Quantum Theory of Many-Particle Systems, Phys. 540

Transcription

1 Quantum Theory of Many-Particle Systems, Phys. 540 IPM? Atoms? Nuclei: more now Other questions about last class? Assignment for next week Wednesday ---> Comments?

2 Nuclear shell structure Ground-state spins and parity of odd nuclei provide further evidence of magic numbers Character of magic numbers may change far from stability (hot) Odd-N even Z T z =/ T z =9/ Odd-N even odd Z T z =0 A. Ozawa et al., Phys. Rev. Lett. 84, 5493 (000) T z =5 N=0 may disappear and N=6 may appear

3 Empirical potential Analogy to atoms suggests finding a sp potential shells + IPM Difference(s) with atoms? Properties of empirical potential overall? size? shape? Consider nuclear charge density Frois & Papanicolas, Ann. Rev. Nucl. Part. Sci. 37, 33 (987)

4 Nuclear density distribution Central density (A/Z* charge density) about the same for nuclei heavier than 6 O, corresponding to 0.6 nucleons/fm 3 Important quantity Shape roughly represented by ch(r) = 0 + exp r c z c z.07a 3 fm 0.55fm Potential similar shape

5 BM U = Vf(r)+V s Empirical potential s r 0 r d dr f(r) Central part roughly follows shape of density Woods-Saxon form Depth radius + neutrons - protons diffuseness f(r) = V = R = r 0 A /3 + exp r R a 5 ± 33 with a =0.67 fm N A Z r 0 =.7 fm MeV

6 Analytically solvable alternative Woods-Saxon (WS) generates finite number of bound states IPM: fill lowest levels nuclear shells magic numbers reasonably approximated by 3D harmonic oscillator 0 U HO (r) = m r V 0 0 A=00 0 H 0 = p m + U HO(r) V(r) (MeV)!0!0!30!40 Eigenstates in spherical basis!50! r (fm) H HO n m m s = (n ) V 0 n m m s

7 Filling of oscillator shells # of quanta N n Harmonic oscillator N =n + # of particles magic # parity

8 Need for another type of sp potential 949 Mayer and Jensen suggest the need of a spin-orbit term Requires a coupled basis n( s)jm j = m m s n m m s ( m sm s jm j ) Use s = (j s ) to show that these are eigenstates s n( s)jm j = j(j + ) ( + ) ( + ) n( s)jm j For j = + eigenvalue while for j = ( + ) so SO splits these levels! and more so with larger

9 Inclusion of SO potential and magic numbers Sign of SO? s V s r 0 V s = 0.44V r Consequence for d dr f(r) 0f 7 0g 9 0h 0i 3 Noticeably shifted Correct magic numbers!

10 08 Pb for example Empirical potential & sp energies Ĥ 0 a 08 Pb g.s. = + E( 08 Pb g.s. ) a 08 Pb g.s. A+: sp energies A-: E A+ n E A 0 directly from experiment Ĥ 0 a 08 Pb g.s. = E( 08 Pb g.s. ) a 08 Pb g.s. also directly from E A 0 E A n Shell filling for nuclei near stability follows empirical potential

11 Comparison with experiment Now how to explain this potential

12 Closed-shells and angular momentum Atoms: consider one closed shell (argument the same for more) n m = m s =,n m = m s =,...n m = m s =,n m = m s = Expect? Example: He (s) = { s s ) s s )} = (s) ; L =0S =0 Consider nuclear closed shell 0 = n( )jm j = j, n( )jm j = j,..., n( )m j = j

13 Angular momentum and second quantization z-component of total angular momentum Action on single closed shell Also Ĵ z = = n jm n n jm So total angular momentum Closed shell atoms j m n jm j z n j m a n jm a n ma n jm a n jm Ĵ z n j; m = j, j +,..., j = = m j m= j j m ma n jm a n jm n j; m = j, j +,..., = j m n j; m = j, j +,..., j = 0 n j; m = j, j +,..., j Ĵ ± n j; m = j, j +,..., j =0 J =0 L =0 S =0

14 Nucleon-nucleon interaction Shell structure in nuclei and lots more to be explained on the basis of how nucleons interact with each other in free space QCD Lattice calculations Effective field theory Exchange of lowest bosonic states Phenomenology Realistic NN interactions: describe NN scattering data up to pion production threshold plus deuteron properties Note: extra energy scale from confinement of nucleons

15 Nuclear masses near stability Data Each A most stable Nuclear Matter 9 M(N,Z) = E(N,Z) B(N,Z) c = Nm n + Zm p c N,Z pair Where fission? Where fusion? B/A (MeV) A

16 Smooth curve volume surface symmetry Coulomb Nuclear Matter B = b vol A b surf A /3 b sym b vol = 5.56 MeV b surf = 7.3 MeV b sym = MeV R c =.4A /3 fm (N Z) Great interest in limit: N=Z; no Coulomb; A Two most important numbers in Nuclear Physics B A 6 MeV fm 3 A 3 5 Z e R c

