new entry ... not to neglect ...


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[QUOTE=science_man_88;273246]catchiest pop song ever scientifically crowned:
[url]http://ca.news.yahoo.com/science-hails-champions-catchiest-pop-song-ever-111119672.html[/url][/QUOTE]

Not to mention the catchiest song performed by scientists:
[url]http://www.youtube.com/watch?v=Hg1yw8D3glE[/url]

(love the older researchers with bow ties)

[YOUTUBE]Hg1yw8D3glE[/YOUTUBE]

[QUOTE=sichase;272574]I have to give you a very unsatisfying answer. For neutrinos, the observed fact is that the mass eigenstates are not the flavor eigenstates. That is, the observed particles don't have exact leptonic flavor, but instead are superpositions of pure-flavor wavefunctions, through a mixing matrix called (I just looked up) the Pontecorvo-Maki-Nakagawa-Sakata matrix. For the other leptons, this is just not so. Electrons, muons and taus are all exact flavor eigenstates.

Given our current level of insight into the structure of the standard model, the best we can say is that we observe that these facts are so, and so write the SM Lagrangian to capture that fact. But why the Lagrangian has this structure, for this feature, or any other (number of families, masses of particles, etc.) is not known.

Some additional insights are available to some of these questions in the various extensions of the SM, but none of them have any evidence in their favor at this point, so I personally prefer "we don't know why" as the best answer.


--Scott[/QUOTE]Thanks for this clarification. By "this is just not so" does that mean that there are sound theoretical believes for that statement (I suspect not, given your subsequent paragraph) or just that there is no experimental evidence to suggest otherwise? If the latter, perhaps it may be possible that the flavor mixing is non-zero but very small for some reason. Do you have any pointers to the experimental evidence easily to hand? If not, I'll search them out, so please don't put yourself to any trouble.

Paul

[QUOTE=Jeff Gilchrist;273283]Not to mention the catchiest song performed by scientists:
[url]http://www.youtube.com/watch?v=Hg1yw8D3glE[/url][/QUOTE]We never danced around like that when I worked in the lab (not that one.)

[QUOTE=sichase;272534]Hi Paul,

Look up "de Broglie wavelength". You'll see that the characteristic quantum wavelength of a non-relativistic electron is dominated by its rest mass in exactly the way you expect that it should. [/QUOTE]I have trouble with that statement.

Particle diffraction experiments have been used for molecular structure determination for many years and with a variety of particles. To be useful, their effective wavelength needs to be of the same order of inter-atomic separations --- about 100pm to the nearest order of magnitude.

For photons, with zero rest mass, everything is straightforward. A photon energy of around 10keV has a wavelength of around 100pm. Such photons are in the (fairly soft) X-ray region.

The mass of a non-relativistic electron is about 500keV (somewhat greater than 511keV to be precise), 50 times larger than the mass of the X-ray photon just described. It should have a de Broglie wavelength of around 2pm, which is much shorter than the inter-atomic spacing in non-degenerate matter.

The mass of a non-relativistic neutron is around 1GeV (~940MeV to be more precise) and so should have a de Broglie wavelength of around 1fm. This is shorter even than the inter-atomic spacing in matter held apart by elctron degeneracy pressure and is utterly insignificant compared to the length typical molecular bond.

However, electron and neutrons diffraction have been used for decades as probes of molecular and crystal structures. As far as I am aware, the particles used in these experiments have been cooled to the point where their [i]classical, or non-relativistic,[/i] momenta [i]in the laboratory rest frame[/i] are comparable with the (necessarily relativistic) momentum of a 10keV photon.

Do you now see why I have difficulties understanding what is going on?


Paul

[QUOTE=xilman;273285]Thanks for this clarification. By "this is just not so" does that mean that there are sound theoretical believes for that statement (I suspect not, given your subsequent paragraph) or just that there is no experimental evidence to suggest otherwise? If the latter, perhaps it may be possible that the flavor mixing is non-zero but very small for some reason. Do you have any pointers to the experimental evidence easily to hand? If not, I'll search them out, so please don't put yourself to any trouble.

Paul[/QUOTE]

Hi Paul. Oscillations amongst the elements of a flavor tuplet is just one possible mechanism for flavor number violation. In this particular case there is a ton of experimental (non)evidence for what is called "lepton number violation". People have been looking for a long time, always unsuccessfully... until neutrino oscillations gave us the first solid evidence of such violation.

Why have people been looking so aggressively? Because, in fact, there is good theoretical reason for believing that at some very small level, it must exist. At sufficiently high energy, in the non-perturbative regime of the electroweak interaction, there are "chiral anomalies". These can't be represented by Feynmann diagrams (because they are non-perturbative) but they exist in the theory and predict lepton number violation at sufficiently high energy.

