Wednesday, November 9, 2011

Adding To the Fundamental Particle And Periodic Tables

The Periodic Table

The periodic table of the elements is old news (some interesting variant typographic representations of it exist, by the way), but I'll summarize it anyway.

Particles with the same number of protons bound together in a nucleus by the nuclear binding force have the same number of electrons in orbits around them, and as a result, have similar chemical properties. This group of similar atoms is called an "element" with a particular atomic number. Different elements can have different numbers of neutrons, which are called "isotypes" of the element. Apart from their mass per atom and their varying likelihoods of decaying into something else, isotypes of the same atom behave identically chemically.

Electrons arrange themselves around atoms in concentric "shells." For elements 1-118, there are up to four subshells (S, F, D, and P) for each of the seven periods. An additional shell (G) is theoretized to exisst for elements of atomic number greater than 118. Hydrogen and helium have only the S shell. Atoms 3-18 (Lithium to Argon) have both S and P shells, and the P shell is the outer shell. Elements 19-54 (Potassium to Xenon) also also have the D shell. Elements 55-118 (all discovered elements from Cesium on up) also have an F shell. The chemical properties of elements are determined mostly by the number of unfilled positions in the outermost (a.k.a. valence) shell.

For example, all elements with full valence shells are noble gases that are extremely unreactive chemically; those with just one electron vacancy in their outermost P shells are highly reactive halogens. Alkali metals have one electron in their outermost S shell, and Alkali earth metals have full outermost S shells. Ordinary transition metals are filling their outermost D shells. The inner transition metals (lanthanides (lanthanoids) and actinides (actinoids)) are in the process of filling their outermost F shells. The post-transition metals, metalloids and non-metals are in the process of filling their outermost P shells.

Some isotypes are unstable and their neutrons have a particular likelihood of decaying into a proton and additional decay products (beta decay), or jettisoning a group of nucleons together (alpha decay), because the nuclear binding force isn't sufficient to hold them together is a stable way. The resulting new number of protons turns the old isotype of one element into an isotype of another element with a lower atomic number. Atomic nuclei can be split in other ways too, which is called nuclear fusion and emit energy if the binding energy of the larger atom is larger than the binding energy of the smaller atom. Atomic nuclei can also be forcibly joined at high energy and if the binding energy of the resulting atom is less than the binding energy of the joined atom nuclear fusion releases atomic energy.

Generally speaking, binding energy per nucleon declines through iron (atomic number 26), and increases thereafter. So, atoms heavier than iron can be split to create nuclear energy, while atoms lighter than iron can be fused to create nuclear energy.

All elements beyond 94, plutonium, do not occur in nature (the last two elements to be discovered in nature were francium discovered in 1940 and plutonium which was synthesized in 1940 but discovered in nature in 1971), although only artificially synthesized examples produced in 1939 or later of the twenty four elements 95-118 and element 93 (neptunium) exist. Elements with atomic numbers greater than 82 (lead), as well as technetium (43) and promethium (61), have no stable isotopes and neither technetium nor promethium are found in nature although they have been synthesized.

There is no generally accepted theoretical limit to the maximum atomic number that a synthetically made atom can form. No chemist or physicist seriously doubts, for example, that we can create element 119.

There is some argument over what chemical properties the heavier synthetic elements would have, particularly beyond atomic number 138, and there is a great deal of interest in locating "islands of stablity" consisting of synthetic elements that are metastable relative to the elements that came before them in the vicinity of atomic number 126, although no one really expects to discover any more completely stable isotypes, in addition to the 255 known stable isotypes of elements greater than 82. Another 84 isotypes are found in nature but have observed radioactivity.

In general, the higher the atomic number an element has the less stable its isotypes, and isotypes tend to be less stable as they acquire more or fewer neutrons than the most stable isotype of an element. Reasoning extrapolating the causes of this growing instablity in fundamental physics have hypothesized that the maximum atomic number may be 130-173. "The light-speed limit on electrons orbiting in ever-bigger electron shells theoretically limits neutral atoms" to an atomic number of approximately 137 in a Bohr model of an electron, although considering the Dirac equation which takes into account relativistic effects, it might be able to have as many as 173 electrons coherently around an atom. Nuclei with more protons than this might be possible only as electrically charged ions, as they couldn't maintain their full electron shells.

But, the issue is largely academic because as the table of nuclides (i.e. isotypes) indicates, atoms with more nucleons become more unstable anyway. No isotype of atomic number 106 or greater has a half life of as much as a day. All but 905 of about 3000 experimentally characterized isotypes have half lives of less than an hour.

The 2400 well characterized isotypes with half lives of less than one hour are all synthetic, as are 556 isotypes with half lives of more than one hour. The longest lived isotype of atoms of atomic number 118, for example, are a bit less than a millisecond (i.e. 10^-3 second). Heavier isotypes would decay more quickly.

