. Clearly, it needs to have a size to be spherical!

http://www3.imperial.ac.uk/newsandeventspggrp/imperialcollege/newssummary/news_26-5-2011-8-58-6

This is a very misleading article, shame on the university media responsible for this, and you bringing it up in this conversation may want to take the original article. You can measure something which can only be non-zero if there is a non-spherical finite shape, yet which would be zero if there was a spherical shape, which includes as a degenerate limit the zero size (trivial spherical symmetry). This is what they get : they set up an upper bound on the electric dipole moment. Their result is consistent with zero and does not imply a finite size. It is in fact consistent with zero size (point-like).

 

[rorix_bw]

I ask because Neil Degrasse Tyson (http://en.wikipedia.org/wiki/Neil_deGrasse_Tyson)
recently said:

Q) What never fails to blow your mind in physics?
A) The fact that an electron has no known size — it’s smaller than the smallest measurement we have ever made of anything.

I wasn't sure if he means it has zero size (point particle?) or if just fantastically small size.

 

[kurros]

He means it is so small that its size is experimentally indistinguishable from zero. We of course cannot ever say that it is exactly zero from an experimental perspective. There could always be some ridiculously tiny string structure or something.

 

[cmb]

This is a very misleading article, shame on the university media responsible for this

So are you asking that I take your word for it over taking Imperial College's article at face value? Why would I do that? Do you have references available [public access and peer reviewed] to back up a contradiction of the article?

Are you going to say 'shame on Nature' for publishing their work, too? http://www.nature.com/nature/journal/v473/n7348/full/nature10104.html

 

[Bill_K]

Electron wavefunctions tend to take the size of their container.

Yes, the wavefunction but not the electron itself, which always remains pointlike. Normal Schrodinger quantum mechanics concerns the motion of a point particle. A quantum object that had a finite size would also have additional degrees of freedom: rotation and vibration. Which are needed for example to describe nuclei. An electron has angular momentum but does not exhibit rigid rotator excitations, as it would if it was finite in size.

One line of research looking at testing the standard model is to determine if the electron is spherical.

The experiment in question was designed to detect if possible an electron's electric dipole moment. The popular idea circulated that this meant "nonspherical", but it was just a very misleading journalist's misunderstanding. A dipole moment says nothing about size. The electron has a magnetic dipole moment and yet remains pointlike.

If you want to pretend an electron is a classical ball of charge, you can come up with the radius 3x10-15 m.

This is known as the "classical electron radius." It was brought up as part of the 1904 Abraham-Lorentz theory that tried to describe an electron as a charged sphere. When quantum mechanics came along the Abraham-Lorentz model had to be discarded. In addition to being inconsistent with quantum mechanics, it was inconsistent with relativity.

 

[cmb]

The experiment in question was designed to detect if possible an electron's electric dipole moment. The popular idea circulated that this meant "nonspherical", but it was just a very misleading journalist's misunderstanding. A dipole moment says nothing about size.

Well, I read what they put in the abstract in a different way:

The electron is predicted to be slightly aspheric, with a distortion characterized by the electric dipole moment (EDM)

If they had meant to say it is the electron dipole moment that is aspheric, then shouldn't they'd have simply said 'The electron dipole moment is predicted to be slightly aspheric.'?

If the EDM is described as a characterisation of the asphericity of an electron, then why would I assume the EDM and the electron shape are the same thing?

A characterisation of a thing is not the thing!

 

[humanino]

So are you asking that I take your word for it over taking Imperial College's article at face value? Why would I do that? Do you have references available [public access and peer reviewed] to back up a contradiction of the article?

This is a public relation advertising. There is not even mention of "electric dipole moment" which what is measured ! I was reading this very day another article where the author complains about public media University services :
String and M-theory: answering the critics
I do not necessarily recommend the entire reading, but if you take a look at the second FAQ

Members of the Imperial media team I dealt with were very professional and sensitive to avoiding such hype. Consequently, I approved a version of the press release [...] headline was added by someone else in the media team, without my knowledge or consent. Then all hell broke loose [...]

This is M. J. Duff, from the Imperial College in London. Take his word if you do not believe me that this kind of occurrence is common practice.

The Nature article is very good. This is the article I was referring to when I said you should have read it.

So again : they find that the electron respects spherical symmetry, which is consistent with zero size.