17 Saturation problem of nuclear matter Given V NN predict correct minimum of E/A in nuclear matter as a function of density inside empirical box 5 Energy/A (MeV) HJ 7% BJ 6.6% 6.% RSC 6.5% AV4 Par 5.8% HM C B 5% Par A B AV4 Sch UNG 4.4% 5 Describe the infinite system of neutrons properties of neutron stars k F (fm )

18 Isospin Shell closures for N and Z the same!! Also m n c m p c MeV vs MeV So strong interaction Hamiltonian (QCD) invariant for p n But weak and electromagnetic interactions are not Strong interaction dominates consequences Notation (for now) p n adds proton adds neutron Anticommutation relations {p,p } =, {n,n } =, All others 0

19 Z proton & N neutron state Exchange all p with n Isospin... Z;... N = p p...p Z n n...n N 0 ˆT + = p n and vice versa Expect ˆT = n p [ĤS, ˆT ± ]=0 Consider ˆT 3 = [ ˆT +, ˆT ]= = will also commute with p n n p p p, n n, H S = n p p n p p n n

20 Check Then operators obey the same algebra as Isospin so spectrum identical and simultaneously diagonal! proton neutron For this doublet and [ ˆT 3, ˆT ± ]=± ˆT ± ˆT = ˆT + + ˆT ˆT ˆT 3 = i rm s p = rm sm t = rm s n = rm s m t = ˆT + ˆT J x,j y,j z States with total isospin constructed as for angular momentum Ĥ S, ˆT, ˆT 3 T rm s m t = ( + ) rm sm t T 3 rm s m t = m t rm s m t

21 Two-particle states and interactions Pauli principle has important effect on possible states Free particles plane waves Eigenstates of notation (isospin) Use box normalization (should be familiar) Nucleons Electrons, 3 He atoms T = p m Bosons with zero spin ( 4 He atoms) p s = m s t = m t pm s m t p s = m s pm s p Use successive basis transformations for two-nucleon states to survey angular momentum restrictions Total spin & isospin; CM and relative momentum; orbital angular momentum relative motion; total angular momentum

22 Start with Antisymmetric two-nucleon states p m s m t ; p m s m t = { p m s m t p m s m t p m s m t p m s m t } = SMS TM T {( m s m s SM S )( m t m t TM T ) p p SM S TM T ) ( m s m s SM S )( m t m t TM T ) p p SM S TM T )} then P = p + p p = (p p ) and use p = p = plm L LM L ˆp = plm L Y LML ( ˆp) LM L LM L plm L LM L p = plm L ( ) L Y LML ( ˆp) LM L LM L Y LML ( p) =Y LML ( p, p + ) =( ) L Y LML ( ˆp) as well as ( m s m s SM S )=( ) + S ( m s m s SM S ) ( m t m t TM T )=( ) + T ( m t m t TM T )

23 Summarize Antisymmetry constraints for two nucleons p m s m t ; p m s m t = ( m s m s SM S )( m t m t TM T ) Y LML ( ˆp) SM S TM T LM L = ( ) L+S+T P p LM L SM S TM T ) ( m s m s SM S )( m t m t TM T ) Y LML ( ˆp) L + S + T SM S TM T LM L JM J (L M L SM S JM J ) ( ) L+S+T P p (LS)JM J TM T ) must be odd! Notation T=0 T= 3 S 3 D P 3 D... S 0 3 P 0 3 P 3 P 3 F D

24 Remove isospin L + S even! Two electrons and two spinless bosons p m s ; p m s = ( m s m s SM S ) Y LML ( ˆp) SM S LM L +( ) L+S P p LM L SM S ) Two spinless bosons p ; p = LM L Y LML ( ˆp) +( ) L P p LML ) L even!

25 Nuclei Different shells only Clebsch-Gordan constraint Uncoupled states in the same shell jm,jm = a jm a jm 0 Coupling Only even total angular momentum With isospin jj,jm = (j mjm JM) jm,jm = (j m jm J M) jm,jm mm mm = ( ) j J (jmjm JM)( ) jm,jm mm = ( ) J (j mjm JM) jm,jm mm = ( ) J jj,jm jj,jm,t M T = (j mjm J M)( m t m t T M T ) jmm t,jm m t mm m t m t = ( ) J+T + jj,jm,t M T J+T odd!

26 40 Ca + two nucleons Spectrum

27 Two fermions outside closed shells Atoms: two different orbits usual Clebsch-Gordan coupling and no other restrictions Two electrons in the same orbit Coupling m m s, m m s = a m m s a m m s 0,LM L,SM S = ( m m LM L )( m s m s S M S ) m m s, m m s L + S even # of states notation (p) m m m s m s = ( ) L+S,LM L,SM S Carbon L=0 S=0 S L= S= 9 3 P g.s. L= S=0 5 D

28 Why S= g.s.? D below S? Carbon atom