Z X with A=Z+N N neutron number 44 N A atomic/weight number 1 - PowerPoint PPT Presentation

INTRODUCTION TO NUCLEAR MODELS Daniel Kollar – Friday Physics Session NUCLEUS BASICS Z – proton number 76 A 32 Ge Z X with A=Z+N N – neutron number 44 N A – atomic/weight number 1 size: R ≅ ≅ R 0 A 0 ≅ R 5 . 1 fm R 1 . 2 fm 3 < + mass: m ( X ) Z m N m (that’s why it holds together) p n [ ] = + − ⋅ 2 binding energy: B ( Z , N ) Z m N m m ( Z , N ) c p n [ ] = + − ⋅ = 76 76 2 B ( Ge ) 36 m 44 m m ( Ge ) c 661 . 6 MeV p n 8 Fe B/A [MeV/nucleon] He U fission 6 4 fusion 2 0 0 50 100 150 200 250 A

NUCLEAR FORCES short range, spin-orbital character YUKAWA THEORY OF NUCLEAR FORCES based on exchange of π 0 , π + , and π – p + n → p + n N + N → N + N consider range of 1.4 fm p n N N ⇒ from uncertainty principle π 0 π – m c r ≈ � → m π ≈ 140 MeV n p N N p n π + Heisenberg Fermi p n n p p n e – e – n p π 0 n p ν n p violates angular n p too weak momentum conservation

NUCLEAR MODELS • total wavefunction of the nucleus is far too complicated to be useful even if it was possible to calculate it (only possible for the lightest nuclei) ⇒ we make use of models and use simple analogies Types of nuclear models Semiclassical Quantum mechanical Fermi gas Shell Independent particle Rotational Liquid drop Collective Vibrational

FERMI GAS MODEL Built on analogy between nucleus and ideal gas • particles don’t interact • particles move independently in the mean field of the nucleus Ground state → particles occupy lowest energy states allowed by the Pauli principle V V – R R – R R Fermi energy E F Fermi sea –V 0 –V 0 neutrons protons

FERMI MODEL 3 2 Vd p dn = Distribution of nucleon momentum states: ( ) 3 π 2 � E F ∫ = n dn • total number of states up to E F : – momentum → energy 0 – n p = Z ; n n = A – Z 3 = π V 4 3 r A – volume ⇒ 0 • Fermi energy: 2 2 π π − 2 2     � 9 Z � 9 A Z 3 3 = ⋅ = ⋅ E p E n     F F 2 2 4 A 4 A     2 mr 2 mr 0 0 → for Z = A – Z = A/2 1 = const ⋅ E F equal for all nuclei m ≈ 0 ≈ E F 30 MeV ( V 40 MeV )

NUCLEAR LIQUID DROP MODEL Weizsäcker formula for the binding energy ( A ≥ 30) condensation energy ∝ V = B ( A , Z ) a V ⋅ A holding nucleus together surface tension ∝ S 2 − a S ⋅ A 3 near-surface nucleons are bound less − 1 − ⋅ − ⋅ a C Z ( Z 1 ) A Coulomb potential 3 − − ⋅ − ⋅ 2 1 a A ( A 2 Z ) A asymmetry − δ  even-even pairing energy  − 1 ∆ = δ ∝ A − ∆ 0  even-odd 2  + δ  odd-odd 1 R ∝ A 3

NUCLEAR LIQUID DROP MODEL Weizsäcker formula for the mass of the nucleus ( ) ( ) = + − − m ( A , Z ) Zm A Z m B A , Z p n ( ) ( ) ( ) − 2 1 2 = + − − + + − + − − + ∆ 1 m ( A , Z ) Zm A Z m a A a A a Z Z 1 A a A 2 Z A 3 3 p n V S C A − δ  ee  for constant A ⇒ m ( A,Z ) is quadratic in Z ∆ = 0 oe   + δ oo  odd A even A − 1 m ( A,Z ) m ( A,Z ) δ ∝ A 2 oo ee β – β + , EC 76 As β – β + , EC β + β – 76 Ge 2 β – Z Z 76 Se

NUCLEAR LIQUID DROP MODEL Valley of stability ( ) ( ) ( ) 2 − 1 2 = + − − + + − + − − + ∆ 1 m ( A , Z ) Zm A Z m a A a A a Z Z 1 A a A 2 Z A 3 3 p n V S C A For fixed A the most stable Z is obtained by differentiating m( A , Z) Z = N PROTON NUMBER Z Z A 1 ≅ ⋅ Z 2 2 + 1 0 . 0075 A P O T O N R N M B E R U 3 82 50 MAGIC NUMBERS: 2, 8, 20, 28, 50, 82, 126 28 not explained by Fermi gas 20 model nor liquid drop model 8 2 2 8 20 28 50 82 136 NEUTRON NUMBER N