In addition, various extensions of the Standard Model, most notably SUSY, give additional mechanisms for lepton number non-conservation. I don't know the details, but the basic idea is that you add additional particles to the theory with coupling such that, e.g., an electron can interact with the new particle in a flavor-number-violating interaction that produces, e.g., a muon without any counterbalancing antineutrinos to make lepton number conservation work out.

In any event, if you search for "lepton number violation" on the web you'll find lots of experimental papers setting limits on different channels.

--Scott

[QUOTE=sichase;273296]In any event, if you search for "lepton number violation" on the web you'll find lots of experimental papers setting limits on different channels.[/QUOTE]Thanks, I'll do that.

Paul

[QUOTE=xilman;273288]I have trouble with that statement.

Particle diffraction experiments have been used for molecular structure determination for many years and with a variety of particles. To be useful, their effective wavelength needs to be of the same order of inter-atomic separations --- about 100pm to the nearest order of magnitude.

For photons, with zero rest mass, everything is straightforward. A photon energy of around 10keV has a wavelength of around 100pm. Such photons are in the (fairly soft) X-ray region.

The mass of a non-relativistic electron is about 500keV (somewhat greater than 511keV to be precise), 50 times larger than the mass of the X-ray photon just described. It should have a de Broglie wavelength of around 2pm, which is much shorter than the inter-atomic spacing in non-degenerate matter.

The mass of a non-relativistic neutron is around 1GeV (~940MeV to be more precise) and so should have a de Broglie wavelength of around 1fm. This is shorter even than the inter-atomic spacing in matter held apart by elctron degeneracy pressure and is utterly insignificant compared to the length typical molecular bond.

However, electron and neutrons diffraction have been used for decades as probes of molecular and crystal structures. As far as I am aware, the particles used in these experiments have been cooled to the point where their [I]classical, or non-relativistic,[/I] momenta [I]in the laboratory rest frame[/I] are comparable with the (necessarily relativistic) momentum of a 10keV photon.

Do you now see why I have difficulties understanding what is going on?

Paul[/QUOTE]

Your numbers are right. The fact that the deBroglie wavelength of an electron is less than a couple of nm is precisely why the limiting resolution of an electron microscope is so much higher than for an optical microscope.

Truthfully, I'm not quite getting where your confusion lies. Are you suggesting that because the electron deBroglie wavelength is small that it should _only_ be able to probe small distances (smaller than its deBroglie wavelength)? This is not true. Electrons are good probes of crystals exactly because their limiting wavelength is less than the typical interatomic spacing. [Similarly, neutrons make excellent nuclear probes, but because they are electrically neutral, interact only very weakly (primarily through their magnetic dipole moment) with orbital electrons.]

Perhaps the simple exercise on this page I found will help reconcile things here - I was unable to properly view the images capturing the formulae in Firefox (the site author likely only tested it under IE), so I translated them into LaTeX in the quote box below (but it seems our site is not set up to be able to inline-render that - If there is a way to do, please let me know). Anyway, it seems the key is that the DeBroglie wavelength formula h/p needs the proper expression for the relativistic momentum p:

[url=http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/debrog2.html]DeBroglie Wavelength[/url]
[quote]A convenient form for the DeBroglie wavelength expression is
[latex]lambda = \frac{hc}{pc}[/latex], where hc = 1239.84 eV nm and pc is
expressed in electron volts.

This is particularly appropriate for comparison with photon wavelengths since for the photon, pc=E and a 1 eV photon is seen immediately to have a wavelength of 1240 nm. For massive particles with kinetic energy KE which is much less than their rest mass energies:

[latex]pc = \sqrt{ 2 * KE * m\sub{0} c\sup{2} }[/latex]

For an electron with KE = 1 eV and rest mass energy 0.511 MeV, the associated DeBroglie wavelength is 1.23 nm, about a thousand times smaller than a 1 eV photon. (This is why the limiting resolution of an electron microscope is much higher than that of an optical microscope.)[/quote]

[QUOTE]A convenient form for the DeBroglie wavelength expression is
[TEX]\lambda = \frac{hc}{pc}[/TEX], where hc = 1239.84 eV nm and pc is
expressed in electron volts.

This is particularly appropriate for comparison with photon wavelengths since for the photon, pc=E and a 1 eV photon is seen immediately to have a wavelength of 1240 nm. For massive particles with kinetic energy KE which is much less than their rest mass energies:

[TEX]pc = \sqrt{ 2 * KE * m_{0} c^{2} }[/TEX]

For an electron with KE = 1 eV and rest mass energy 0.511 MeV, the associated DeBroglie wavelength is 1.23 nm, about a thousand times smaller than a 1 eV photon. (This is why the limiting resolution of an electron microscope is much higher than that of an optical microscope.)[/QUOTE]

made a few changes.

Ah, so we don't need the [latex] tags around here - cool. The resulting images still render borderline-unreadably for me using FF, however.