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You often see it written or hear it said that the interior of atoms is mostly empty. This is an attempt to convey something about the nature of atoms to a non-expert audience. But is it right? Isn't it rather misleading? Isn't the interior of an atom pretty full up really? (Full up with electrons I mean).

This question is not important to research physics, but it has some educational significance. It is important to the goal of conveying correct physical intuition and thus encouraging correct physical insight.

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  • $\begingroup$ Related: physics.stackexchange.com/q/165721/2451 and links therein. $\endgroup$ – Qmechanic Jun 7 at 15:40
  • $\begingroup$ Also related: physics.stackexchange.com/q/126512/50583. Technically not a duplicate since that question takes the atom being "empty space" as its premise, but the answers nevertheless discuss the (un)truth of that premise, too. $\endgroup$ – ACuriousMind Jun 10 at 10:46
  • $\begingroup$ I’ve calculated the atom radio here under paragraph 3. The calculation is as simple as effective, the derived electron density matches the covalent radii. A naive work with surprising results. $\endgroup$ – HolgerFiedler Jul 30 at 8:13
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This is an educational exercise; I am hereby posting an answer to my own question, but I hope others will also respond.

I will answer the question by comparing empty space and vacuum with the properties of the interior of an atom. I will write at the level of high school physics in order to make my answer accessible. My conclusion will be that the answer is "no": it is not true that the interior of an atom is mostly vacuum or empty space, and furthermore such an idea conveys a thoroughly misleading picture of the nature of atoms. The main point is that the electrons are smoothly spread throughout the interior of each atom, and they carry enough mass and charge to make it misleading to compare the situation to empty space.

First let's see what we mean by empty space. We mean of course "nothing is there". To flesh this out a little, consider the case of an ordinary gas at standard temperature and pressure (STP). This is not a vacuum, clearly, since the pressure is quite high. We might speak of "vacuum" once the pressure is below a millibar (100 Pa); at STP the pressure is about a bar ($10^5$ Pa). On the other hand it is true to say that such a gas is "mostly empty space" in that the mass is concentrated in the molecules, with almost no mass in between the molecules, and the volume occupied by the molecules is a small fraction (about a thousandth) of the total. On the other hand, if you place a mass-detector anywhere in the gas, then it will register some mass very quickly, because the molecules will soon hit it. So to say that a gas is "mostly empty space" is helpful to get the intuition that the molecules can move freely with a large mean free path, but it can mislead in some respects.

The density of an ordinary gas at STP is about 1 kg/m$^3$.

Now let's think about the interior of an atom. I have in mind some ordinary atom such as carbon, and ordinary locations inside the atom, so not at the nucleus and not far away; one could pick a location about half a Bohr radius from the centre, for example. Let's calculate some properties.

First, the mass density. This is the mass density owing to the electrons which are present. Their mass is spread throughout the atom via their extended wavefunctions, and the average mass density can be estimated by noting that an electron is about 2000 times lighter than a proton, and a typical atomic nucleus has as many neutrons as protons, so the electrons contribute about one part in 4000 of the total mass. The density of a solid element such as carbon is about 2000 kg/m$^3$ so we can estimate that the density owing to the electrons at a typical place in an atom is about $0.5$ kg/m$^3$. An estimate using the atomic radius of carbon gives the value 8 kg/m$^3$, suggesting that our previous value was an underestimate because there is some space between the atoms. Anyway the main conclusion here is that mass density at a typical spot inside an atom is similar to the average mass density of a gas at STP.

To get the charge density, we multiply the mass density by $q/m_e$, the charge to mass ratio of an electron, which gives about $10^{12}$ coulombs per cubic metre. This is a huge charge density by everyday standards. (For comparison, a typical 1 microfarad capacitor charged to 1 volt carries a micro-Coulomb in a volume of order $10^{-7}$ m$^3$, giving a charge density $10$ C/m$^3$.)