 

[cmb]

This is a public relation advertising. There is not even mention of "electric dipole moment" which what is measured !

See my post #11.

 

[humanino]

See my post #11.

Oh I understand now. Initially you thought they measured "the electron", instead of its electric dipole moment...

 

[cmb]

Oh I understand now. Initially you thought they measured "the electron", instead of its electric dipole moment...

I read that to mean they are measuring the EDM as a proxy for the electron shape. Like measuring the shape of the Earth's atmosphere to get an idea of the shape of the earth.

Isn't this what it means?

 

[humanino]

The electric charge and magnetic current distributions are encoded in the electron-photon vertex form factors. Up to linear combinations $^{(1)}$, form factors are Fourier transforms of those distributions. To calculate the electromagnetic current carried by a spin 1/2 particle, you sandwich between the initial and final spinors an operator with a vector index $\Gamma^\mu$, which can be in all generality (Lorentz invariance, current conservation and hermiticity) written as :
$$\Gamma^\mu=\gamma^\mu F_1(q^2) + \frac{\sigma^{\mu\nu} q_{\nu}}{2m}\left[i F_2(q^2) - \gamma^5 F_3(q^2)\right]$$

where $q_\nu$ is the 4-vector difference between initial and final momenta, and is the momentum of the exchanged (virtual) photon (in the one photon exchange approximation) ; the $\gamma$s are the usual Dirac matrices ; the $\sigma$s are i/2 times the commutator of the $\gamma$s (usual spin operator).

Those form factors are measurable directly in elastic scattering. One can show using states closed to rest (non-relativistic)

$e F_1(0) =$ electric charge

$\frac{e}{2m} \left[ F_1(0)+F_2(0) \right] =$ magnetic moment (c.f. magnetic coupling $-\mu \, \vec{\sigma}\cdot\vec{B}$)

$\frac{e}{2m} F_3(0) =$ electric dipole moment (c.f. electric dipole coupling $-d\,\vec{\sigma}\cdot\vec{E}$)

For the electron, up to quantum corrections (loop), $F_1=1$, $F_2=0$ and $F_3=0$. That means it acts as pointlike electric charge down to the smallest scales we could probe (the magnetic current distribution vanishes, and the electric charge distribution is the Fourier transform of a constant, which is a "delta function"). That also means we went at great length to recover the usual electron-photon vertex $\Gamma^\mu=\gamma^\mu$

Note
(1) Which linear combination depends whether one uses the rest frame or an infinite momentum frame. There are technical advantages to using infinite momentum frames, in which case the Dirac and Pauli form factors $F_1$ and $F_2$ are directly Fourier transform of the electric charge and magnetic current distributions. In the rest frame, one needs linear combinations which are usually called Sachs form factors. The Sachs form factors have the advantage to separate neatly in the final cross-section.

source
Mostly Itzikson & Zuber

 

[A. Neumaier]

The size of an electron is zero. Period. Quantum field theory describes particles that are pointlike.

Being point-like doesn't imply being a point.

QED, or relativistic quantum field theory in general, is not based on the notion of ''point particles'', as one sees stated so often and yet so erroneously.

(emphasis as in the original; see p.2 of the book
O. Steinmann, Perturbative quantum electrodynamics and axiomatic field theory, Springer, Berlin 2000)

It is often said that the electron is a point particle without structure in contrast to the proton, for example. We will see in this section that this is not true. The electromagnetic structure of the electron is contained in the form factors

(beginning of Section 3.9 of
G. Scharf, Finite Quantum Electrodynamics: The Causal Approach, 2nd ed., Springer, New York 1995.)

In the book
S. Weinberg,The quantum theory of fields, Vol. I,Cambridge University Press, 1995,
Weinberg defines and explicitly computes in (11.3.33) a formula for the charge radius of a physical electron, though he does not carry the computation far enough to get a numerical value.

 

[cmb]

The electric charge and magnetic current distributions are encoded in the electron-photon vertex form factors.

That's waaaay over my head. I'm guessing next you'll be telling me I have to read 20 books and study for 15 years to understand it.

Bottom line is this - I do not accept that an infinite energy density can exist in a stable state. It is beyond my comprehension that all the energy of an electron can exist in a space of zero size.

