User:Tohline/Appendix/Ramblings/StrongNuclearForce
Radial Dependance of the Strong Nuclear Force
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Tidbits
From an online chat:
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From the study of the spectrum of quarkonium (bound system of quark and antiquark) and the comparison with positronium one finds as potential for the strong force,
<math>~V(r)</math>
<math>~=</math>
<math>~ - \frac{4}{3} \cdot \frac{\alpha_s(r) \hbar c}{r} + kr \, , </math>
where, the constant <math>~k</math> determines the field energy per unit length and is called string tension. For short distances this resembles the Coulomb law, while for large distances the <math>~kr</math> factor dominates (confinement). It is important to note that the coupling <math>~\alpha_s</math> also depends on the distance between the quarks.
This formula is valid and in agreement with theoretical predictions only for the quarkonium system and its typical energies and distances. For example charmonium: <math>~r \approx 0.4~\mathrm{fm}</math>.
- Of course, the "breaking of the flux tube" has no classical or semi-classical analogue, making this formulation better for hand waving than calculation.
- This is fine for the quark-qark interaction, but people reading this answer should be careful not to interpret it as a nucleon-nucleon interaction.
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At the level of quantum hadron dynamics (i.e., the level of nuclear physics, not the level of particle physics where the real strong force lives) one can talk about a Yukawa potential of the form,
<math>~V(r)</math>
<math>~=</math>
<math>~ - \frac{g^2}{4\pi c^2} \cdot \frac{e^{-mr}}{r} \, , </math>
where <math>~m</math> is roughly the pion mass and <math>~g</math> is an effective coupling constant. To get the force related to this you would take the derivative in <math>~r</math>.
This is a semi-classical approximation, but it is good enough that Walecka used it briefly in his book.
- The nuclear force is now understood as a residual effect of the even more powerful strong force, or strong interaction, which is the attractive force that binds particles called quarks together, to form the nucleons themselves. This more powerful force is mediated by particles called gluons. Gluons hold quarks together with a force like that of electric charge but of far greater power. Marek is talking of the strong force that binds the quarks within the protons and neutrons. There are charges, called colored charges on the quarks, but protons and neutrons are color neutral. Nuclei are bound by the interplay between the residual strong force, the part that is not shielded by the color neutrality of the nucleons, and the electro magnetic force due to the charge of the protons. That also cannot be simply described. Various potentials are used to calculate nuclear interactions.
Pointers from Richard Imlay circa 1983
When I asked Richard Imlay (high-energy experimentalist at LSU) for a reference to high-energy physics articles in which quark-quark interactions have been expressed in terms of a logarithmic potential, he pointed me to the following:
- Quigg, C. & Rosner, J. L. (1977, Physics Letters, 71B, pp. 153-157), Quarkonium level spacings
- Tuts, P. Michael (1983), Proceedings of the 1983 International Symposium on Lepton and Photon Interactions at High Energies, edited by D. G. Cassel & D. Kreinick (Cornell University, Ithaca, 1983), p. 284, Experimental results in heavy quarkonia
Cosmologies
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Derivation of the Friedmann Equations in the context of our discussion Newtonian free-fall collapse.
Newtonian Description of Pressure-Free Collapse
<math>~\biggl( \frac{\dot{R}}{R} \biggr)^2</math>
<math>~=</math>
<math>~\frac{8}{3}\pi G \rho - \frac{k(R_i, v_i)}{R^2} \, ,</math>
<math>~\frac{\ddot{R}}{R}</math>
<math>~=</math>
<math>~- \frac{4}{3}\pi G \rho \, ,</math>
where, <math>~k(R_i,v_i)</math>
<math>~=</math>
<math>~\frac{8}{3}\pi G \rho_i R_i^2 - v_i^2 \, .</math>
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Frieman, Turner & Huterer (2008, ARAA, 46, 385 - 432) provide an excellent, very readable review of dark matter and dark energy in the context of various cosmologies.
Friedmann equations:
Field equations of GR applied to the FRW metric<math>~H^2 = \biggl( \frac{\dot{a}}{a} \biggr)^2</math>
<math>~=</math>
<math>~\frac{8}{3}\pi G \rho - \frac{k}{a^2} + \frac{\Lambda c^2}{3}\, ,</math>
<math>~\frac{\ddot{a}}{a}</math>
<math>~=</math>
<math>~- \frac{4}{3}\pi G (\rho + 3p) + \frac{\Lambda c^2}{3} \, .</math>
- Nothing new …
Potentially Useful References
- Wikipedia — Semiempirical Formula for the Nuclear Binding Energy
- Walecka, John Dirk, Theoretical Nuclear and Subnuclear Physics, World Scientific (2004)
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