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Lectures 4&5 the nuclear force & the shell model 11 Nov 2004, Lecture 4&5 Nuclear Physics Lectures, Dr. Armin Reichold 1 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 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 choosing a potential L*S couplings odd and even A mirror nuclei 4.4 The collective model 11 Nov 2004, Lecture 4&5 2 4.2 Shortcomings of the SEMF 11 Nov 2004, Lecture 4&5 Nuclear Physics Lectures, Dr. Armin Reichold 3 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 (N,Z) (10,10) (6,6) (2,2) (8,8) 2*(2,2) = Be(4,4) Ea-a=94keV 4 Z 4.2 Shortcomings of the SEMF Neutron Magic Numbers (magic numbers in numbers of stable isotopes and isotones) Proton Magic Numbers • Magic Proton Numbers (stable isotopes) • Magic Neutron Numbers (stable isotones) N 11 Nov 2004, Lecture 4&5 Nuclear Physics Lectures, Dr. Armin Reichold 5 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 11 Nov 2004, Lecture 4&5 Ba Neutron separation energy in MeV Nuclear Physics Lectures, Dr. Armin Reichold 6 4.2 Shortcomings of the SEMF (abundances of elements in the solar system) Complex plot due to dynamics of creation, see lecture on nucleosynthesis Z=82 N=126 N=82 Z=50 N=50 iron mountain no A=5 or 8 7 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!] 11 Nov 2004, Lecture 4&5 Nuclear Physics Lectures, Dr. Armin Reichold 208Pb 8 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 11 Nov 2004, Lecture 4&5 Nuclear Physics Lectures, Dr. Armin Reichold 9 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 11 Nov 2004, Lecture 4&5 Nuclear Physics Lectures, Dr. Armin Reichold 10 4.3 The nuclear shell model 11 Nov 2004, Lecture 4&5 Nuclear Physics Lectures, Dr. Armin Reichold 11 4.3.1 Making a shell model (Assumptions) Assumptions: Each nucleon moves in an averaged potential Each nucleon moves in single particle orbit corresponding to 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) 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 11 Nov 2004, Lecture 4&5 Nuclear Physics Lectures, Dr. Armin Reichold 12 4.3.1 Making a shell model (without thinking, just compute) Try some potentials; motto: “Eat what you know” desired magic numbers 126 82 50 28 20 8 2 Coulomb infin. square harmonic 13 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) R ≈ Nuclear Radius d ≈ width of the edge 11 Nov 2004, Lecture 4&5 Nuclear Physics Lectures, Dr. Armin Reichold 14 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 Radial Wavefunction U(r)=R(r)*r for the finite square well (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!! 15 4.3.1 Making a shell model The “well improvement program” (with thinking) Harmonic is bad 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 16 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 1 dV (r ) W ( r ) : strongest in non symmetric environment r dr V (r ) V (r ) W (r )L gS 2 h 1 dV (r ) Dimension: L2 where: W (r ) = - VLS compensate 1/r * d/dr m c r dr 1 and VLS = VLS (E nucleon ) and V (r ) = (Woods-Saxon) r a 1 exp d with a = R 0 A 1 `3 11 Nov 2004, Lecture 4&5 and R 0 1.2 fm and d 0.75 fm = "thickness of edge" Nuclear Physics Lectures, Dr. Armin Reichold 17 4.3.1 Making a shell model (spin orbit terms) Good quantum numbers without LS term : 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 ] l, lz & s=½ , sz from operators L2, Lz, S2, Sz with Eigenvalues of l(l+1)ħ2, s(s+1)ħ2, lzħ, szħ ½[j(j+1)-l(l+1)-s(s+1)]ħ2 So potential becomes: V(r) + ½l ħ2 W(r) V(r) - ½(l+1) ħ2 W(r) 11 Nov 2004, Lecture 4&5 for j=l+½ for j=l -½ Nuclear Physics Lectures, Dr. Armin