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Lectures 4&5

Lectures 4&5. the nuclear force & the shell model. 4.1 Overview. 4.2 Shortcomings of the SEMF magic numbers for N and Z spin & parity of nuclei unexplained magnetic moments of nuclei value of nuclear density values of the SEMF coefficients 4.3 The nuclear shell model

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Lectures 4&5

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  1. Lectures 4&5 the nuclear force & the shell model Nuclear Physics Lectures, Dr. Armin Reichold

  2. 4.1 Overview • 4.2 Shortcomings of the SEMF • magic numbers for N and Z • spin & parity of nuclei unexplained • magnetic moments of nuclei • value of nuclear density • values of the SEMF coefficients • 4.3 The nuclear shell model • 4.3.1 making a shell model • choosing a potential • L*S couplings • 4.3.2 predictions from the shell model • magic numbers • spins and parities of ground state nuclei • “simple” excited states in mirror nuclei • collective excitations • 4.3.3 Excited Nuclei • odd and even A mirror nuclei • 4.4 The collective model

  3. 4.2 Shortcomings of the SEMF Nuclear Physics Lectures, Dr. Armin Reichold

  4. (10,10) (N,Z) (6,6) (2,2) (8,8) 2*(2,2) = Be(4,4) Ea-a=94keV 4.2 Shortcomings of the SEMF(magic numbers in Ebind/A) • SEMF does not apply for A<20 • There are deviations from SEMF for A>20

  5. Neutron Magic Numbers Z Proton Magic Numbers N 4.2 Shortcomings of the SEMF(magic numbers in numbers of stable isotopes and isotones) • Magic Proton Numbers (stable isotopes) • Magic Neutron Numbers (stable isotones) Nuclear Physics Lectures, Dr. Armin Reichold

  6. 4.2 Shortcomings of the SEMF(magic numbers in separation energies) • Neutron separation energies • saw tooth from pairing term • step down when N goes across magic number at 82 Ba Neutron separation energy in MeV Nuclear Physics Lectures, Dr. Armin Reichold

  7. Z=82 N=126 N=82 Z=50 N=50 iron mountain 4.2 Shortcomings of the SEMF(abundances of elements in the solar system) • Complex plot due to dynamics of creation, see lecture on nucleosynthesis no A=5 or 8

  8. 4.2 Shortcomings of the SEMF(other evidence for magic numbers) • Nuclei with N=magic have abnormally small n-capture cross sections (they don’t like n’s) First excitation energy • Close to magic numbers nuclei can have “long lived” excited states (tg>O(10-6 s) called “isomers”. One speaks of “islands of isomerism” [Don’t make hollidays there!] 208Pb Nuclear Physics Lectures, Dr. Armin Reichold

  9. 4.2 Shortcomings of the SEMF(others) • spin & parity of nuclei do not fit into a drop model • magnetic moments of nuclei are incompatible with drops • value of nuclear density is unpredicted • values of the SEMF coefficients are completely empirical Nuclear Physics Lectures, Dr. Armin Reichold

  10. The nuclear shell model • How to get to a quantum mechanical model of the nucleus? • Can’t just solve the n-body problem because: • we don’t know the two body potentials • and if we did, we could not even solve a three body problem • But we can solve a two body problem! • Need simplifying assumptions Nuclear Physics Lectures, Dr. Armin Reichold

  11. 4.3 The nuclear shell model Nuclear Physics Lectures, Dr. Armin Reichold

  12. 4.3.1 Making a shell model(Assumptions) • Assumptions: • Each nucleon moves in an averaged potential • neutron see average of all nucleon-nucleon nuclear interactions • protons see same as neutrons plus proton-proton electric repulsion • the two potentials are wells of some form (nucleons are bound) • Each nucleon moves in single particle orbit corresponding to potential •  We are making a single particle shell model • Q: why does this make sense if nucleus full of nucleons and typical mean free paths of nuclear scattering projectiles = O(2fm) • A: Because nucleons are fermions and stack up. They can not loose energy in collisions since there is no state to drop into after collision • Use Schroedinger Equation to compute Energies (i.e. non-relativistic), justified by simple infinite square well Energies • Aim to get the correct magic numbers (shell closures) and be content Nuclear Physics Lectures, Dr. Armin Reichold

  13. infin. square Coulomb harmonic 4.3.1 Making a shell model (without thinking, just compute) desired magic numbers • Try some potentials; motto: “Eat what you know” 126 82 50 28 20 8 2

