Distances within the atom

Factual Questions
1

I heard that a proton is about 1800 or 2010 times the diameter of the electron. Let’s say I was making an actual model, and made the proton 12 inches in diameter in a hydrogen atom. How big would the electron be and how far would it be from the proton in inches, or feet?
2)Now say I was making a model of a more complicated atom with lots of protons and neutrons in it, would the electrons still start at the same distance from, say, the center of the nucleus, as the little electron in the hydrogen atom was at?
Like what if in the hydrogen atom the electron was 400 ft away from the nucleus, would the first bunch of electrons be revolving around the nucleus at 400 ft also?

 3) On this same scale, how big is a meson
 4) I discovered the concept of the Fermi Surface but I can't understand it.  However, there are some beautiful pictures of it on this one website.  They look like those strange volumes (3D) that they have in topology or in what they call the manifolds, but inside of a outlined 3D polygon like a crystal.  Are these pologons the shape of, like a crystal of Ni for instance?  And does a Fermi Surface mean that's where the electrons are?  This is on http://www.phys.ufl.edu/fermisurface/jpg/Niup.jpg

for the Ni picture of Niobium or Nitrogen

2

If I remember correctly, the electron (and all other “elementary” particles) are suspected to be true point particles – at least, in the Standard Model. Of course, you can’t prove, other than mathematically, that a particle has no size – you can only measure and show that it is smaller than some size – but so far, it has come back smaller than every attempt to measure a radius. Things may have changed in recent years, or String Theory or M-Theory may have a different opinion on the matter. If so, hopefully Chronos or Ring will come along shortly to let us know.

3

Well, in QM electrons are probabilty distributions not point particles. Long time since I did chem. , but you need to be looking at things like ‘atomic radii’ and ‘ionic radii’.

4

bryanmcc is correct that electrons appear to have zero size. Certainly, they’re much smaller than would be expected from back-of-the-envelope calculations: If you assume that the entire mass of the electron is due to electrostatic potential energy, then you can “calculate” a minimum radius for an electron… But the electron is smaller than that. Which means that more than all of the mass of the electron is electrostatic, or alternately, that the “bare mass” of the electron (the mass it would have if it didn’t have a charge) is negative (and possibly infinite). The actual size of an electron (and of the various other “fundamental” particles) may well end up being the Planck scale, but we’re nowhere near being able to measure at that scale.

As for spacing: The figure I remember is that if you scaled a hydrogen atom to the size of a football stadium, then the single proton of the nucleus would be about the size of a pea sitting on the 50-yard line. The overall size of an atom depends on where it is on the periodic table: As you go down the columns, they tend to get larger (because there are more electrons), but as go right along the rows, the atoms tend to get smaller (because the electrons are held in more tightly).

5

As the others have already noted, electrons appear to be point particles.
How big a proton is is a slightly delicate question, since it appears to be slightly different in size depending on how you’re trying to measure it. In practice it’d done by scattering other particles off it and the results will depend on how those particles interact with the proton. But the usual rough number is that the radius is 8 x 10[sup]-16[/sup] m (i.e. 0.8 fm).
Electrons in a hydrogen atom are roughly a Bohr radius away from the proton nucleus, i.e. about 5 x 10[sup]-11[/sup] m. So for a proton a foot across, the electron is about a million inches away.

Answered by Chronos.

The equivalent of the 0.8 fm radius for the proton for a pion is 0.64 fm. So a shade under 10 inches.

Argghh, solid state physics …

6

There’s your problem. It’s mass not diameter that has this ratio.

The mass of a proton is 1836 times the mass of an electron. The mass of a neutron is 1842 times the mass of an electron.

7

The best way to look at this is figuring out how big the atom would be in relation to the proton. Ie, you can figure out roughly the space that the electron(s) would take up, but you can’t figure out where they are or how big they are (I believe the mentions of point charges is correct).

