Relativistically harmless

Every now and again when I’m studying I discover something which I think is just cool. And relativistic quantum chemistry — is just cool! Quantum mechanics and general relativity aren’t the best of friends. This much is pretty well known, and a great many people (physicists and otherwise) have waxed lyrical about it. Finding a way of combining the effects of the two into a single theory is the goal of any theorist working towards the so-called “theory of everything.” General relativity has difficulty fitting into the world of particles. Special relativity, on the other hand, bears no such hindrance.

In fact, special relativity underpins a huge amount of modern physics. It shouldn’t come as any surprise, then, that relativity rears its head in physical chemistry too. A lot of chemists may not need to care about relativistic effects, particularly organic chemists. It only really becomes an issue with the heavyweights near the bottom of the periodic table.

Dating back to Bertha Swirles’ 1935 paper, “The relativistic self-consistent field,” the idea is quite intuitive. Special relativity asserts that the faster an object travels, the greater its effective mass. The key to all of this is in the ever-curious fine structure constant, α;

α is a ubiquitous little thing. It seems to crop up all over physics, but in this case it denotes something quite simple — the speed of an electron orbiting a hydrogen nucleus. Hydrogen is the simplest atom. One proton and one electron. In a hydrogen nucleus, an electron will travel at roughly 137th the speed of light.

Increase the size of the atomic nucleus though, and you increase its charge. The increase in charge causes the electrons to accelerate to increasingly higher speeds. Nuclear charge is usually denoted by a Z, and it’s blissfully easy to plug this into the equation. Just multiply by Z to find the speed of the innermost electron in the shell. Like so;

For instance, that atomic nucleus up at the top of this post is uranium. Everyone’s favourite actinide. Uranium has 92 protons and thus, a charge of 92 on its nucleus. As a result, its innermost electrons will be travelling at around 67% of the speed of light. This increases their mass by around 34%, causing those electrons to orbit sligtly closer to the atomic nucleus. In turn, this has a knock-on effect, causing the entire atom to shrink slightly!

Interestingly enough, a whole range of effects are opened up by this relativistic malarkey. It contributes to many of the heavier elements having smaller atomic radii than they should. It explains why lead doesn’t have the same crystal structure as diamond. It explains why gold is actually gold! The electronic energy levels in gold are shifted by the relativistic effect, causing it to absorb the right optical frequencies to make it appear golden.

Most dramatically, it explains why mercury is liquid at room temperature. Relativistic effects cause the contraction of mercury’s 6s2 orbital, severely weakening its ability to form bonds. Subsequently, mercury remains liquid down to nearly -39°C. When it boils at around 357°C, it’s atoms drift away independently as a monatomic gas. Gaseous mercury is thus sometimes referred to as a “pseudo-noble gas.”

Ahhh. I love finding things out.

…wait, what do you mean quantum chemistry isn’t cool?

About Invader Xan

Molecular astrophysicist, usually found writing frenziedly, staring at the sky, or drinking mojitos.
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14 Responses to Relativistically harmless

  1. dr_psycho says:

    Hmmm…..
    The Invariant Set Postulate: A New Geometric Framework for the Foundations of Quantum Theory and the Role Played by Gravity
    Authors: T.N.Palmer. December 2008.
    http://arxiv.org/abs/0812.1148

  2. invaderxan says:

    Hmmm… I’m afraid I’m not sure without actually calculating it. And I suspect the calculations would be non-trivial!
    It would be to do with calculating the bohr radius of the electron shell, while accounting for the relativistic increase in mass. The electron orbital will be comparable to the size of the atomic nucleus when its cross-sectional area is on the order of one barn (which is the cross-sectional area of a uranium nucleus).
    As for what would happen to the electrons? Okay, this is speculation…
    As the first 1s electron is captured by a proton, the nuclear charge will decrease by 1. This would slow the orbiting electrons, causing the whole cloud to expand slightly. (Statistically, it’s possible that both 1s electrons would be captured, decreasing the nuclear charge and relaxing the electron cloud even more.)
    This would leave an open innermost shell, which would be equivalent to the whole atom being in an excited state. I’d assume, then, that the outer electrons would fall down to fill the space. And with a lot of electrons, there may be a number of transitions before the atom reached a relaxed state. With all those electron transitions, the atom would almost certainly fluoresce furiously while it was doing this. At what frequency? That I don’t know. Most likely in the visible/ultraviolet.