Next let's consider the flux of matter---the rate at which mass will approach and hit a detector if we were able to place a mass-detector inside our atom. The electrons have speeds of order a few times $\alpha c$ where $\alpha \simeq 1/137$ is the fine structure constant and $c$ is the speed of light. The flux (mass crossing unit area per unit time) is therefore around $8 \times 3\times 10^8 / 137 \simeq 10^7$ kg per second per square metre. Needless to say, this is a huge value in everyday terms.

Next let's enquire into whether or not there is "empty space" in the sense that there is room to put stuff inside an atom. The original statement perhaps comes from a desire to compare an atom to a gas, using some notion that electrons are point-like in some sense, with room in between them.

To address this question we need some more advanced physical ideas, to do with the Pauli exclusion principle. This is an important result in quantum physics, which says that particles such as electrons cannot overlap one another completely. To be precise, in any given spatial situation there can be at most two electrons having that particular combination of position and momentum.

What this means in practice is that there is no more room for low-energy electrons in any atom. If the atom is a hotel, then all the rooms on the lower floors are occupied---completely full up. Thus the space inside an atom is completely unavailable to further electrons unless they move quickly. This is about as far from "empty space" as you can get. It is "complete and utterly full-up space", as far as low-energy electrons are concerned. But this does not exclude electrons altogether, as I already said. If they are moving quickly then there is room for further electrons to get into an atom. They won't stay there---they would have to form a beam passing through; they are visitors to the guests staying in the hotel. For an extra electron bound to an atom (making a negatively charged ion), the wavefunction of the extra electron does get inside the atom a bit (it penetrates the core, as we say), and this can be compared with a visitor rapidly visiting again and again.

What about other types of particle---say, neutrons? They can more easily enter an atom. But is the experience of a neutron sitting inside an atom anything like the experience of a neutron sitting in empty space? Hardly. They would be continually bombarded by that high flux of electrons we calculated just now, and they would notice because although they carry no electric charge, neutrons carry a substantial magnetism, and this leads to an electromagnetic interaction between the neutron and all the nearby electrons.

Now let's summarize.

Electrons in atoms behave in ways that classical physics cannot account for; we require quantum physics. As a result, when we talk about atoms in everyday language, we are trying to convey in everyday terms what quantum physics says is going on. Among the things that quantum physics tells us about the interior of an atom is that the electrons are smoothly spread out, such that the probability of an electron being present at any given moment is non-zero throughout the interior of an atom. We can flesh this out a little by calculating properties such as mass density and charge density and flux. The mass density of the electron cloud of a typical atom is similar to the average mass density of an ordinary gas at standard temperature and pressure. The flux is huge and the charge density is enormous. Also, it is strictly impossible to introduce further slow-moving electrons into the inside of an atom, but it is possible for fast-moving electrons to pass through. Neutrons can also enter atoms, and when inside they interact with the electrons which are there.

In view of the above, it seems to me that it is misleading to say that the interior of atoms is anything like either vacuum or empty space. It really isn't. But it seems that this idea has become lodged into popular presentations of science. It will take some effort to dislodge it.

I now wonder where this idea came from in the first place. I think possibly it might have originated in the early attempts to model atoms via classical point-like particles, or perhaps it is descended from the "fly in a cathedral" image, which is a correct statement about the relative sizes of the atomic nucleus and the whole atom. The "fly in a cathedral" seems to imply that the rest of the "cathedral" is empty, but it is not. It is full of electrons.

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  • $\begingroup$ Most of this answer is accessible to an A-level student. It could also be presented in part as guided exercises with a clear and worthwhile purpose. $\endgroup$ – Philip Wood Jun 7 at 10:45
  • $\begingroup$ Nice. As far from where or how the idea was developed I think is shifting from Thomson to Rutherford model. In the latter nothing was well defined about electrons in terms of their motion, but it opened the room for an empty space inside the atom. Nice Q and A. $\endgroup$ – Alchimista Jun 7 at 11:16
  • $\begingroup$ What about the viewpoint that electrons inhabit specific shells, spreading out on each shell, but with empty space between the shells? $\endgroup$ – D. Halsey Jun 7 at 17:37
  • $\begingroup$ @Halsey In fact the wavefunction associated with each shell overlaps quite a lot with the others; there are no gaps. But thanks for this and I might add a sentence on it to my answer. $\endgroup$ – Andrew Steane Jun 7 at 17:50
  • $\begingroup$ There is no need to justify a self-answer: the process is not only sanctioned, but explicitly supported by the UI. $\endgroup$ – dmckee Jun 7 at 22:53
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I would take your sentence

It is important to the goal of conveying correct physical intuition and thus encouraging correct physical insight.

as a central guideline in my answer. As you'll see, my conclusions are quite different.