Fundamental Particles

The known synthetic elements, all made up of ordinary protons and neutrons, last much longer than any of the unstable fundamental particles (i.e. second or third generation fermions, W bosons and Z bosons), the longest lived of which, the muon, has a mean lifetime of about 10^-6 seconds (although in fairness, accurate estimates for muon neutrinos and tau neutrinos are not available). No hadron other than a proton or a neutron lasts longer than about 10^-8 seconds.

Most "exotic" atoms, such as muons hydrogen, are as unstable as their exotic component, although in principle, anti-hydrogen made up entirely of antiparticles, ought to be more stable so long as it is kept away from ordinary matter.

Fourth generation quarks (conventionally labeled t' and b') would presumably be significantly heavier than the top quark (about 174 GeV) and they are experimentally excluded for the b' below 199 Gev and for the t' below 256 GeV, would presumably have a half life of quite a bit less than 7*10-25 seconds and thus would presumably not hadronize, and presumably, like the top quark, the t' would almost always and instantly decay to a W+ boson and a bottom quark, while the b' would almost always and instantly decay to a W- boson and a top quark (with the antiparticles, of course, experiencing the reverse reaction).

Thus, in a collider, a t' looks just like a top quark decay with additional very energetic W+ and W- boson decays (although not energetic enough to give rise to t' and anti-t' pairs), and a b' looks just like a top quark decay with an additional very energetic W- boson decay (although not energetic enough to give rise to t' and anti-t' or b' and anti-b' pairs). Neither the t' nor the b' would give us anything more interesting than these decays, although W bosons of these energies might be expected to produce fourth generations leptons as well.

If the t' to t quark mass ratio were similar to the t quark to c quark mass ratio, one would expect the t' quark to have a mass of 20 TeV; the b quark to s quark ratio would imply a t' quark mass of 7 Tev; the s quark to u quark mass ratio would imply a t' quark mass of about 3.5 Tev; the tau to muon mass ratio would imply a t' mass of about 3 TeV. The ratio of c quark mass to u quark mass, or of muon to electron mass would suggest an even greater t' mass.

Of the b/c, c/s, and s/d mass ratios, none are smaller than a factor of about 3, so one would expect a b' mass of not less than about 522 GeV and much heavier masses on the order of 1 TeV or more would be plausible given expectations for the t' mass.

The lower experimental bound on a tau prime (i.e. fourth generation lepton) mass is about 100 GeV, which is about 55 times the tau mass, and d/s/b type mesons appear to tend to be within an order of magnitude of mass of their same generation charged lepton, so a mass in the high hundreds of GeV would not be particularly unexpected.

These values are not updated for the latest 2011 LHC bounds, which are higher. LHC puts a lower bound on the b' mass of 385 GeV, and the puts a reasonably expected t' mass values, if there is a t' quark, higher.

One possibility that could explain the thus far observed three and only three generations of fundamental fermions rule in particle physics is that there are deep reasons that prohibit particles that decay faster than W boson, a bound that the top quark already approaches and that any heavier quark would likely exceed.

Another possibility would be that there is some fundamental limit on the amount of energy that a W boson can hold. For example, the combination of the short lifetime of a W boson and its high mass translates into a maximum amount of kinetic energy or wavelength for the W boson (does a W boson or Z boson or gluon have a frequency in the way that a photon does?), that might be on the order of c (3*10^8 m/s) times the half life of a W boson (3*10^-25 s), which would be about 10^-18 meters, which is the approximate effective range of the weak force.

We know that a Z boson can give rise to a top/antitop pair with a combined mass of 348 GeV which far exceeds the 90 GeV of the Z boson rest mass. But, perhaps at some point there is a limit, and if that limitation is less than the correct mass for fourth generation fermion if there was one, then that limit would prevent fourth generation fermions from arising.

Honestly, the greatest bound on fourth generation fermions, in my mind, is the fact that we have seen Z boson decays energetic enough to produce top/antitop pairs, but have not seen any evidence of a fourth generation neutrino in the Z boson decays, which would seem to be energetically permitted up to 173-174 GeV neutrinos. All previous experience leads us to think that generations of fermions come in fours, and a 173 GeV or heavier neutrino would be so far outside the range of what we would expect given the experimental bounds on the first three generations of neutrinos, that it would seem to rule out a fourth generation of fermions entirely, and at least, would seem to rule out a fourth generation of fermions with masses low enough to be experimentally detected any time in the next century or so. The conventional statement of the experimental limitation on fourth generation neutrino mass from precision electroweak measurements is 45 GeV, which is still 2*10^6 times the approximate bound on third generation neutrino mass, when no other one generation mass increase of the same kind of fermion increases by even a factor of 10^3. (Of course, this limitation wouldn't apply to a sterile neutrino that doesn't interact with the weak force, for example, because it is is a right handed, non-antiparticle.)

If we knew the true laws of quantum mechanics better and knew they existed, we probably wouldn't care that much about finding the t' or b' or a fourth generation charged lepton experimentally. The stakes have more to do with learning the rules of the game by extending the available patterns than they do with finding that incredibly hard to create, anti-social and extremely ephemeral particles themselves. But, since we have lots of question marks about how the Standard Model functions at high energy levels, we look anyway.

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