  3. nedu says:

    Any idea how high the atomic number would get for that to happen? Actually, what would happen to the atom anyway? I assume that only the 1s electrons would hit the nucleus. But then what would become of all the other electrons?

  4. invaderxan says:

    Hmmm… I’m a little skeptical of that website, TBH. Either it’s oversimplifying and explaining things poorly, or a few things don’t really make sense…
    “…general relativity comes to the rescue and saves element 137 from having electrons moving faster than the speed of light. However, even with general relativity, element 139 would still have electrons moving faster than light.”
    I’m not entirely sure what this means. General relativity? How? Are they suggesting reference frame dragging around an atomic nucleus? Because I’m fairly sure that’s one of those problems I was talking about in my first paragraph up there…
    Without a clearer explanation, I’m not entirely sure what all of this is trying to say… :
    Mind you, I do think it’s entirely possible that there could be a maximum speed at which electrons could orbit a nucleus. Presumably it would be where the high speed made the electron so massive that its orbit wasn’t significantly greater than the atomic nucleus itself.
    Interestingly, if that was the case, the electron could possibly be captured by a proton, forming a neutron (inverse beta decay). Subsequently, I suppose, any element larger than this would spontaneously decay by this mechanism.
    …But don’t take my word on that — it’s just speculation. ;)

  5. invaderxan says:

    Interestingly, that pretty much sums up my feelings towards neurology. :P
    I wouldn’t say frustrating as such. Though it can be incredibly confusing…

  6. Anonymous says:

    Real Theory of Everything
    As an alternative to Quantum Theory there is a new theory that describes and explains the mysteries of physical reality. While not disrespecting the value of Quantum Mechanics as a tool to explain the role of quanta in our universe. This theory states that there is also a classical explanation for the paradoxes such as EPR and the Wave-Particle Duality. The Theory is called the Theory of Super Relativity and is located at: Super Relativity
    This theory is a philosophical attempt to reconnect the physical universe to realism and deterministic concepts. It explains the mysterious.
    Quantum theory is incomplete since it is blind to the intricate structure of the invariant set.

  7. I have to say – I’ve always thought quantum chemistry sounded terrifically neat, but always in that ‘I’m glad it’s someone else‘s field’ way. I imagine it has to be terribly frustrating.

  8. nedu says:

    I can’t found where I originally read it, but looking for it has turned up some mentions. There’s this one: http://www.fotuva.org/online/frameload.htm?/online/137.htm
    Which, in the midst of talking about element 137 briefly mentions…
    “An electron in the ground state will orbit at the speed of light. This is the electromagnetic equivalent of a black hole. For gravitational black hole, general relativity comes to the rescue to prevent planets from orbiting at the speed of light and beyond. For an electromagnetic black hole, general relativity comes to the rescue and saves element 137 from having electrons moving faster than the speed of light. However, even with general relativity, element 139 would still have electrons moving faster than light. According to Einstein, this is an impossibility. Thus proving that we still don’t understand 137.”
    I guess that’s why it flew over my head before. I’ve never taken a physics class in my life (but I hopefully will next year).

  9. invaderxan says:

    Re: Awsome!
    Yeah, no kidding. The most interesting bits are always the ones you have to find out for yourself! :P

  10. invaderxan says:

    139? That’s interesting… I don’t suppose you recall where you read that, by any chance…?

  11. invaderxan says:

    Hmmm… Seems already answered this better than I could.
    Unless the nuclear theorists are right about the island of stability, I’m not sure how much heavier an atomic nucleus could be. Many of the bottom row of the Periodic table have a half life of seconds as it is.
    Although I do wonder about what might lurk inside neutron stars

  12. maxdwolf says:

    Awsome!
    Why did they leave this stuff out of chemistry class? This is way cool!

  13. nedu says:

    The FTL problem has been pointed out. However, I’ve read that some stuff I don’t understand pushes the actual limit to 139, rather than 137 as previously thought.

  14. Does that mean that atomic numbers can’t exceed 137, since otherwise the innermost electrons would be FTL? Or does that ration an approximation that doesn’t work as well for large atomic numbers?

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