The origin of the statement about the almost empty interior of an atom comes from the thouroughful Rutherford's analysis (1911) [1] of the Geiger and Marsden experiment of scattering of alpha particles by thin gold foils. Rutherford does not writes explicitly such expression, but the key point of his analysis was to show that a model of the atom made by a spatially concentrated heavy particle whose charge should be proportional to the atomic weight was able to account for the experimental data much better than a diffuse density of the scattering component. Therefore, the emphasize in Rutherford's paper was more on the "concentrated" character of the scattering component than on the presence of an "empty space".

However, such last inference was quite natural once one considers the positive nuclei as the main source of scattering, if the second atomic component, the electrons are considered point-like particles. An unavoidable step more than 10 years before the birth of wave-mechanics.

So, from an historical perspective, the sentence corresponds to a well definite and historically justifiable point of view.

Now, let's examine the situation with today eyes. What could be a correct statement? The interior of the atom is empty? is full of something? electrons? fields?

Here, I do not agree with your idea that

The main point is that the electrons are smoothly spread throughout the interior of each atom, and they carry enough mass and charge to make it misleading to compare the situation to empty space.

I am afraid that such a statement would enforce a common misconception which tries to maintain the original idea of the "waves of matter" proposed by de Broglie and eliminated from the possible interpretations of QM since the analysis of scattering processes made by Born and resulting in the present day probabilistic interpretation of QM.

The key point, from a pedagogical point of view, is to insist that all the existing experiments agree on a point-like structure of the electron. Measuring charge or mass density, would be misleading, since it would convey the idea that it is possible to detect part of an electron.

In summary, I would not insist too much on the sentence about "empty space" in the atom, if not in connection with Rutherford's analysis. But certainly I would not substitute it with potentially wrong ideas about an extended electron.


[1] Professor E. Rutherford F.R.S. (1911) LXXIX. The scattering of α and β particles by matter and the structure of the atom, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 21:125, 669-688, DOI: 10.1080/14786440508637080

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  • $\begingroup$ Thanks for this. I agree one should be cautious about implying the electron is extended, and there was a (quiet) hint of such caution in my answer, but this does not change the main point that atomic interiors really are very unlike empty space. The plane wave modes employed in quantum field theory are de Broglie waves; they have not been abandoned so much as learned about more. The probability of detecting an electron at any given place inside an atom is non-zero at all times and places; the charge density is used in atomic structure calculations. $\endgroup$ – Andrew Steane Jun 10 at 8:16
  • $\begingroup$ @AndrewSteane I know very well that charge density is used in electronic structure calculations, and "observed" in STM experiments. However, I keep thinking that it may be misleading to insist on something that could easily be confused with a continuum description. Plane waves modes used in QFT are quite more sophisticated than de Broglie waves, IMO. But, a part these divergencies of opinion, I would put the emphasize on the difference that the concept of "empty' or "crowded" may have in the present context. At the end of the day in the hydrogen atom there is just one electron and one proton. $\endgroup$ – GiorgioP Jun 10 at 11:26
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The intention of the following, apart from directly answering the question as it is asked, and as one judges it to be intended, is also to introduce a different approach to such questions about the notion of 'empty space' and vacua' generally; do please forgive me if it is a little too 'through the looking-glass'; or tangential to the immediate question.

I might have desisted were it not for the recognition that your question and answer attempts to transcend the increasingly archaic notion of the 'point particle'; which while remaining adequate to practical purposes, has become a serious impediment to understanding, not of reality itself which requires no 'physics', but of the very models of theoretical physics themselves, primarily of QED and of its real physical basis, the basis of intrinsic spin in particular.

