electron degeneracy, neutron degeneracy, and pressure?

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electron degeneracy, neutron degeneracy, and pressure?

Postby >-) » Sat Oct 05, 2013 6:31 pm UTC

So on wikipedia here it says the formula for electron degeneracy, which is dependent on a few constants and the mass of the proton and electron:

http://en.wikipedia.org/wiki/Electron_d ... y_pressure

i'm also interested in neutron degeneracy pressure. however, this time i could not find anything at all. i did however, stumble upon this page which just gives a general pressure for degenerate fermion matter. http://quantummechanics.ucsd.edu/ph130a ... de204.html

my question is, why is electron degeneracy pressure dependent on the mass of the proton and electron while the fermion degeneracy pressure is only dependent on a single mass. is one or both of the equations incorrect? can neutron degeneracy pressure be calculated with the second equation?

one possible cause could be that electrons and protons have opposite charges, so they attract, unlike neutrons. is this the case?

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Re: electron degeneracy, neutron degeneracy, and pressure?

Postby VrookLamar » Sat Oct 05, 2013 9:50 pm UTC

It's because if there are electrons in your white dwarf, there are also protons to keep the total charge neutral. So both the electron mass and the proton mass will matter. In neutron star matter, the protons and electrons have 'fused' to neutrons so they are the only relevant particle.

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Re: electron degeneracy, neutron degeneracy, and pressure?

Postby >-) » Sat Oct 05, 2013 10:09 pm UTC

that seems to make sense. thanks!

about the 'fusing' of protons and electrons, if i want to calculate the neutron degeneracy pressure of a single helium-4 atom, i should use 4 as the number of neutrons?.

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Re: electron degeneracy, neutron degeneracy, and pressure?

Postby thoughtfully » Sat Oct 05, 2013 10:49 pm UTC

>-) wrote:that seems to make sense. thanks!

about the 'fusing' of protons and electrons, if i want to calculate the neutron degeneracy pressure of a single helium-4 atom, i should use 4 as the number of neutrons?.

Since they're bosons, I'm not sure there's any degeneracy pressure. But I'm more wiseguy than smartypants on this subject matter.
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Re: electron degeneracy, neutron degeneracy, and pressure?

Postby >-) » Sat Oct 05, 2013 11:07 pm UTC

according to this wikipedia chart they fall under baryons/hardons. however, you're still right that they're not fermions: http://i.snag.gy/xybGC.jpg
they should have degeneracy pressure : http://en.wikipedia.org/wiki/Degenerate ... degeneracy
"Neutron degeneracy is analogous to electron degeneracy and is demonstrated in neutron stars, which are primarily supported by the pressure from a degenerate neutron gas.[7]"

i'm quite interested in finding an equation that can describe neutron degeneracy pressure. does anyone have any ideas?

edit: ah-ha! the term 'neutron gas' did it. i found this via google: http://es.ucsc.edu/~glatz/astr_112/lectures/notes18.pdf which gives a pressure formula similar to that of fermions.

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Re: electron degeneracy, neutron degeneracy, and pressure?

Postby gmalivuk » Sun Oct 06, 2013 3:53 am UTC

Given that the size limit on neutron stars has been calculated as between 1.5 and 3.0 solar masses, I suspect the formula for neutron degeneracy pressure is not known nearly as precisely as that for electrons, or else they'd be able to calculate a tighter limit.
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Re: electron degeneracy, neutron degeneracy, and pressure?

Postby starslayer » Sun Oct 06, 2013 8:21 pm UTC

Yeah, we don't know the nuclear matter equation of state yet. It's one of the big unknowns in neutron star physics, and it almost certainly does not have an analytic form. People have been doing/trying to do numerical simulations of it for 30-40 years now, but still haven't had definitive success.

>-) wrote:according to this wikipedia chart they fall under baryons/hardons. however, you're still right that they're not fermions: http://i.snag.gy/xybGC.jpg


You missed thoughtfully's point. He-4 nuclei are bosons, because they are spin 0. Bosons are particles with integer spin, fermions have half-integer spin. The Wiki page lists fundamental bosons and fermions, not composite ones.

The neutron degeneracy pressure formula you found has exactly the same form as the one for electrons, because they are both fermions. When deriving the degeneracy pressure, you start by dealing with identical fermions of some mass, charge, etc. Then you can plug and play for gases composed of different particles. This is exactly what the second link you posted showed. Because bosons do not follow the Pauli Exclusion Principle, they do not have degeneracy pressure.

However, as those lecture notes hint at, this is not the whole story when it comes to neutrons. It turns out that the equation given is a non-relativistic, low-density approximation to the real thing. At neutron star densities, you must take relativity and the nuclear forces between neutrons into account. This is why we have known the general properties of white dwarf interiors for a long time, but still know next to nothing about the deep interiors of neutron stars.

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Re: electron degeneracy, neutron degeneracy, and pressure?

