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Structure of nuclei. 3224 Nuclear and Particle Physics Ruben Saakyan UCL. Fermi gas model. Assumptions.
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Structure of nuclei 3224 Nuclear and Particle Physics Ruben Saakyan UCL
Fermi gas model. Assumptions • The potential that an individual nucleon feels is the superposition of the potentials of other nucleons. This potential has the shape of a sphere of radius R=R0A1/3 fm, equivalent to a 3-D square potential well with radius R • Nucleons move freely (like gas) inside the nucleus, i.e. inside the sphere of radius R. • Nucleons fill energy levels in the well up to the “Fermi energy” EF • Potential wells for protons and neutrons can be different • If the Fermi energy were different for protons and neutrons, the nucleus would undergo b-decay into an energetically more favourable state • Generally stable heavy nuclei have a surplus of neutrons • Therefore the well for the neutron gas has to be deeper than for the proton gas • Protons are therefore on average less strongly bound than neutrons (Coulomb repulsion) • 2 protons/2 neutrons per energy level, since spins can be
Fermi momentum and Fermi energy • The number of possible states available to a nucleon inside a volume V and a momentum region dp is • In the nuclear ground state all states up to a maximum momentum, the Fermi momentum pF, will be occupied. Integration leads to the following number of states n. Since every state can contain two fermions, the number of protons Z and neutrons N are also given: The nuclear volume V is given as from electron scattering
Fermi momentum and Fermi energy • Assuming the depths of the neutron and proton wells are the same and Z = N = A/2, the Fermi momentum • The energy of the highest occupied state, the Fermi energy is
Fermi gas model. Potential • The difference between the Fermi energy and the top of the potential well is the binding energy B’ = 7-8 MeV/nucleon that we already know from the liquid drop model • The depth of the potential well V0 is to a good extent independent of the mass number A:
Nuclear Models • The liquid drop model allows reasonably good descriptions of the binding energy. It also gives a qualitative explanation for spontaneous fission. • The Fermi gas model, assuming a simple 3D well potential (different for protons and neutrons) explained the terms in SEMF that were not derived from the liquid drop model. • Nucleons can move freely inside the nucleus. This agrees with the idea that they experience an overall effective potential created by the sum of the other nucleons • There are things which the Fermi gas model can not explain. This will lead us to the Shell Model.
Basics • The Shell Model is based very closely on the ideas from atomic physics: orbital structure of atomic electrons • Atomic energy levels n = 1, 2, 3,… In nuclear physics we are not dealing with the same simple Coulomb potential: radial node quantum number n • Atomic Physics: for any n there are energy-degenerate levels with orbital angular momentum l= 0,1,2,…,(n-1) • For any l there are (2l+1) sub-states with different values of the projection of l along any chosen axis ml = -l, -l+1,…,0,1,…,l-1,l – magnetic quantum number Due to rotational symmetry of Coulomb potential these sub-states will be degenerate in energy
Basics • Since electrons have spin-1/2, each of the states above can be occupied by 2 electrons with , corresponding to the spin-projection number ms=1/2. Again both states will have the same energy. • Summarizing, any energy eigenstate in, say, H2 atom has quantum numbers (n, l, ml , ms ) and for any n there will be nd degenerate states
Basics • This degeneracy can be broken if there is a preferred direction in space (magnetic field). Recall spin-orbit coupling and fine structure. • Going beyond H2 atom one has to introduce electron-electron Coulomb interaction. • This introduces splitting to any level n according to l. The degeneracies in ml and ms are unchanged. • If a shell or sub-shell is filled, then In this case Pauli principle implies L = S = 0 and J = L + S = 0 Such atoms (with paired off electrons) are chemically inert Z = 2, 10, 18, 36, 54
Shell Model – Energy Levels Observed magic numbers 2 8 20 28 50 82 126 …
Spins in the Shell Model • Shell model can be used to make predictions about the spins of ground states • A filled sub-shell must have J=0 • This means that, since magic number nuclides have closed sub-shells, the contribution to the nuclear spin from protons/neutrons with magic number must be zero • Hence doubly magic nuclei are predicted to have zero nuclear spin (observed experimentally)
