One of my old high school friends on Facebook followed up my recent graphene post with a question I found intriguing:
Is a dimensional plane defined as the width of an atom?
The reason for this question comes from the wording of the recent 2010 Nobel Prize in Physics, awarded last Tuesday to Andre Geim and Konstantin Novoselov "for groundbreaking experiments regarding the two-dimensional material graphene."
Graphene is a sheet of carbon only one atom thick.
Source: University of Manchester
However, the problem is that graphene isn't really two-dimensional. It has length and width, but each carbon atom in the sheet has a height, so it takes up three space dimensions.
My first answer tended toward the glib side:
It depends on whether you ask a physicist or a mathematician.
The point of the comment is that physicists tend to make approximations (there's even a joke, very funny in physics circles, that ends with the punchline, "Assume the cow is a sphere"). To a mathematician, however, a 2-dimensional plane will have no third dimensional length at all. This is fundamental to the definition of 2-dimensional.
The glib answer didn't fly, as I realized when I received the reply:
I disagree, as does my wife the Physics teacher. Graphene is described as being "so dense that even helium cannot pass through it." Something that is two-dimensional cannot have density, as density is defined as mass per unit volume, and you cannot have a volume without three dimensions
So, my old high school friends (both he and his wife) weren't going to let me off so easily. For a second, I thought about saying something about surface density, but that doesn't really get to the heart of the issue: that the graphene does really occupy three dimensions.
Then, I struck on an argument which might work on both physicists and mathematicians. As with many things in physics (and mathematics) it depends on how precisely you define your space.
If you are talking about a physical space, then even the single atom exists in a 3-dimensional world right out of the gate. Of course, in this space, even a world without any objects in it is three dimensional right off the bat - you've defined it that way.
A zero-dimensional space therefore does not relate to "nothing," but to a single atom ... just as zero-dimensions in mathematics relates to a single point.
A 1-dimensional space relates to a series of atoms in a line. Note that this space is non-continuous, like the integer space as opposed to the real number space, because you cannot have 1.5 atoms.
Expanding this into another direction we now have a 2-dimensional lattice of atoms that is one atom high ... such as graphene.
A space without any atoms isn't just zero-dimensional ... it's non-existent. (This would be like a number system that contains no numbers, not even 0.)
Clearly, the Physics Nobel Committee (which, to my knowledge, is composed of physicists) who presented the award considered it close enough to identify graphene as "two-dimensional material" in the official prize wording ... but the discussion is quite illuminating, as it makes clear the importance of precisely defined terminology.
Comments
People don’t call a drawing on a piece of paper 3D even though the atoms of graphite or ink will be distributed three dimensionally.
Right, the language used in the prize was a semantic description. Physicists know what a two-dimensional surface is, and they know an atom is not. They call that a “brane.” I’m not sure this commentary was fully thought through.
Besides which, graphene is not actually one atom thick, is it? Carbon doesn’t quite line up in a plane in hexagonal rings, at least with substances like benzene and sugar. Each bond is at a slight angle, making a bumpy hexagon. Is it the same way in graphene? If the angles making bumps, it would be a little more than one atom thick.
As far as being “dense” so that helium cannot pass through it, that doesn’t have as much to do with the positions of the atomic nuclei as much as the valence field between them, through which their electrons are changing places in one gigantic sheet wave.
I wonder how much electron exchange there is in the valence shell from one side of the sheet to the other. Do half the electrons stay on one side, and the other half on the other side? That might make the valence field on each side of the sheet extremely smooth and even, which would repel any other negative charge that came close, i.e. a helium valence shell, no matter whether the other object was headed toward a nucleus or a hole.
The point of being two-dimensional is not about being one atom thick, it’s about being in the lowest quantum state in the third direction, so that electrons cannot move in that dimension. This is a reasonable definition of a physical system with reduced dimension. For instance, the two-dimensional electron gas in a semiconductor heterostructure, used in the quantum Hall effect, is much thicker than one atom but is still considered two dimensional because the wave function is in the lowest quantum state in the third direction.
I agree with Brian. In art and many other things “two dimensional” objects are much, much thicker than just one atom.
Your comment:
“A space without any atoms isn’t just zero-dimensional … it’s non-existent.”
Without a void of some kind, atoms would have nowhere within which to exist. It can therefore be concluded that space is on the one hand conceptual, but on the other hand an existant unmeasurable entity.
I’m sorry to be pedantic, but:
“A space without any atoms isn’t just zero-dimensional … it’s non-existent.”
If something has zero dimensions, it cannot be called a “space”…
Pedantry aside though, I’m afraid I thought the exact same things as your old High School friend does. I find myself in the remarkable position of thinking the Nobel Committee got it wrong. I do not believe their statement can be explained away so easily as you cannot divorce Physics from Mathematics. Saying graphene is two-dimensional might be Physics “shorthand” for saying a single atom (actually, to be pedantic again, it’s a molecule, not an atom) “thick”. But it is incorrect.
Atoms (molecules) might have “negligable” thickness, but they do have thickness – One that can be measured and is sometimes needed in both Chemistry and Physics at times.
You did a very good job of trying to find an explanation that reconciles both viewpoints though and I commend you for it. I just wish I could get over the feeling that the Committee used language in a very imprecise way.
Graphene would be a single sheet of our old friend, graphite, as an extended single layer. As such , many of the probable qualities and reactions of the material probably can be explained, or guessed at, from organic chemistry considerations.
That is, it could be considered as naphthalene, anthracene, etc. attached together indefinitely. It would be interesting to hear more about how large a units of this material have been produced. Its possible use as a component for electronic use would seem very interesting as the material would have very possible “two dimensional super-conductivity ” in the bonding plane,
with transister-type activity at right angles to the bonding plane.l
Presumably graphene could be twisted into the classic }two dimensional” infinity strip, or into the corresponing bottle just as can a piece of paper in which case, the resulting continued surface could be considered “two-dimensional.”
However, I agree that the Nobel announcement calling Graphene a two-dimensional substance was in error.
‘Tis not the first time the Nobel Prize people have been in error. After all, they gave that “House of Cards” that goes by the name of the “Standard Model of Particle Physics” a prize, and versions of Quantum Mechanics got prizes two years in a row….I’m sure other people have their ideas as to what they think the Nobel Prize people have done in the way of interesting “boo-boos” over the years.
TeeK, sorry to be equally pendantic, but your following quote is wrong:
If something has zero dimensions, it cannot be called a “space”…
The zero dimensional space that contains the point 0 is a complete space in the mathematical definition of the term. It’s a “trivial” space, of course, but it’s still a space.
@Physics
TeeK, sorry to be equally pendantic, but your following quote is wrong:
Physics, you really missed the point, if you start redefining terms for Physics and Mathematics, you end up with incompatable words. I was talking English Language – Of course you can have a zero-dimentional space in mathematics, but that is acceptable use of the word “space” only in Mathematics.
I fear you may be falling into the same trap as the Committee – Also, I did point out it was being pedantic as a bit of humor!
@(6) Teek:
“”I’m sorry to be pedantic, but:
“A space without any atoms isn’t just zero-dimensional … it’s non-existent.”
If something has zero dimensions, it cannot be called a “space”…”"
I’m sorry to be pedantic, but something that is non-existent, cannot be called. Period. That is his point.
Wow. Brian, Nath, you’re wrong. Mathematically its 2d because its on a 2d plane, but nothing can possible exist in this world with no height.