Global properties of atomic nuclei Binding 1 1 m(N,Z) = E(N,Z) = NM + ZM − B(N, Z) c2 n H c 2 The binding energy contributes significantly (~1%) to the mass of a nucleus. This implies that the constituents of two (or more) nuclei can be rearranged to yield a different and perhaps greater binding energy and thus points towards the existence of nuclear reactions in close analogy with chemical reactions amongst atoms.
B is positive for a bound system!
€ The sharp rise of B/A for light nuclei comes from increasing the number of nucleonic pairs. Note that the values are larger for the 4n nuclei (α-particle clusters!). For those nuclei, the difference 4He
ΔE ≡ B(N,Z) − nB(2, 2)
divided by the number of alpha-particle pairs, n(n-1)/2, is roughly constant (around 2 (MeV). This is nice example of the saturation of nuclear force. The associated symmetry is known as SU(4), or Wigner supermultiplet symmetry.
Consequence: possible alpha clustering in low-density regions
€ Odd-even effect Binding (summary) • For most nuclei, the binding energy per nucleon is about 8MeV. • Binding is less for light nuclei (these are mostly surface) but there are peaks for A in multiples of 4. (But note that the peak for 8Be is slightly lower than that for 4He. • The most stable nuclei are in the A~60 mass region • Light nuclei can gain binding energy per nucleon by fusing; heavy nuclei by fission. • The decrease in binding energy per nucleon for A>60 can be ascribed to the repulsion between the (charged) protons in the nucleus: the Coulomb energy grows in proportion to the number of possible pairs of protons in the nucleus Z(Z-1)/2 • The binding energy for massive nuclei (A>60) grows roughly as A; if the nuclear force were long range, one would expect a variation in proportion to the number of possible pairs of nucleons, i.e. as A(A-1)/2. The variation as A suggests that the force is saturated; the effect of the interaction is only felt in a neighborhood of the nucleon.
http://amdc.in2p3.fr/web/masseval.html American Journal of Physics -- June 1989 -- Volume 57, Issue 6, p. 552 Nuclear liquid drop The semi-empirical mass formula, based on the liquid drop model, considers five contributions to the binding energy (Bethe-Weizacker 1935/36) (N − Z) 2 Z 2 B = a A − a A2/3 − a − a −δ(A) vol surf sym A C A1/3 15.68 18.56 28.1 0.717
pairing term
$ −34 A−3 / 4 for even - even & δ(A) = % 0 for even - odd & −3/4 '& 34A for odd - odd
Leptodermous expansion B. Grammaticos, Ann. Phys. (NY) 139, 1 (1982)
€ € The semi-empirical mass formula, based on the liquid drop model, compared to the data # ∂B & stability valley proton % ( = 0 $ ∂N ' A= const drip line isotones # ∂B& % ( = 0 isobars $ ∂Z' N = const
neutron drip line
# ∂B & % ( = 0 isotopes $ ∂N ' Z = const
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€ Separation energies
A = 21 isobaric chain one-nucleon separation energies
two-nucleon separation energies
Using http://www.nndc.bnl.gov/chart/ find one- and two-nucleon separation energies of 8He, 11Li, 48Ni, 48Ca, and 141Ho. Discuss the result. Odd-even effect
J. Phys. G 39 093101 (2012) chemical potential
Pairing energy
N-odd B(N +1, Z)+ B(N −1, Z) Δ = − B(N, Z) n 2 B(N, Z +1)+ B(N, Z −1) Δ = − B(N, Z) p 2 Z-odd
A common phenomenon in mesoscopic systems! Example: Cooper electron pairs in superconductors…
Find the relation between the neutron pairing gap and one-neutron separation energies. Using http://www.nndc.bnl.gov/chart/ plot neutron pairing gaps for Yr (Z=70) isotopes as a function of N. I. Tanihata et al., PRL 55 (1985) 2676 Halos
For super achievers: Neutron Drip line nuclei HUGE D i f f u s e d PA IR ED
4He 5He 6He 7He 8He 9He 10He Pairing and binding
2 8
2 protons 20 8 28 neutrons known nuclei known The Grand Nuclear Landscape Nuclear The Grand 20 (finite nuclei + extended nucleonic matter) nucleonic + extended nuclei (finite 28 probably known only up to oxygen 50 known up to Z=91 82 stable nuclei 82
50 Z=118, A=294 126
neutron stars neutron terra incognita superheavy nuclei The limits: Skyrme-DFT Benchmark 2012 120 stable nuclei 288 known nuclei ~3,000 drip line two-proton drip line N=258 80 S2n = 2 MeV Z=82 SV-min two-neutron drip line N=184 Asymptotic freedom ?
110 Z=50 40 N=126 100
proton number Z=28
N=82 proton number Z=20 90 230 244 N=50 232 240 248 256 N=28 Nuclear Landscape 2012 neutron number 0 N=20 0 40 80 120 160 200 240 280 fromneutron B. Sherrill number How many protons and neutrons can be bound in a nucleus?
Literature: 5,000-12,000 Erler et al. Skyrme-DFT: 6,900±500 Nature 486, 509 (2012) syst http://massexplorer.frib.msu.edu
Neutron star, a bold explanation
A lone neutron star, as seen by NASA's Hubble Space Telescope
2 2 2/3 (N − Z) Z 3 G 2 B = avolA − asurf A − asym − aC 1/3 −δ(A) + 1/3 M A A 5 r0 A Let us consider a giant neutron-rich nucleus. We neglect Coulomb, surface, and pairing energies. Can such an object exist?
3 G 2 B = avolA − asym A + 1/3 (mn A) = 0 limiting condition 5 r0 A 3 G m2 A2/3 = 7.5MeV ⇒ A ≅ 5×1055, R ≅ 4.3 km, M ≅ 0.045M n ⊙ 5 r0
More precise calculations give M(min) of about 0.1 solar mass (M⊙). Must neutron stars have R ≅10 km, M ≅1.4 M⊙
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€ Nuclear isomers
The most stable nuclear isomer occurring in nature is 180mTa. Its half-life is at least 1015 years, markedly longer than the age of the universe. This remarkable persistence results from the fact that the excitation energy of the isomeric state is low, and both gamma de- excitation to the 180Ta ground state (which itself is radioactive by beta decay, with a half- life of only 8 hours), and direct beta decay to hafnium or tungsten are all suppressed, owing to spin mismatches. The origin of this isomer is mysterious, though it is believed to have been formed in supernovae (as are most other heavy elements). When it relaxes to its ground state, it releases a photon with an energy of 75 keV.
Discovery of a Nuclear Isomer in 65Fe with Penning Trap Mass Spectrometry http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.100.132501