Now, if the energy of the electron exists probabilistically in a space around a 'location', and that, actually, that is all there is to an electron and there is no physical structure to it, then, OK, that begins to look like a comprehensible answer. So its size could then be defined as, for example, 99.9999% likely to be within a radius of X from a location. I don't know if that description is reasonable or not, because so far all I have read here is that it is 'point-like' which makes no sense to me. Are there other 'point-like' particles, and if so then why do their properties differ if they have no physical shape or structure that 'causes' the energy around that location to be as it is?

Can someone please provide an 'Einstein' answer that a 6 year old would understand?

 

[chrisbaird]

All elementary particles are point particles, otherwise they wouldn't be elementary. From the wikipedia article on elementary particles:

"In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles... For mathematical purposes, elementary particles are normally treated as point particles, although some particle theories such as string theory posit a physical dimension."

 

[cmb]

"In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles.

I do not see how indivisibility implies an object has zero dimension.

A description devised 'For mathematical purposes..' means nothing, in the context of the question.

 

[humanino]

Being point-like doesn't imply being a point.

It is well understood. However, hiding philosophical prejudices there where nobody can see them leads to pointless discussion just as well.

The fact of the matter is that quantum field theory uses point like particles and does not have a successor in phenomenological success so far.

S. Weinberg,The quantum theory of fields, Vol. I,Cambridge University Press, 1995,
Weinberg defines and explicitly computes in (11.3.33) a formula for the charge radius of a physical electron, though he does not carry the computation far enough to get a numerical value.

The calculation Weinberg explains there are probably the most famous ones in the matter of quantum (or loop) corrections : anomalous magnetic moment and all that jazz. These form factors induced by quantum (loop, radiative) corrections are what I was mentioning earlier

up to quantum corrections (loop)

Take the quarks for instance. They acquire an effective mass in the nuclear medium. We say that they "dress". Their constituent mass is about 300 MeV (surprise surprise, 1/3 the proton mass). Now, we understand very well that this 300 is not the quark mass. The up and down quark mass is much less, few MeV only.

Say you are walking in a medium at high viscosity. Your effective inertia is much greater than the one outside the medium. You would not confuse your own mass for this "renormalized" mass in the medium. Can we take out quarks of their nuclear medium ? You bet : just measure them at higher energy and they become free.

By the same token, and I really mean that the renormalized mass of the walker in honey is the same mathematical procedure, the electron dresses, it polarizes the vacuum around itself. The fact that an effective radius appear at low energy should not confuse us as to whether the electron has a finite size.

I will try to think of a dumbed down explanation which is relevant anyway. Learning quantum field theory does not take 15 years. If you know quantum mechanics and special relativity, university curriculum is usually half a year to a year, that leads you into calculations. A couple more year to mature can lead to a decent understanding.

 

[A. Neumaier]

so far all I have read here is that it is 'point-like' which makes no sense to me. Are there other 'point-like' particles, and if so then why do their properties differ if they have no physical shape or structure that 'causes' the energy around that location to be as it is?

Can someone please provide an 'Einstein' answer that a 6 year old would understand?

Associated to every charged particle are (2 times spin plus 1) functions called form factors that tell how it responds to an electromagnetic field and summarize the experimentally or theoretically available information about a particle. The (observable) charge radius of a composite particle such as the proton is determined by its form factors.

A point particle is a particle whose form factors are those of a free field, hence no substructure at all.

Pointlike means that one starts with a free field (which consists of ''bare'', nonphysical point particles)
and then imposes renormalizable interactions. The renormalization ''dresses'' the point particle, turning it into a physical, ''point-like'' particle whose internal structure is now given by nontrivial form factors computable in perturbation theory. Thus the dressing process imposes a substructure. From these, one can derive properties like a charge radius just as in case of composite particles.

 

[torquil]

In short, it depends on the definition of "electron" and "size". :-)

 

[zahero_2007]

Can't one say that the actual size of the electron is not less than the Planck scale or that it must have finite size in a theory of quantum gravity?

 

[Hans de Vries]

The term "point like" stems from the form of the electron propagator. This propagator
does not have separate terms to describe for instance:

- The amplitude that the "middle-point" of the electron goes from A to B.
- The amplitude that a certain point on the "surface of the electron" goes from A to B.
- The amplitude that ...