  14. R ≈ Nuclear Radius d ≈ width of the edge 4.3.1 Making a shell model (with thinking) • We know how potential should look like! • It must be of finite depth and … • If we have short range nucl.-nucl. potential • Average potential must be like the density • flat in the middle (you don’t know where the middle is if you are surrounded by nucleons) • steep at the edge (due to short range nucl.-nucl. potential) Nuclear Physics Lectures, Dr. Armin Reichold

  15. 4.3.1 Making a shell model (what to expect when rounding off a potential well) • Higher L solutions get larger “angular momentum barrier  • Higher L wave functions are “localised” at larger r and thus closer to “edge” • Clipping the edge (finite size and rounded) affects high L states most because they are closer to the edge then low L ones. • High L states drop in energy because • can now spill out across the “edge” • this reduces their curvature • which reduces their energy • So high L states drop when clipping and rounding the well!! Radial Wavefunction U(r)=R(r)*r for the finite square well

  16. 4.3.1 Making a shell model(with thinking) • Harmonic is bad The “well improvement program” • Even realistic well does not match magic numbers • Need more shift of high L states • Include spin-orbit coupling a’la atomic • magnetic coupling much too weak and wrong sign • Two-nucleon potential has nuclear spin orbit term • deep in nucleus it averages away • at the edge it has biggest effect • the higher L the bigger the shift

  17. Dimension: L2 compensate 1/r * d/dr 4.3.1 Making a shell model(spin orbit terms) • Q: how does the spin orbit term look like? • Spin S and orbit L are that of single nucleon in average potential • strongest in non symmetric environment  Nuclear Physics Lectures, Dr. Armin Reichold

  18. 4.3.1 Making a shell model(spin orbit terms) • Good quantum numbers without LS term : • l, lz & s=½ , sz from operators L2, Lz, S2, Sz with Eigenvalues of l(l+1)ħ2, s(s+1)ħ2, lzħ, szħ • With LS term need operators commuting with new H • J=L+S & Jz=Lz+Sz with quantum numbers j, jz, l, s • Since s=½ one gets j=l+½ or j=l-½ (l≠0) • Giving eigenvalues of LS [ LS=(L+S)2-L2-S2 ] • ½[j(j+1)-l(l+1)-s(s+1)]ħ2 • So potential becomes: • V(r) + ½l ħ2 W(r) for j=l+½ • V(r) - ½(l+1) ħ2 W(r) for j=l -½ Nuclear Physics Lectures, Dr. Armin Reichold

  19. 4.3.1 Making a shell model(fine print) • There are of course two wells with different potentials for n and p • The shape of the well depends on the size of the nucleus and this will shift energy levels as one adds more nucleons • This is too long winded for us though perfectly doable • So lets not use this model to precisely predict exact energy levels but to make magic numbers and … Nuclear Physics Lectures, Dr. Armin Reichold

  20. 4.3.2 predictions from the shell model Nuclear Physics Lectures, Dr. Armin Reichold

  21. 4.3.2 Predictions from the shell model (total nuclear “spin” in groundstates) • Total nuclear angular momentum is called nuclear spin = Jtot • Just a few empirical rules on how to add up all nucleon J’s to give Jtot of the whole nucleus • Two identical nucleons occupying same level (same n,j,l) couple their J’s to give J(pair)=0 Jtot(even-even ground states) = 0 Jtot(odd-A; i.e. one unpaired nucleon) = J(unpaired nucleon) Carefull: Need to know which level nucleon occupies. I.e. accurate shell model wanted! |Junpaired-n-Junpaired-p|<Jtot(odd-odd)< Junpaired-n+Junpaired-p there is no rule on how to combine the two unpaired J’s Nuclear Physics Lectures, Dr. Armin Reichold

  22. 4.3.2 Predictions from the shell model (nuclear parity in groundstates) • Parity of a compound system (nucleus): • P(even-even groundstates) = +1 because all levels occupied by two nucleons • P(odd-A groundstates) = P(unpaired nucleon) • No prediction for parity of odd-odd nuclei Nuclear Physics Lectures, Dr. Armin Reichold

  23. 4.3.2 Predictions from the shell model (magnetic dipole moments) • The truth: Nobody can really predict nuclear magnetic moments! • But: we should at least find out what a single particle shell model would predict because … • Nuclear magnetic moments very important in (amongst other things) Nuclear Magnetic Resonance (Imaging) NMR • Q: What is special about nuclear magnetic moments compared to atomic magnetic moments? • A: Nuclei don’t collide with each other and are shielded by electrons  Precession of magnetic moments in external B-fields (excited by RF pulses) are nearly undamped Q=108  Even smallest frequency shifts give information about chemical surroundings of magnetic moment See Minor Option on Medical & Environmental Physics Nuclear Physics Lectures, Dr. Armin Reichold