Spins in the Shell Model • All even-even nuclei have zero nuclear spin • Pairing hypothesis: For ground state nuclei, pairs of n and p in a given sub-shell always couple to give a combined angular momentum of zero, even when the sub-shell is not filled. • Last neutron/proton determines the net nuclear spin. • In odd-A there is only one unpaired nucleon. Net spin can be determined precisely • In even-A odd-Z/odd-N nuclides we have an unpaired p and an unpaired n. Hence the nuclear spin will lie in the range |jp-jn| to (jp+jn)
Parities in the Shell Model • The parity of a single-particle quantum state depends exclusively on l with P = (-1)l • P = Pi . A pair of particles with the same l will always have P = +1 • From pairing hypothesis we have: • Pnucleus = Plast_p Plast_n • The parity of any nuclide (including odd-odd) can be predicted (confirmed by experiment)
Magnetic moments in the Shell Model • m = gj j mN, mN – nuclear magneton, gj – Lande g-factor • For odd-odd nuclei we have to consider an unpaired n and an unpaired p • For even-odd nuclei one has to “only” find out orbital and intrinsic components of magnetic moment of the single unpaired nucleon
Magnetic moments in the Shell Model We need to combine gs s and gl l
Magnetic moments in the Shell Model Since gl = 1 for p and gl = 0 for n, gs +5.6 for p and gs -3.8 for n For a given j the measured moments lie between j = l -1/2 and j = l+1/2 but beyond that the model does not predict the moments accurately Unlike spin and parities the Shell Model does not predict magnetic moments very well
Excited states in the Shell Model • First one or two excited states can be predicted relatively easily • Consider 178O protons: (1s1/2)2 (1p3/2)4 (1p1/2)2 neutrons: (1s1/2)2 (1p3/2)4 (1p1/2)2 (1d5/2)1 • 3 possibilities for 1st excited state • One of the 1p1/2 protons to 1d5/2, giving (1p1/2)-1(1d5/2)1 • One of the 1p1/2 neutrons to 1d5/2, giving (1p1/2)-1(1d5/2)2 • 1d5/2 neutron to next level, 2s1/2 or 1d3/2 giving (2s1/2)1or(1d3/2)1 • The 3d possibility corresponds to the smallest energy shift and therefore it is favourable
Excited states in the Shell Model • Comparing the above predictions with experimental results it was found that the expected excited states do exist but not always in precisely the order anticipated • Higher excited states calculation is much more complicated • Collective model is an attempt to bring together shell and liquid drop models • Recent encouraging developments in nuclear calculations due to progress in computing power
b-decay. Fermi theory • W, Z, quarks were not known. Theory based on general principles and analogy with QED • Fermi’s Second Golden Rule • - transition rate, |M| - matrix element, n(E) – density of states (phase space determined by the decay’s kinematics)
b-decay. Fermi theory • g – dimensionless coupling constant, O - five basic classes of Lorentz invariant interaction operators • scalar S, pseudo-scalar P, vector V, axial-vector A, tensor T • The main difference is the effect on the spin states of the particles • Fermi guessed that O should be of vector type (EM interaction transmitted by photon with spin-1)
Fermi coupling constant • If we do not consider particle spins matrix element can be thought in terms of a classical weak interaction potential, like the Yukawa potential • Point-like interaction. Matrix element in this case is just a constant M = GF/V • GF – Fermi coupling constant • Can be applied to any weak process provided the energy is not too great • Extracted from muon decay GF = 90 eV fm3 • Often quoted as
b-decay. Electron momentum distribution Fermi screening factors F(Z, Ee) Spectra shifted for b+ w.r.t. b-! Possible changes of nuclear spins are not taken into account If the change is > 1, the decay is suppressed
Kurie plots and the neutrino mass • Studying b-spectrum around the end point can be used to measure mne • Kurie plots are the most obvious Qb F(Z, Ee) and constants are here
3H spectrum and the neutrino mass • 3H most suitable isotope • Low Qb, Qb E0 = 18.6 keV • Simple atom World’s best result (Mainz, Troitsk) mne < 2.2 eV/c2 Future experiment: KATRIN Sensitivity: mne ~ 0.2 eV/c2 Probably the lowest possible limit for this technique
Katrin detector transportation from Deggendorf to Karlsruhe (400km away) but had to make a detour of…9000 km