The simple answer to your question is first to deny the possibility in metaphysics of 'empty space', and then to proceed to address, according to a specific perspective in metaphysics, the question of the 'energetic vacuum' which you most astutely describe in its sub-atomic context in terms of the 'pressures' involved; which I suppose is just another way of conceiving energy in a molecular theory of thermodynamics; or as suggested in the following, a particular way of conceiving the distribution of an essential force implicit in a singular universal reality, which is then capable of description by a practical molecular theory, or indeed electro-dynamic theory of 'pressure'.

Regarding the vacuum more generally, one proceeds as follows to suggest that at a more fundamental level than currently admitted by physics, all of reality, matter and space, even the very mind, is composed of the same singular substance, whose dynamic behaviour translates to what we loosely call 'energy'. Energy, in whatever more-or-less strict sense we define it, is ubiquitous in reality, conserved at every turn as if indeed it were the very embodiment of an innate impetus to efficiency in the dynamic of that substance: this follows from that conception of a singular substance itself.

That we are told in QFT that even the vacua of its quantum fields are continuously energetic would appear to bear this intuition out, that there is no state or place which can realistically be said to be without energy, but this conclusion remains uncertain without a concession a priori to the metaphysical postulate of a singular substance. Nonetheless, how then might we explain this ubiquity; and if indeed energy is a function of motion, what ultimately is moving?

One is entitled to ask then, since energy is classically and in any real context only the motion of a force over a spatial distance--i.e. the spatial integration of the force--, what force may be conceived to act ubiquitously in this way to imply ubiquitous energy; and it is not unreasonable to suggest here that orthodox physics is effectively stalled at an impasse between the perspectives on this question embodied in QFT and GRT, essentially because it prohibits itself from formulating precisely and explicitly such a question; the arcane musings of M-theory notwithstanding.

In order to address such a question however, what is required--and what the sciences have come to regard as anathema-- is first simply to propose a priori the existence of a unitary universal substance or fabric, and to understand that because we too are comprised of it, not only is it impossible to discern its ultimate nature, but that due to this very constraint, its essence becomes effectively equivalent to the ubiquitous force holding the universal unity and entirety together. All we are capable of perceiving, through the mechanism of resonance implicit in such an inviolate dynamic universal unity is this 'cohesive force'.

That is, were one to suppose that the universe is composed entirely and exclusively of a singular inviolate substance, and given that it would therefore be quite impossible for us, also comprised of it, to determine its ultimate nature, since we are nevertheless embroiled in its dynamics, we are therefore capable of, if also constrained to, an understanding of its principles --operating continuously all around and within--, commencing with some meditation on the self-evident postulate, more properly an innate perception, that a unitary force must bind that inviolate universal unity or substance together.

In such a conception then, all of reality is effectively only the dynamic action of this singular 'cohesive force' in various contexts of its more-or-less local distribution in space, including what is inevitably an 'interior spatial dimension', of 'spatial depth', necessary to permit such a conception; and it is thus that we find a way to transcend the obstacle of the 'point particle'. One then proceeds to consider how it is that the existence of such a force acting everywhere at once permits the world of matter and space, perpetually inflated, as we perceive it; and that the very idea of a true 'vacuum' or 'empty space'is only confusing to the understanding.

There, the question is answered; the very idea of 'empty space' is a nonsense; and it is not necessary to proceed further with the argument unless one is interested in how the appearance and physical behaviour of vacua may be IMAGINED to come about. It is insufficient for example to speak of electrons whizzing about if one cannot say what an 'electron' fundamentally is; accordingly, despite its pretension, the question is one in metaphysics.

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  • $\begingroup$ Please do try not to be churllsh in the face of unfamiliar ideas, however difficult they may at first appear, and indeed unfounded in one's own familiar reality. They are ideas and should be welcomed, refugees as they are from distant climes. Sometimes they may turn out to be very useful to one's deeper understanding of the familiar, as might be expected in any of life's encounters. Thank you for your forbearance and open-mindedness, and for remembering the importance of the 'First Amendment' in any and all contexts of human interplay. $\endgroup$ – jeremiah Jun 10 at 11:48

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