NUCLEAR SHELL MODEL – WHY? New model needed to explain discontinuities of several nuclear properties → binding energy → high relative abundances → low n-capture cross section → high excitation energies → … ? Fe MAGIC U 8 NUMBERS B/A [MeV/nucleon] He fission 6 4 fusion semi-empirical experimental 2 Magic number = 0 closed shell 0 50 100 150 200 250 A Magic numbers indicate similarity of nucleus to electron shells of atom, BUT still different from “Atomic magic numbers” (2, 10, 18, 36, 54, 86)

NUCLEAR SHELL MODEL AIM → Explain the magic numbers ASSUMPTION → Interactions between nucleons are neglected → Each nucleon can move independently in the nuclear potential STEPS → Find the potential well that resembles the nuclear density → Consider the spin-orbit coupling

NUCLEAR SHELL MODEL Potential well   ( ) ∑ ∑ ∑ = + + λ − H T V r ( ) v r ( ) V r ( )   Hamiltonian of a nucleus: i i ij i   ≠ i i j i , , j i central potential residual potential ⇒ λ → 0 Potential well candidates Central potential � Residual potential V ( r ) R r 1. Square Well 2. Harmonic Oscillator V 0 3. Woods-Saxon Solve Schrödinger equation Potential ( )   − 2 l l 1 � 2 d 2 M + − − = R  E V r ( )  R 0   nl nl nl 2 2 2 dr r � 2 Mr  

NUCLEAR SHELL MODEL Square well potential − ≤  V r R ( ) =  0 V r ⇒ no analytical solution 0 > 0 r R  V ( r ) R r Occupation Total 1g 18 58 2p 6 40 1f 14 34 2s 2 20 1d 10 18 1p 6 8 1s 2 2 -V 0 Closed Shell ≠ Magic Number

NUCLEAR SHELL MODEL Harmonic potential 1 ( ) = − + ω 2 2 V r V M r ⇒ analytical solution possible 0 0 2 V ( r ) r Occupation Total 1i,2g,3d,4s 56 168 1h,2f,3p 42 112 1g,2d,3s 30 70 1f,2p 20 40 1d,2s 12 20 1p 6 8 1s 2 2 -V 0 Closed Shell ≠ Magic Number

NUCLEAR SHELL MODEL Woods-Saxon potential resembles the nuclear density from scattering measurements V ( ) = − + V r 0 ⇒ no analytical solution ( ) 0 1 exp − r R a V ( r ) R r Occupation Total 1i,2g,3d,4s 56 168 1h,2f,3p 42 112 1g,2d,3s 30 70 1f,2p 20 40 1d,2s 12 20 1p 6 8 1s 2 2 -V 0 Closed Shell ≠ Magic Number

NUCLEAR SHELL MODEL Spin-orbit coupling contribution Maria Mayer ( Physical Review 78 (1950), 16 ) suggested: 1. There should be a non-central component 2. It should have a magnitude which depends on S & L 1 d ( ) ( ) ( ) ( ) ( ) = + ⋅ = V r V r V r L s V r V f r with 0 s s 0 s r dr non-central potential Woods-Saxon shape Results in energy splitting of individual levels for given J (angular momentum) j = l – ½ for l > 0 1d 3/2 ∆ E j = l +/- ½ e.g. ⇒ 1d 1d 5/2 j = l + ½

NUCLEAR SHELL MODEL Level splitting 4s 6 3d 3 3d 4s 2 4s 1 2 3d 2g 72 2g 2g 1i 11 2 3d 5 2 REMARKS 2g 92 5 3p 1i 126 1i 13 1i 2 The essential features are given 3p 1 3p 2f 2 3 3p 2 2f 5 2 by any potential of the form 2f 2f 72 1h 92 3s 4 1h 1h 82 ( ) ( ) ( ) = + ⋅ V r V r V r L σ 1h 11 2d 3s 2 3s 1 0 s 2 2d 3 2d 2 1g 2d 5 2 1g 7 2 1g 50 3 Energies of levels are parameter 2p 1g 9 2 2p 1 2p 2 dependent 1f 5 2 1f 1f 3 2p 2 28 1f 7 2 2 2s Shell model fails when dealing 20 1d 3 2s 1d 2 2s 1 2 with deformed nuclei, i.e., nuclei 1d 1d 5 2 far from magic numbers 8 1 1p 1p 1p 1 2 1p 3 2 2 1s 1s Collective models: 0 1s 1 2 plus square harmonic Woods-Saxon spin-orbit rotational, vibrational well oscillator potential coupling

Other models Close to CLOSED-SHELL nuclei well described by shell model However, most of the nuclear properties are indeed determined by nucleons outside the closed shells Collective models → treating the closed shells as inert and only dealing with the rest Models not mentioned (but used): 1. rotational model → rotations of permanently deformed nuclei 2. vibrational model → excitations within shell – multipole account 3. Nilsson model → shell model with deformed potential 4. α -particle model → α -particle clusters inside the nucleus 5. interacting boson model → considering pairs of nucleons as bosons