Symmetry energy

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In nuclear physics, the symmetry energy reflects the variation of the binding energy of the nucleons in the nuclear matter depending on its neutron to proton ratio as a function of baryon density. Symmetry energy is an important parameter in the equation of state describing the nuclear structure of heavy nuclei and neutron stars.[1][2][3][4]

## Definition

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Let  n p {\displaystyle n_{p}} and  n n {\displaystyle n_{n}} be the number density of protons and neutrons in nuclear matter, and  n = n p + n n {\displaystyle n=n_{p}+n_{n}} . Let  E 0 ( n ) {\displaystyle E_{0}(n)} be the binding energy per nucleon in symmetric matter, with equally many protons as neutrons, as a function of density. The binding energy per nucleon  E {\displaystyle E} of non-symmetric matter is then a function that also depends on the isospin asymmetry,

δ = n p − n n n {\displaystyle \delta ={\frac {n_{p}-n_{n}}{n}}}

so to lowest order the energy per baryon is

E ( n , δ ) = E 0 ( n ) + S ( n ) δ 2 + O ( δ 4 ) , {\displaystyle E(n,\delta )=E_{0}(n)+S(n)\delta ^{2}+O(\delta ^{4}),}

where  S {\displaystyle S} is the symmetry energy.[2] There are no odd powers of  δ {\displaystyle \delta } in the expansion because the nuclear force acts the same between two protons as between two neutrons.[5] At saturation density n 0 {\displaystyle n_{0}} , the symmetry energy is 32.0±1.1 MeV.[4]

## References

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1. ^ Baldo, M.; Burgio, G. F. (November 2016). "The nuclear symmetry energy". Progress in Particle and Nuclear Physics. 91: 203–258\. arXiv:1606.08838. Bibcode:2016PrPNP..91..203B. doi:10.1016/j.ppnp.2016.06.006. S2CID 119216703.
2. ^ a b Tsang, M. B.; Zhang, Y.; Danielewicz, P.; Famiano, M.; Li, Z.; Lynch, W. G.; Steiner, A. W. (2009). "Constraints on the Density Dependence of the Symmetry Energy". Physical Review Letters. 102 (12): 122701. arXiv:0811.3107. Bibcode:2009PhRvL.102l2701T. doi:10.1103/PhysRevLett.102.122701. PMID 19392271.
3. ^ Tsang, M. B.; et al. (September 2010). "Constraints on the Density Dependence of the Symmetry Energy". International Journal of Modern Physics E. 19 (8n09): 1631–1638\. arXiv:0811.3107. Bibcode:2010IJMPE..19.1631T. doi:10.1142/S0218301310016041. ISSN 0218-3013.
4. ^ a b Lattimer, J. M. (January 2023). "Constraints on Nuclear Symmetry Energy Parameters". Particles. 6 (12): 30–56\. arXiv:2301.03666. Bibcode:2023Parti...6...30L. doi:10.3390/particles6010003.
5. ^ Zamora, Juan Carlos; Giraud, Simon (18 June 2024). "Monopole Excitation and Nuclear Compressibility: Present and Future Perspectives". Oxford Research Encyclopedia of Physics. arXiv:2406.16217. doi:10.1093/acrefore/9780190871994.013.115.

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