Postby >-) » Tue Oct 08, 2013 2:08 am UTC

i see what you mean by the composite and fundamental bosons/fermions now.

when a he-4 atom is subjected to pressures where neutron degeneracy pressure is significant, the protons and electrons have broken down so that they're all fermions with degeneracy pressure, correct?

So, when computing the neutron degeneracy pressure of a mol of helium-4 per nm^3, should i use 4 mols of neutrons per nm^3 since the protons and electrons have 'merged'?

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Re: electron degeneracy, neutron degeneracy, and pressure?

Postby PM 2Ring » Tue Oct 08, 2013 6:17 am UTC

>-) wrote:when a he-4 atom is subjected to pressures where neutron degeneracy pressure is significant, the protons and electrons have broken down so that they're all fermions with degeneracy pressure, correct?


According to the Standard Model electrons don't break down since they're fundamental particles. (OTOH, there are some speculations that they aren't truly fundamental).

The relevant equation is:

proton + electron + energy → neutron + neutrino

This is just a rearrangement of the equation for the decay of a free neutron via the weak nuclear force, aka the beta decay equation:

neutron → proton + electron + antineutrino + energy

where the released energy is generally in the form of kinetic energy of the resulting particles; from conservation of momentum it can easily be shown that most of the KE goes into the electron & antineutrino.
Spoiler:
Sometimes, (about one decay per 1000 events,) energy is released in the form of a high energy photon, aka a gamma ray. See http://en.wikipedia.org/wiki/Free_neutr ... tron_decay for details.

However, although protons and neutrons are fermions, they aren't fundamental, since they're composed of quarks, which are fundamental fermions.

At the quark level, the neutron decay equation becomes

down → up + W- + energy
where down & up are quarks and W- is a heavy boson associated with the weak nuclear force.
The W- has a very short half-life, so it quickly decomposes:
W-→ electron + antineutrino

Here's a diagram, from the above Wiki link.
Image

From http://en.wikipedia.org/wiki/Free_neutr ... tron_decay
When bound inside of a nucleus, the energetic instability of a single neutron to beta decay is balanced against the instability that would be acquired by the nucleus as a whole if an additional proton were to appear by beta decay, and thus participate in repulsive interactions with the other protons that are already present in the nucleus. As such, although free neutrons are unstable, bound neutrons in a nucleus are not necessarily so. The same reasoning explains why protons, which are stable in empty space, may transform into neutrons when bound inside of a nucleus.


Also see http://en.wikipedia.org/wiki/Weak_decay

For a neutron gas, the situation is slightly different to what happens in a nucleus: the neutrons "want" to decay, but the huge amount of energy present favours the reverse reaction. Roughly speaking, the rate that neutrons are being produced by inverse beta decay overpowers the rate that they're decaying through beta decay.


>-) wrote:So, when computing the neutron degeneracy pressure of a mol of helium-4 per nm^3, should i use 4 mols of neutrons per nm^3 since the protons and electrons have 'merged'?

Well, as I (hopefully) explained above, the protons and electrons haven't actually merged, per se, but yes, each 4He atom gets converted into 4 neutrons (and 2 neutrinos).

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Re: electron degeneracy, neutron degeneracy, and pressure?

Postby >-) » Sat Oct 12, 2013 2:42 am UTC

that's a fascinating in-depth explanation. thank you!

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Re: electron degeneracy, neutron degeneracy, and pressure?

Postby Minerva » Thu Oct 17, 2013 7:02 am UTC

The hydrostatic equation that describes neutron-degenerate stuff is the Tolman–Oppenheimer–Volkoff equation, which is analogous to the hydrostatic equation derived by Chandrasekhar for electron-degenerate matter. This gives rise to the Tolman–Oppenheimer–Volkoff limit to the maximum mass of a neutron star. http://en.wikipedia.org/wiki/Tolman%E2% ... f_equation

The TOV limit for the maximum mass of a neutron star has a fair bit of uncertainty in it, compared to the Chandrasekhar limit for the limiting mass of a white dwarf, because an accurate equation of state for neutron degenerate matter doesn't exist.
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Re: electron degeneracy, neutron degeneracy, and pressure?

Postby Dopefish » Thu Oct 17, 2013 4:56 pm UTC

Minerva wrote:The hydrostatic equation that describes neutron-degenerate stuff is the Tolman–Oppenheimer–Volkoff equation, which is analogous to the hydrostatic equation derived by Chandrasekhar for electron-degenerate matter. This gives rise to the Tolman–Oppenheimer–Volkoff limit to the maximum mass of a neutron star. http://en.wikipedia.org/wiki/Tolman%E2% ... f_equation

The TOV limit for the maximum mass of a neutron star has a fair bit of uncertainty in it, compared to the Chandrasekhar limit for the limiting mass of a white dwarf, because an accurate equation of state for neutron degenerate matter doesn't exist.


If this was any other forum, I would attribute this to crazy sciency words just being strung together for Sci Fi purposes. Actual science is crazy.


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