Difference between revisions of "User:Tohline/Appendix/Ramblings/StrongNuclearForce"
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==Tidbits== | |||
From an [https://physics.stackexchange.com/questions/8452/is-there-an-equation-for-the-strong-nuclear-force online chat]: | |||
<ul> | |||
<li> | |||
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, | |||
<div align="center"> | |||
<table border="0" cellpadding="5" align="center"> | |||
<tr> | |||
<td align="right"> | |||
<math>~V(r)</math> | |||
</td> | |||
<td align="center"> | |||
<math>~=</math> | |||
</td> | |||
<td align="left"> | |||
<math>~ | |||
- \frac{4}{3} \cdot \frac{\alpha_s(r) \hbar c}{r} + kr \, , | |||
</math> | |||
</td> | |||
</tr> | |||
</table> | |||
</div> | |||
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>. | |||
</li> | |||
<ul> | |||
<li> | |||
Of course, the "breaking of the flux tube" has no classical or semi-classical analogue, making this formulation better for hand waving than calculation. | |||
</li> | |||
<li> | |||
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. | |||
</li> | |||
</ul> | |||
<li> | |||
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, | |||
<div align="center"> | |||
<table border="0" cellpadding="5" align="center"> | |||
<tr> | |||
<td align="right"> | |||
<math>~V(r)</math> | |||
</td> | |||
<td align="center"> | |||
<math>~=</math> | |||
</td> | |||
<td align="left"> | |||
<math>~ | |||
- \frac{g^2}{4\pi c^2} \cdot \frac{e^{-mr}}{r} \, , | |||
</math> | |||
</td> | |||
</tr> | |||
</table> | |||
</div> | |||
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. | |||
</li> | |||
</ul> | |||
=Potentially Useful References= | =Potentially Useful References= | ||
Revision as of 19:13, 23 February 2018
Radial Dependance of the Strong Nuclear Force
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Tidbits
From an online chat:
-
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.
-
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.
Potentially Useful References
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