According to a calc that a friend and I did while working on a textbook a few years ago, a better analogy than the standard “baseball in the center of {insert local baseball stadium}” that textbooks like to use is a lightbulb on a marquee (the nucleus of a hydrogen atom) in the center of Las Vegas (the electron cloud surrounding the nucleus). Sorry, I can’t find the numbers right now (something like a 5 inch diameter lightbulb, and I want to save LV is about 10 miles in diameter), but that’s the order-of-magnitude relationship.

8

Quick question. Does that mean that electrons are effectively a black hole?(Admittedly with an event horizon so small that pretty much everything misses it.)

9

The image to which you linked is rich with complexity. I’ll try to give you an idea of what the picture is. The “why’s” and “how’s” of the picture are beyond the scope of this post.

Take a crystal lattice. The lattice has a certain periodicity to it in various directions. Because of that periodicity, you can create imaginary planes (“Bragg” planes) defined by the lattice points. For a simple square lattice:



square lattice (continues in all directions):

     0 0 0 0 0

     0 0 0 0 0

     0 0 0 0 0
example Bragg planes:

    -0-0-0-0-0-

    -0-0-0-0-0-

    -0-0-0-0-0-
    \ \ \ \ \
     0 0 0 0 0
    \ \ \ \ \ \
     0 0 0 0 0
    \ \ \ \ \ \
     0 0 0 0 0
      \ \ \ \ \

    .   .   .
     0 0 0 0 0
    . `-. `-. `-.
     0 0 0 0 0
    . `-. `-. `-.
     0 0 0 0 0
      `   `   `

That last one is hard to draw with ASCII characters, but imagine planes (or lines, really, since we’re in 2D) that go down one unit, over two units, down one unit, over two units, etc.

Any one set of these planes can be described by a vector whose direction is perpendicular to the planes and whose magnitude is related to the plane separation. (The useful choice is to have the magnitude be inversely proportional to the plane separation.) Thus, we can describe all the Bragg planes by a set of vectors (or, as they’re usually called for reasons discussed below: “wavevectors” or “wavenumbers”). What’s more is that this set of wavenumbers contains all the information needed to reconstruct the original lattice.

Now, these wavenumbers describe another lattice called the reciprocal lattice. That is:

  • choose a starting position
  • take a wavenumber from the list (recall that these are vectors)
  • move away from the starting position according to that vector’s direction and magnitude
  • draw a point where you end up
  • repeat until every one of the infinity of vectors is used

The space of points you just drew is this new reciprocal lattice.

(We’re almost there… hang on…)

The reciprocal lattice has its own perodicity properties. In particular, it has (like all lattices do) a unit cell. A unit cell is just the building block – the smallest tile, if you will – of the lattice. (Think of a complex but repeating kitchen tile. There is a smallest shape in there that is just repeated over and over.) For the square lattice:



the unit cell is the box:

   0   0   0
     .---.
   0 | 0 | 0 
     `---'
   0   0   0

Since the lattice is just an infinite number of unit cells stacked together, most of the interesting physical properties of the system can be discussed in terms of a single unit cell. The polygon in the picture in your link is the unit cell of the reciprocal lattice of the crystal those guys are studying.

WHEW!

Now, the colored surface in the picture…

The quantum states available to electrons in a lattice can be characterized (in part) by a momentum vector. This momentum vector can be written in terms of a wavenumber k: p=(hbar)k. These wavenumbers are related to the reciprocal lattice, and we can imagine the reciprocal space as an enumeration or organization of these possible electron states. (It is this context that produces the name “wavenumber” for the reciprocal vectors.)

Now, electrons are fermions, so the Pauli exclusion principle applies to them: no two electrons can occupy the same state. At absolute zero, the electrons will occupy the lowest energy states available. The energy of an electron in the lattice is proportional to the square of the momentum, and thus, is proportional to the square of the wavenumber. So at absolute zero, states with small wavenumber magnitudes will be filled first. Once all electrons are accounted for, we can draw an imaginary surface in k-space (reciprocal space) that separates those states which are occupied (i.e., those with small k[sup]2[/sup]) and those which are not. That imaginary surface is the Fermi surface.

WHEW! (again)

Now, having said all that, the concept of a Fermi surface exists outside of crystals and solid state physics. Any space which 1) represents the states of a fermion system, and 2) has a sense of energy ordering (for example, energy increases as you move to states that lie further from the origin) can have a Fermi surface defined in it.

