Intro to Nuclear and Particle Physics (5110)

Transcription

1 Intro to Nuclear and Particle Physics (5110) March 13, 009 Nuclear Shell Model continued 3/13/009 1

2 Atomic Physics Nuclear Physics V = V r f r L r S r Tot Spin-Orbit Interaction ( ) ( ) Spin of e magnetic dipole moment, interacts with EM field (magnetic field component in e rest frame). Nucleon-nucleon strong interaction postulated to have spin-orbit term. Nucleon deep in nucleus feels no effect (symmetry), but one near surface feels net interaction with all others. Differs from atomic: f(r) to be chosen Sign chosen to match observed level splitting Next step is to apply what we know about angular momentum and deduce the form of the resulting level splitting. 3/13/009

3 V = V r f r L r S r Tot ( ) ( ) J r = L r + S r r r r r r J = L + S + L S L r S r commutes with J r, L r, S r, Jz good q. nos. are l, s, j, m j r r h L S = j j+ l l+ s s+ r r h 3 L S = j( j+ 1) l( l+ 1) 4 ( 1) ( 1) ( 1) h 1 r r l j = l L S = h 1 ( l + 1 ) j = l + Compute the energy shifts (perturbatively) = expectation values of the spin-orbit term 1 h = + = 3 Enl j l d rψ nl r f r h = = ( l + 1) ( ) ( ) 1 3 r Enl j l d rψ nl f r Total splitting: 1 = + r h ( ) ( ) 3/13/009 3 r ( ) ( ) 3 l d rψ nl r f r Key observation: Splitting grows with l can reorder the levels. As it turns out: f ( r) 1 dv ( r) = r dr

4 And it does Even-Even Nuclei J P = 0 + Odd-A Spin-Parity Assignments p and n levels fill independently paired in every level Last unpaired nucleon s j determines nuclear spin and its l determines nuclear parity (-1) l Example : C 6 protons, all paired no effect neutrons: S 1 1 P 3 1P 1 The SPSM has been coaxed to do what we need: provide a physics rationale for observed level ordering in the nucleus. Deduce consequences! j = 1/ l = 1 odd parity 1 = Other successes magnetic moments Omission odd-odd nuclei 3/13/009 4 J P

5 z r Parity? θ ( π -θ ) ( π+ φ ) the Parity y transformation P φ change r into r x r In the spherical coordinates it is to change θ π θ ϕ π + ϕ and leave the radial coordinate r unchanged. i.e. ( r, θ, ϕ) ( r, π θ, π + ϕ) 3/13/009 5

6 In describing the hydrogen atom we solve the Schrödinger equation and come up the solution which is written as ψ nlm ( r ) = ψ ( r, θ, φ) = R ( r) Y ( θ, φ) nlm where n, l, and m are quantum numbers. R nl (r) is the radial part of the wave function and Y lm ( θ, φ ) is the angular part of the wave function. Y lm ( θ, φ ) is generally referred as the spherical harmonics nl lm r In this case ψ ( r ) = ψ ( r, π θ, π + φ) = R ( r) Y ( π θ, π + φ) nlm Y lm nlm l ( π + θ, π φ) = ( 1) Y ( θ, φ) nl lm lm θ ( π + φ) y therefore if l = even then ψ nlm (r r ) is a function of even parity. x -r φ if l = odd then ψ nlm(r r ) is a function of odd parity. 3/13/009 6

7 Determination of Nuclear Spin & Parity? Experimentally: Can use nuclear magnetic resonance to determine nuclear spin (look for transitions between the different m J states, which split in energy in a magnetic field) Spin and parity can be determined from angular correlations in nuclear reactions. Example: angular correlations of the gammas from (p,γγ) reaction 61 Ni (p,γγ) 61 Cu which in normal physics notation is: p+ 61 Ni 61 Cu+γ+γ This technique can be used to probe excited states as well. 3/13/009 7

8 16 p : 8O Examples (SPSM Predictions): 1 8 Z=8, N=8: so called doubly-closed shell nucleus n : 4 ( 1S ) ( 1P ) ( ) 1/ 3/ 1P1/ 4 ( 1S ) ( 1P ) ( ) 1/ 3/ 1P1/ Even-Even nucleus: Ground State: J Π = 0 + Standard notation A Z X N 3/13/009 8

9 17 8O 9 Z=8, N=9 p : n : 4 ( 1S ) ( ) ( ) 1/ 1P3/ 1P1/ 4 ( 1S ) ( 1P ) ( 1P ) ( ) 1 1/ 3/ 1/ 1D5/ Example Odd A nucleus: unpaired neutron has L=, J=5/ Ground State: J Π = 5/ + 3/13/009 9

