Nuclear Structure III: What to Do in Heavy Nuclei

Transcription

1 Nuclear Structure III: What to Do in Heavy Nuclei J. Engel University of North Carolina June 15, 2005

2 Outline 1 Hartree-Fock 2 History 3 Results 4 Collective Excitations

3 Outline 1 Hartree-Fock 2 History 3 Results 4 Collective Excitations

4 The Situation in Heavy Nuclei Above A 100, shell model is usually unworkable; need too large a valence space. Main alternative is mean-field theory and extensions. Let s begin with Hartree-Fock theory. Call the Hamiltonian H (it won t be the NN interaction itself). The Hartree-Fock ground state is the Slater determinant with the lowest expectation value H. Employ: Theorem (Thouless) Suppose φ a 1 a F 0 is a Slater determinant. The most general Slater determinant not orthogonal to φ can be written φ = exp( C mi a ma i ) φ = [1 + C mi a ma i + O(C 2 )] φ m,i m>f,i<f

5 Variational Procedure Find best Slater det. φ by minimizing H φ H φ / φ φ : H C nj = φ Ha na j φ = 0 n > F, j F (1) Write H as H = p 2 α 2m + V αβ = T ab a aa b 1 4 V ab,cda aa b a ca d, α α<β a,b a,b,c,d where T ab = a p2 2m b and V ab,cd = ab V 12 cd ab V 12 dc. Then equation (1) gives h nj T nj + k<f V jk,nk = 0 n > F, j F This will certainly be true if we can find a single particle basis in which h is diagonal, i.e. solve the Hartree-Fock equations h ab T ab + k F V ak,bk = δ ab ɛ a a, b. (2)

6 Self Consistency Note that in equation (2) the potential-energy term depends on all the occupied levels. So do the eigenvalues ɛ a, therefore, and Solutions are self-consistent To solve equations: 1 Start with a set of basis states a, b, c... and calculate the matrix elements of h according to equation (2) 2 Diagonalize h to obtain a new set of basis states a, b... 3 Repeat steps 1 and 2 until you get essentially the same basis out of step 2 as you put into step 1.

7 Coordinate Space In coordinate space, equations are 2 2m φ a(r) + dr V ( r r ) j Fφ j(r )φ j (r ) }{{} φ a(r) ρ(r ) [ ] dr V ( r r )φ j(r )φ a (r ) φ j (r) j F = ɛ a φ a (r) First potential term involves the direct (intuitive) potential U d (r) dr V ( r r )ρ(r ). Second term contains the nonlocal exchange potential U e (r, r ) j FV ( r r )φ j(r )φ j (r). Self consistency means that these potentials produce wave s.p. wave functions that in turn regenerate the same potentials.

8 Outline 1 Hartree-Fock 2 History 3 Results 4 Collective Excitations

9 3 Tried to include three-body interaction approximately as density-dependent two-body interaction, in the same way as the two-body interaction is approximately a density-dependent mean field. This gave better results and had convenient zero-range approximation. Hartree-Fock History Results Collective Excitations Brief History of Mean-Field Theory 1 Big problem early: HF doesn t work with realistic NN potentials because of hard core, which isn t reflected in Slater determinants. 2 Included hard core implicitly through effective interaction: Brueckner G matrix, the solution to Bloch-Horowitz equations for a nucleon pair in the presence of other nucleons. Still didn t work perfectly. G = V V above Fermi surface V V V V

10 4 Phenomenology successfully evolved toward zero-range density-dependent (Skyrme) interactions, with H = t 0 (1 + x 0 ˆ P σ ) δ(r 1 r 2 ) t 1 (1 + x 1 ˆ P σ ) [ ( 1 2 ) 2 δ(r 1 r 2 ) + h.c. ] +t 2 (1 + x 2 ˆ P σ ) ( 1 2 ) δ(r 1 r 2 )( 1 2 ) t 3 (1 + x 3 ˆ P σ ) δ(r 1 r 2 )ρ α ([r 1 + r 2 ]/2) +iw 0 (σ 1 + σ 2 ) ( 1 2 ) δ(r 1 r 2 )( 1 2 ), where ˆP σ = 1 + σ 1 σ 2, 2 and t i, x i, W 0, and α are adjustable parameters. Abandoning first principles leads to still better accuracy.

11 5 Convenient because exchange potential is local; easy to solve. Also, variational principal can be reformulated in terms of a local energy-density functional. Defining ρ ab = X b φ i φ i a, ρ(r) = X ρ rs,r s = X φ i(r, s) 2 i F s i F,s τ(r) = X X φ i(r, s) 2, J(r) = i φ i(r, s)[ φ i(r, s ) σ ss ] i F,s i F,s,s and you find E = R dr [ 2 2n τ t0ρ ρ3 + 1 (3t1 + 5t2)ρτ (9t1 5t2)( ρ) W0ρ J (t1 t2)j2 ] (E i ɛ iρ ii ) ρ ab = h ab ɛ a δ ab = 0, a, b i.e. the HF equations. Density dependence makes h more complicated than what you d get by just varying φ.

12 6 Shoot, we can include more correlations, get back to first principles, if we mess with the density functional via: Theorem (Hohenberg-Kohn and Kohn-Sham, vulgarized) universal functional of the density that, together with a simple one depending only on external potentials, gives the exact ground-state energy and density when minimized through Hartree-like equations. (Finding the functional is up to you!) At least two recent EFT-like approaches to constructing functional Power counting used to identify important terms; coefficients calculated from first principals (Furnstahl et al). Expansion in local-density scheme with coefficients fit to data (Dobaczewski et al). Both approaches have a way to go. In mean time we have pretty good empirical functionals, with parameters fit in nuclei near closed shells. More sophisticated version of Bethe-Weiszacker.