10

Man that was some post Pasta.

If the audience will suspend their QM nit pics This
is a pretty good lay man’s description.

11

Not necessarily.
Superficially, the big problem with the idea is Hawking radiation. Looked at naively, very small black holes radiate their mass away very quickly in an intense blast of radiation. Quite unlike what we see electrons doing. However, the kicker is that a black hole of electron mass would only be 10[sup]-47[/sup] m across, which is about a trillion times smaller than the Planck length. So even if you can get black holes of this mass, you’ll need a good handle on quantum gravity in order to say anything about them. Including deciding whether they’ll radiate or not. Basically, we don’t yet know enough about physics at distances this small to say much about either particles or black holes of this size.

That said, about ten years ago Gerard 't Hooft, who’s since won the Nobel prize for his earlier work, was trying to push the idea that fundamental particles might be little black holes, but I don’t think he got anywhere with it. Indeed, I’m not sure he ever even published anything on this research.

12

Chronos already addressed this to some extent, but I just wanted to add that yes, as the nuclear charge (the number of protons), Z, increases, the distance of the first electron from the nucleus decreases.

One of the major factors in determining the radius of an atom is “shielding”. Shielding occurs when inner-shell electrons reduce the effective charge of the nucleus, Z[sub]eff[/sub] (i.e., Z[sub]eff[/sub]<=Z). With the weaker pull that results, the electrons move farther from the nucleus. Of course, the lowest shell electrons have no other electrons between them an the nucleus, so they feel the full effect of the nuclear charge. This serves to bring them closer to the nucleus than in atoms with a lower Z.

Of course, this ignores factors like electron-electron repulsion, amongst others, but I’m just trying to present a simplistic picture of one of the elements (no pun intended) involved here.

13

All very helpful, thanks everyone. One question to Sacroiliac, shouldn’t the golf ball be the electron and the baseball the proton?
How did the electron get bigger than the proton in size? And for that matter shouldn’t, according to some other input here, shouldn’t the electron be like a pinpoint or a point of mathematics with no size at all and then the proton would be, say maybe a golfball? But I like that thing about the electron going so fast that it effectively makes like a solid sphere around the proton imagery. Now I am wondering just what path it takes when it goes around its sphere. Like would it start by making a little circle at the south pole so to speak and then a slightly larger circle and so forth on up the sphere or does it go randomly around? But even if it is random something must make it go so it couldn’t be going randomly. I thought nothing was random when you looked closely at it. In chaos theory they don’t really mean chaos, they mean deterministic chaos which looks lilke chaos because it is so complicated (to us at our level).
Then there’s the problem of those experiments that proved that electrons and/or photons were particles because of the slits they go through but also it was proved that they are waves. So they are sort of both but in any case they each must have some size and mass or the experiment that showed they are particles wouldn’t have come out.
–Although I gather that someone said above that it’s the EFFECT of the particle that counts, and that EFFECT might be much bigger than the particle (I mean it always is bigger than the particle). Like that electron going around in its sphere is on the one hand tiny but on the other hand it’s the whole sphere because of what I call its EFFECT.

14

That’s where all the “QM nit pics” come in. You can talk about the size of a proton, since it’s a composite particle, but not about the size of an electron (although you can talk about the size of an electron orbital). It’s not correct to think of an electron as a little sphere (perhaps with an “e” or an “-” printed on it ;)) zipping around the nucleus. The electron just occupies the orbital. If thinking of it that way helps, go for it, but like an analogy, you can push the picture too far. For example, when you ask:

you’ve pushed the imagery too far.

15

You can’t say what path the electron takes. So far as we know, the question is completely meaningless. At any given time, you can measure the electron’s momentum (and hence velocity) as precisely as you like, but that doesn’t mean it would have had that momentum had you not measured it.

And the electron isn’t just a little sphere with a minus sign and an e printed on it; it’s a little blue sphere with a minus sign and an e printed on it. Positively charged particles, of course, are little red spheres with plus signs ;).