10 15 8O 7 Z=8, N=7 p : n : 4 ( 1S ) ( ) ( ) 1/ 1P3/ 1P1/ 4 ( 1S ) ( 1P ) ( ) 1 1/ 3/ 1P1/ Example 3 Odd A nucleus: unpaired neutron has L=1, J=1/ Ground State: J Π = 1/ 3/13/009 10

11 15 N 7 8 Z=8, N=7 p : n : 4 ( 1S ) ( ) ( ) 1 1/ 1P3/ 1P1/ 4 ( 1S ) ( 1P ) ( ) 1/ 3/ 1P1/ Example 4 Odd A nucleus: unpaired proton has L=1, J=1/ Ground State: J Π = 1/ O-15 and N-15 are known as mirror-nuclei 3/13/009 11

12 Example 5 (a bit more far out) 08 8 Pb Z=8, 16 p : n : ( 3S ) ( ) 1 1/ 1H 11/ 6 ( F ) ( ) 1 5/ 3P 1/ Even-Even nucleus: Ground State: J Π = 0 + N=16 Doubly- Closed Shell Observed Magic Numbers: Z or N =, 8, 0, 8, 50, 8, (16) 3/13/009 1

13 Example 6 (a bit more far out) 07 8 Pb Z=8, 15 N=15 p : n : ( 3S ) ( ) 1 1/ 1H11/ 6 ( F ) ( ) 1 5/ 3P 1/ Odd A nucleus: unpaired proton has L=1, J=1/ Ground State: J Π = 1/ 3/13/009 13

14 17 8O Excited States from the SPSM? 9 Z=8, N=9 p : n : 4 ( 1S ) ( ) ( ) 1/ 1P3/ 1P1/ 4 ( 1S ) ( 1P ) ( 1P ) ( ) 1 1/ 3/ 1/ S1/ 1st Excited State: promote 1D 5/ unpaired neutron to the 1S 1/ state State # (1 st excited state): J Π = 1/ + Actually correct 3/13/009 14

15 17 8O 9 Z=8, N=9 p : n : 4 ( 1S ) ( ) ( ) 1/ 1P3/ 1P1/ 4 ( 1S ) ( 1P ) ( 1P ) ( ) 1 1/ 3/ 1/ S1/ 1st Excited State: promote 1D 5/ unpaired neutron to the 1D 3/ state State # (1 st excited state): J Π = 3/ + WRONG! Should be 1/ nd Excited State? 3/13/009 15

16 17 p : 8O 9 Z=8, N=9 n : 4 ( 1S ) ( ) ( ) 1/ 1P3/ 1P1/ 4 1 ( 1S ) ( 1P ) ( 1P ) ( ) 1/ 3/ 1/ 1D5/ nd Excited State: promote 1P 1 unpaired neutron to the 1D 5/ state Unpaired neutron is now in the 1P 1/ state State # (1 st excited state): J Π = 1/ This is still a one particle state of sorts (a one neutron hole state) How do we get 1/ -??? 3/13/009 16

17 The single-particle (or hole) excited states do exist But the energy ordering is not obvious from the diagram we use to predict the magic numbers And there exist collective excited states Limitations of the SPSM 3/13/009 17

18 Excited States in mirror nuclei It appears, however, that mirror nuclei do have single-particle excited states that indeed mirror one another This too is a prediction of the Single-Particle Shell Model 3/13/009 18

19 Recap 3D spherical well 3D harmonic oscillator potential Reasonable nuclear potential Reasonable nuclear potential + spin-orbit = Single Particle Shell Model with tensor (L S) term Observed Magic Numbers: Z or N =, 8, 0, 8, 50, 8, (16) 3/13/009 19

20 Nuclear Physics? A pastiche of slightly related ideas and techniques Assessment of theoretical nuclear physics by W.S.C. Williams, an experimental nuclear physicist Models do not represent a coherent, fundamental framework, because Underlying interaction (strong force) is not well understood. Many-body effects are very difficult to handle, central to collective models of nuclear matter, but Still, it works quite well on an empirical level Choose your functions intelligently and give them enough, but not too many, free parameters and you get a phenomenology with many impressive applications Pragmatism: Use the models and empirical tools to analyze behavior of nuclei, especially radioactive decay, but also the energy-producing processes of fission and fusion 3/13/009 0