Reichold 18 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 … 11 Nov 2004, Lecture 4&5 Nuclear Physics Lectures, Dr. Armin Reichold 19 4.3.2 predictions from the shell model 11 Nov 2004, Lecture 4&5 Nuclear Physics Lectures, Dr. Armin Reichold 20 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 11 Nov 2004, Lecture 4&5 Nuclear Physics Lectures, Dr. Armin Reichold 21 4.3.2 Predictions from the shell model (nuclear parity in groundstates) Parity of a compound system (nucleus): P ( A -nucleon system) = Pintrinsic (nucleon i ) P i (nucleon i ) i A i A =1, =1, where P i = (-1)l i and Pintrinsic (nucleon i ) = 1 P ( A -nucleon system) = ( -1)l i A i =1, 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 11 Nov 2004, Lecture 4&5 Nuclear Physics Lectures, Dr. Armin Reichold 22 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 11 Nov 2004, Lecture 4&5 Nuclear Physics Lectures, Dr. Armin Reichold 23 4.3.2 Predictions from the shell model (magnetic dipole moments) Units: Nuclear Magneton mnucl mnucl eh eh = analogue to atomic Bohr Magneton mBohr = 2M p 2M e nucleons have intrinsic magnetic moment from ms spin = g s where: s mnucl ms ( proton ) = 2.7928 mnucl ms (neutron ) = -1.9130 mnucl s = 1 gsp = 2.7928 2 and gns = -1.9130 2 2 11 Nov 2004, Lecture 4&5 Nuclear Physics Lectures, Dr. Armin Reichold 24 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 mj = g j j where: mnucl j ( j 1) - l (l 1) s (s 1) j ( j 1) l (l 1) - s (s 1) g j = gs gl 2 j ( j 1) 2 j ( j 1) if j = l s and s = 1 2 mj = gl l gs 1 2 mnucl Schmidt Values if j = l - s and s = 1 mj = gl mnucl 2 1 l g l -1 s 2 2 l -1 2 25 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 11 Nov 2004, Lecture 4&5 Nuclear Physics Lectures, Dr. Armin Reichold 26 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” (± exchange) giving extra currents! nuclei are not really spherical as assumed (see collective model) 11 Nov 2004, Lecture 4&5 Nuclear Physics Lectures, Dr. Armin Reichold 27 4.3.3 Excited nuclei (simplest ones = odd-A mirror nuclei) p 7 n n 3Li = 3p + 4n 7 p 4Li = 4p + 3n MeV J MeV 7.46 5/2- 7.21 6.68 5/2- 6.73 4 3 2 4.63 7/2- 4.57 2 1 0.48 1/2- 0.43 1 0.00 3/2- 0.00 4 3 7 3Li 7 4Be 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 28 4.3.3 Excited nuclei (simple ones = even-A mirror nuclei) • Non analogues states in Na called isoscalar singlet. • Unpaired nucleons in spin-space symmetric between n and p • can not be occupied by (n,n) or (p,p), would violate Pauli principle isovector multiplet MeV 3.357 mirror p n p n n JP 4+ p 8 6 10Ne 10p+12n 12Na 22 11Mg 11p+11n 12p+10n accumulated occupancy per well 2 22 MeV 3.308 JP 4+ 1.246 2+ 0.000 0+ 7 more states without analogue in Ne or Mg 14 22 JP 4+ MeV 4.071 1.275 0.000 22 2+ 0+ 10 Ne 1.984 1.952 1.937 3+ 2+ 1+ 1.528 5+ 0.891 0.657 0.583 4+ 0+ 1+ 0.000 3+ 22 11Na 22 12Mg 29 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 11 Nov 2004, Lecture 4&5 Dipole mode: isovector only! large electric dipole moment. E0≈77MeV/A1/3 Quadrupole mode: Octupole mode: fundamental, leads E0≈32MeV/A1/3 to fission instability E0≈63MeV/A1/3 Nuclear Physics Lectures, Dr. Armin Reichold 30 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! 11 Nov 2004, Lecture 4&5 Nuclear Physics Lectures, Dr. Armin Reichold 31 4.4 The collective model (rotations) Many nucleons participate in rotations can treat them quasi classically Classical energies: E=I2/2J where Quantum mechanical: J = moment of inertia and I = angular momentum 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. 11 Nov 2004, Lecture 4&5 Nuclear Physics Lectures, Dr. Armin Reichold 32