Transcript HW 18
The Deuteron • Deuterium (atom). •The only bound state of two nucleons simplest bound state • Neither di-proton nor di-neutron are stable. Why? • Experimentally 2.224 MeV (Recoil..!). n H H • Also inverse (,n) reaction using Bremsstrahlung (Recoil…!). 1 2 •mc2 = 2.224…??…MeV Very weakly bound. • Compare n-p to n-n and p-p Charge independence of nuclear force. • Only ground state. (There is an additional virtual state). Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh). 1 The Deuteron V(r) = -V0 r < R =0 r>R • Oversimplified. HW 17 Show that V0 35 MeV. (Follow Krane Ch.4 and Problem 4.6), or similarly any other reference. • Really weakly bound. • What if the force were a bit weaker…? Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh). 2 The Deuteron Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh). 3 The Deuteron • Experiment deuteron is in triplet state = 1. • Experiment even parity. • = l + sn + sp parity = (-1)l • Adding spins of proton and neutron gives: s = 0 (antiparallel) or s = 1 (parallel). • For = 1 parallel s-state even parallel p-state odd antiparallel p-state odd parallel d-state even Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh). 4 The Deuteron • Experiment = 0.8574376 N spins are aligned…..But.? • Direct addition 0.8798038 N. • Direct addition of spin components assumes s-state (no orbital component). • Discrepancy d-state admixture. = a 0 0 + a 2 2 = a020 + a222 HW 18 In solving HW 17 you assumed an s-state. How good was that assumption? Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh). 5 The Deuteron • S-state No quadrupole moment. • Experiment +0.00288 b. HW 18 Discuss this discrepancy. • From and Q, is it really admixture? • What about other effects? • Important to know the d-state wavefunction. Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh). 6 Nuclear Force Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh). 7 Nuclear Models • Nuclear force is not yet fully understood. • No absolutely satisfying model, but models. • Specific experimental data specific model. • Model success in a certain range. • Some are: Individual particle model. (No interaction, E. states, static properties, …). Liquid drop model. (Strong force, B.E., Fission, …). Collective model. -particle model. Optical model. others ….. Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh). 8 Shell model • Electron configuration…. 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 …. • Atomic magic numbers: 2, 10, 18, 36, 54, … Common center of “external” attraction. Well understood Coulomb force. One kind of particles. Clear meaning for electron orbits. … • Nuclear magic numbers: 2, 8, 20, 28, 50, 82,126, … Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh). 9 Shell model Evidence: 1) End of radioactive series: 208Pb thorium series uranium series 206Pb actinium series 207Pb neptunium series 209Bi 2) At Z and N mn’s there are relatively large numbers of isotopes and isotones. Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh). 10 Shell model Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh). 11 Shell model Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh). 12 Shell model 3) Natural abundances. Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh). 13 Shell model NEUTRON CAPTURE CROSS SECTION 4) Neutron capture cross section. NEUTRON NUMBER Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh). 14 Shell model 5) Binding energy of the last neutron (Separation Energy). (The measured values are plotted relative to the calculations without ). Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh). 15 Shell model 6) Excited states. Pb (even-A) isotopes. Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh). 16 Shell model 7) Quadrupole moments ….. ? HW 19 Work out more examples for the above evidences. For example, take part of a plot and work on a group of relevant nuclides. Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh). 17 Shell model • Nucleons are in definite states of energy and angular momentum. • Nucleon orbit ?? Continuous scattering expected ..!! • No vacancy for scattering at low energy levels. • Potential of all other nucleons. • Infinite square well: rR 0 V rR • Harmonic oscillator: V 1 m r 2 2 2 Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh). 18 Shell model • More realistic: • Finite square well potential: V0 V 0 rR rR • Rounded well potential: V (r) V0 r R / a V 0 ~ 57 MeV 1 e • Correction for asymmetry (n-p has more possibilities than n-n or p-p) and Coulomb repulsion. Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh). 19 Shell model • Separation of variables: ( r , , ) R ( r ) ( ) ( ) R ( r ) Y l ( , ) m • For a given spherically symmetric potential V(r), the bound-state energy levels can be calculated from radial wave equation for a particular orbital angular momentum l. • Notice the important centrifugal potential. ml ms 1s 1p 1d 2s 1f 2p 1g 2d 3s 2(2l +1) 2 6 10 2 14 6 18 10 2 Total 2 8 18 20 34 40 58 68 70 Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh). 20 Shell model centrifugal potential Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh). 21 Shell model Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh). 22 Shell model ? ? ? 2(2l + 1) accounts correctly for the number of nucleons in each level. But what about magic numbers? Infinite spherical well Harmonic oscillator (R=8F) E ( 3 2 ) Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh). E nl ( 2 n l 1 2 ) 23 Shell model • So far, 2(2l + 1) accounts correctly for the number of nucleons in each level, since we already considered both orbital angular momentum, and spin, but still not for closed shells. m m l, m l , s, m s Yl s l Spherical Harmonics, Eigenfunctions of L2 and Lz. s S s s ( s 1) s Szs m s s 2 ms ms 2 Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh). ms ms s 1 2 s ms s 24 Shell model • 2, 8, 20 ok. • What about other magic numbers? • Situation does not improve with other potentials. • Something very fundamental about the single-particle interaction picture is missing in the description…..!!!!! • Spin-orbit coupling. Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh). 25 Shell model Spin-Orbit Coupling • M. G. Mayer and independently Haxel, Jensen, and Suess. • Spin-Orbit term added to the Hamiltonian: H p 2 2m V ( r ) V SO ( r ) S . L Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh). 26 Shell model S .L ( J 2 S L )/2 2 L 2 S LL antiparallel J J 2 J SL jm j ls j ( j 1) 2 jm j ls , J z jm j ls m j jm j ls , L 2 jm j ls l ( l 1) S 2 jm j ls s ( s 1) 2 jm j ls , 2 jm j ls , Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh). UL parallel ls jls jmj j l 0 ,1, 2 ,.... s 1 2 27 Shell model 2j+1 2(2x3 + 1) = 14 1f7/2 First time l=3 j Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh). 28 Shell model HW 20 L.S 1 [ j ( j 1) l ( l 1) s ( s 1)] 2 2 gap 1 ( 2 l 1) , 2 l0 2 Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh). 29