13 Outline 1 Hartree-Fock 2 History 3 Results 4 Collective Excitations

14 Modern Results: Shell Structure Near Neutron Drip Line H. Sagawa, Phys. Rev. C65, (2002)

15 Shell Stucture Summary From J. Dobaczewski et al., Phys. Rev. C 53, 2809 (1996).

16 Densities Near Drip Lines This and next 3 slides from J. Dobacewski, dobaczew/ria.summer.lectures/slajd45.html -3 ) Particle density (fm (n) Sn (p) R (fm) (n) 100 Zn (p)

17 Two-Neutron Separation Energies Experiment Theory

18 Deformation

19 ces ic s, as t- 12:06:49 PM DRAFT-4 RIA Theory Blue Book Hartree-Fock History Results Collective Excitations Range of Predictions for Drip Line i- l Book 6.pdf n- Figure 3. Comparison of two-neutron separation ener-

20 Outline 1 Hartree-Fock 2 History 3 Results 4 Collective Excitations

21 Collective Excited States Can do time-dependent Hartree-Fock in an external potential f(r, t) = f(r)e iωt + f (r)e iωt. TDHF equation is: i dρ ab dt = E[ρ] ρ ab Assuming small amplitude oscillations + f ab (t) ρ = ρ 0 + δρe iωt + δρ e iωt gives iωδρ mi = n>f,j F h mi ρ nj δρ nj + h mi ρ jn δρ jn + f mi Setting f = 0 gives the condition for a resonance an oscillation that persists in the limit of no forcing which corresponds to an excited stationary state (a pole in the response function) E = ω. The resulting δρ ρ tr (E), is the transition density to E. Small amplitude approximation is usually called the random phase approximation (RPA).

22 Ground-State Density, Transition Density, etc. Here, very explicitly, are the various types of densities we ve been discussing. The density operator itself in first quantization is ρ op (r) = α δ(r r op α ) The ground-state and transition densities are then ρ(r) = 0 ρ op (r) 0, ρ tr (E, r) = E ρ op (r) 0. To calculate ground-state expectation values or transition matrix elements of an operator: 0 f(r op ) 0 = drf(r)ρ(r), E f(r op ) 0 = drf(r)ρ tr (E, r). Finally, the one-body density matrix we used for the variational principle, which is more general than Slater determinants, is (in second quantization) ρ ab 0 a b a a 0.

23 RPA Collectivization of Transition Strength Full response Hartree response 132 Sn Transition strength R(E) for an operator f: R (e 2 fm 2 /MeV) E (MeV) R(E) = E f 0 2 Figure shows the effects of the residual N N interaction that is, its effects beyond the mean-field approximation on the isovector dipole strength (f = e p z p). This and next 3 slides from N. Paar, paar/rrpa.html

24 ÓÐÙØ ÓÒ Ó Ø Hartree-Fock History Results Collective Excitations More Isovector Dipole in RPA ÁÎ ÔÓÐ ØÖ Ò Ø Ò ËÒ ÓØÓÔ 8 Strength Distribution 132 Sn MeV Transition Densities neutrons protons E=14.04 MeV R [e 2 fm 2 ] MeV r 2 δρ [fm -1 ] E=11.71 MeV 7.60 MeV E=7.60 MeV E [MeV] r [fm]

25 Pygmy Resonances Near the Drip Line? ³ ² O ÁÎ ÔÓÐ ØÖ Ò Ø ÁÎ ÔÓÐ ØÖ Ò Ø Ò ÓÜÝ Ò ÓØÓÔ ÚÓÐÙØ ÓÒ Ó Ø 0.2 R[e 2 fm 2 ] O 18 O O self-consistent RQRPA no dynamical pairing E[MeV]

26 Quadrupole Phonons Near the Drip Line ÉÊÈ Ó Ð Ö Ò ÓÚ ØÓÖ R [e 2 fm 4 ] isoscalar full pairing 22 isovector no dynamical pairing O no pairing quadrupole phonon 2.95 MeV R [e 2 fm 4 ] ÕÙ ÖÙÔÓÐ Ü Ø Ø ÓÒ È Ö Ò «Ø ÓÒ Ø ÐÓÛ¹ÐÝ Ò Ì ¼ ¾ Ø Ø Transition Density 120 R [e 2 fm 4 ] O r 2 δρ[fm -1 ] full pairing neutrons no dynamical protons pairing no pairing E=2.95 MeV E [MeV] E [MeV] E [MeV] r [fm]

27 Smalll Amplitude TDHF: Dipole Resonance in 8 Be This slide and next courtesy of T. Nakatsukasa

28 Surface Octupole Vibration in 16 O

29 Beyond Mean-Field Theory Project deformed Slater determinants onto states with good angular momentum and/or mix Slater determinants: One-dimensional energy surface, now different for each state in the rotational band. Ψ = dq g(q) φ(q) The coordinate q can be a deformation parameter like β or an orientation angle θ (for projection). Mean Field 16 O From M. Bender and P.-H. Heenen, Nuclear Physics A713, 390 (2003)

30 Challenges and the Road Ahead Better understanding of how much physics can be subsumed into a Kohn-Sham density functional (i.e. into mean-field equations) Development of methods that include the things mean-field theory cannot Better connection with the bare NN interaction Quantitative predictions in neutron-rich nuclei, before there s a RIA (important for nucleosynthesis even after RIA)

31 Next... Double-Beta Decay and Nuclear Structure