There is only a single amplitude, the amplitude that the electron field goes from A to B.

In theory it would therefor be possible to contain an electron field in an infinitely small
volume. It would take an infinitely high potential barrier however to do so.

We should not forget that the (Dirac) field of the electron is used to determine the
interaction in both QFT and molecular modelling. The field describes the charge-current
density, the spin-density, the magnetization tensor and so on. We can even split the
charge-current density into several different parts due to charge and spin using Gordon
decomposition.

Hans

 

[humanino]

We can even split the
charge-current density into several different parts due to charge and spin using Gordon
decomposition.

which is exactly what was done earlier in this thread with the explicit full vertex operator including the third form factor for the putative electric dipole moment.

 

[cmb]

..the explicit full vertex operator including the third form factor for the putative electric dipole moment.

??

In short, it depends on the definition of "electron" and "size". :-)

..in my mind at least, there also appears to be dependence on the definition of 'physical'.

It's looking like the question itself doesn't make sense.

 

[humanino]

??

It was done here

$$\Gamma^\mu=\gamma^\mu F_1(q^2) + \frac{\sigma^{\mu\nu} q_{\nu}}{2m}\left[i F_2(q^2) - \gamma^5 F_3(q^2)\right]$$

You just have to stick in the left and right the appropriate final and initial spinors to obtain the EM current (a 4-vector with the index $\mu$).

It's looking like the question itself doesn't make sense.

The question "does the electron have a finite size ?" does make sense, and the way we are trying to explain it is

• the same which is done to describe the finite size of e.g. the proton
• the one actually used by high-energy experiments to claim that the electron is pointlike down to $10^{-18}$ m or so.

 

[tasp77]

Not to go to far afield, but how identical are electrons?

Are they 'infinitely' identical, or perhaps, only as identical as Heisenberg might expect them to be? In other words, are their properties 'fixed' or can they 'briefly' vary?

 

[humanino]

Franck Wilzeck tells this story, that for a while he would ask colleagues at conferences "What is the one most important thing that quantum field theory taught us, which we did not already know with quantum mechanics and special relativity ?". And according to him, only Freeman Dyson answered without a shred of hesitation "Why, but that all electrons are the same of course !". All electrons are but excitation of the same field. If you have two electrons, you can never know whether they are really one and the same, which came back from the future in the form of a positron. You can even imagine that all the electrons in the observable universe are one the same, and go back to the past at a great distance from us in the form positrons. How cool is that ? It does not have to be true, but it could be in principle, and as a consequence, no two electrons can ever have distinguishing properties.

 

[tasp77]

Deep, man. Really, really deep.


Thanx.

 

[cmb]

All electrons are but excitation of the same field.

Does this imply an electron 'ether'?

 

[cmb]

you can never know whether they are really one and the same, which came back from the future in the form of a positron.

Can I just make sure I understand the implication of this proposition? Is this implying that, taking the whole 'time-line' into account, that every electron has a positron that it is going to annihilate with it in the future, and vice-versa? Each electron has the 'name' of another positron already on it, so to speak, which is essentially moving the opposite way down the timeline to an inevitable annihilation with that given electron?

And, by implication, in the course of the universe there will have been as many electrons as positrons?

 

[A. Neumaier]

Can I just make sure I understand the implication of this proposition? Is this implying that, taking the whole 'time-line' into account, that every electron has a positron that it is going to annihilate with it in the future, and vice-versa?

As we are in the realm of quantum physics, talking about trajectories of electrons (positions changing with time) must be taken with a large grain of salt.

According to QED, we have an electron/positron field and an electromagnetic field (and, in the extension of QED as used by chemists, one field for every kind of nucleus). These are the real things. (They do not deserve being called ether; the ether of old times has no energy density.)

Trajectories make sense only if one looks at the fields from a coarse-grained point of view and studies an approximate dynamics for distinguishable local aggregations of field strength that may be called particles.
And they are not more accurate than the extensions of these local aggregations - their physical size.

 

[Hans de Vries]

which is exactly what was done earlier in this thread with the explicit full vertex operator including the third form factor for the putative electric dipole moment.

Yes, The Gordon decomposition is used in the vertex calculations (In the specific form of
the interaction current of two plane waves with different momenta). Itzykson and Zuber
is a good source. Their treatment is more extensive as most of the others.

Hans.