  24. 4.3.2 Predictions from the shell model (magnetic dipole moments) • Units: Nuclear Magneton mnucl • nucleons have intrinsic magnetic moment from spin Nuclear Physics Lectures, Dr. Armin Reichold

  25. Schmidt Values 4.3.2 Predictions from the shell model (magnetic dipole moments) • Angular momentum also gives magnetic moment for net-charged particle (protons only, gln=0) • Total contribution from each unpaired nucleon mj

  26. max Schmidt Value min Schmidt Value 4.3.2 Predictions from the shell model (magnetic dipole moments) • Q: So how does this compare to reality? (odd-A) • A: Can just about determine L of unpaired nucleon Nuclear Physics Lectures, Dr. Armin Reichold

  27. 4.3.2 Predictions from the shell model (magnetic dipole moments) • Predictions are “pretty bad”! Why? • intrinsic nucleon magnetic moment influenced by environment (compound nucleons) • Configuration mixing: The pairing of spins is not exact. Many configurations possible with slightly “unpaired” nucleons to give one effective unpaired nucleon • Nucleon-Nucleon interaction has “charged component” (p± exchange) giving extra currents! • nuclei are not really spherical as assumed (see collective model) Nuclear Physics Lectures, Dr. Armin Reichold

  28. MeV 7.46 6.68 4.63 0.48 0.00 J 5/2- 5/2- 7/2- 1/2- 3/2- MeV 7.21 6.73 4.57 0.43 0.00 p n p n 73Li74Be 73Li = 3p + 4n 74Li = 4p + 3n 4.3.3 Excited nuclei (simplest ones = odd-A mirror nuclei) 4 4 3 3 2 2 1 1 • Energy levels very similar  charge independence of nuclear force • Spin in ground state and first exited state is easy: One unpaired nucleon! • Predicted first excitations • Second excitation is only reconstructable not predictable

  29. isovector multiplet n p p n n p mirror 14 8 6 2 accumulated occupancy per well 2212Na 11p+11n 2211Mg 12p+10n 2210Ne 10p+12n • Non analogues states in Na called isoscalar singlet. • Unpaired nucleons in spin-space symmetric Y between n and p •  can not be occupied by (n,n) or (p,p), would violate Pauli principle 4.3.3 Excited nuclei (simple ones = even-A mirror nuclei) JP4+ 2+ 0+ JP4+ 3+2+1+ 5+ 4+0+1+ 3+ MeV4.071 1.9841.9521.937 1.528 0.8910.6570.583 0.000 MeV3.357 1.275 0.000 MeV3.308 1.246 0.000 JP4+ 2+ 0+ 7 more states without analogue in Ne or Mg 2210Ne2211Na2212Mg

  30. 4.4 The collective model(vibrations) • From liquid drop model might expect collective vibrations of nuclei • Classify them by multipolarity of mode and by: • isoscalar (n’s move with p’s) or • isovector (n’s move against p’s) Breathing or Monopole Mode: compresses nuclear matter  high excitation energy. E0≈80MeV/A1/3 Dipole mode: isovector only! large electric dipole moment. E0≈77MeV/A1/3 Quadrupole mode: fundamental, leads to fission instability E0≈63MeV/A1/3 Octupole mode: E0≈32MeV/A1/3 Nuclear Physics Lectures, Dr. Armin Reichold

  31. 4.4 The collective model(rotations) • Need non-spherical nuclei to excite rotations! • Observation: asphericity (electric quadrupole moment) largest when many nucleons far away from shell closure (150<A<190 & A>220) • How does this happen? • some non-closed shell nucleons have non spherical wavefunctions • these can distort the potential well for the complete nucleus • E(distorted nucleus) < E(undistorted nucleus) distortion will happen • Most large A distortions are prolate! Nuclear Physics Lectures, Dr. Armin Reichold

  32. 4.4 The collective model(rotations) • Many nucleons participate in rotations •  can treat them quasi classically • Classical energies: E=I2/2Jwhere • J = moment of inertia and • I = angular momentum • Quantum mechanical: • I2=j(j+1)ħ2 but: look only at even-even nuclei  j=0,2,4,6 • Fits observed even-even states well • But: most cases are more complex. Combinations of rot. & vib. & single particle excitations. Nuclear Physics Lectures, Dr. Armin Reichold

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