Short-Ranged Central and Tensor Correlations. Nuclear Many-Body Systems. Reaction Theory for Nuclei far from INT Seattle

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1 Short-Ranged Central and Tensor Correlations in Nuclear Many-Body Systems Reaction Theory for Nuclei far from INT Seattle September 6-, Hans Feldmeier, Thomas Neff, Robert Roth

2 Contents Motivation Unitary Correlation Operator Central Correlations Tensor Correlations Determination of Correlator Many-Body Calculations Interaction in Momentum-Space Momentum Distributions No-Core Shell Model Calculations Fermionic Molecular Dynamics Summary and Outlook

3 Motivation Aim describe basic properties of nuclear system in terms of a realistic nucleon-nucleon interaction Many-Body State Φ realistic interaction H meson-exchange (Bonn), phenomenological (Argonne), χ-pt reproduce scattering data and deuteron properties feature short-ranged repulsion and strong tensor force mean-field calculations (with effective interactions) Φ H eff Φ (Slater determinant Φ ) describe bulk properties (energies, radii) well realistic interactions induce short-ranged central and tensor correlations, Slater determinants cannot describe these use realistic interactions Ansatz and include correlations with unitary correlation operator C in simple many-body states (shell-model, FMD) Φ C H C Φ

4 Unitary Transformation Unitary Transformation transform eigenvalue problem H ˆΨ n = En ˆΨ n with the unitary operator C ˆΨ n = C Ψ n, C = C pre-diagonalization include typical effects common to all states into the equivalent eigenvalue problem Ĥ Ψ n = (C H C ) Ψ n = En Ψ n Nuclear System unitary correlator admixes components from outside the model space Ψ n does not project on model space the nuclear system has different scales: long range (low momenta) can be described by mean-field (Slater determinant) short-range (high momenta) cannot be described by mean-field include short-range correlations by unitary transformation

5 The Unitary Correlation Operator Two-Body Correlations two-body generator C = e ig, G = i< j Cluster Expansion g i j correlated operators  = C A C are no longer operators with definite particle number decompose correlated operator into irreducible k-body operators  =  [] +  [] + Two-Body Approximation ˆT C = ˆT [] + ˆT [], ˆV C [] = ˆV correlation range should be smaller than mean distance of nucleons Correlator C should conserve translational, rotational and Galilei invariance cluster decomposition principle should be fulfilled Spin-Isospin Dependence nuclear interaction strongly depends on spin and isospin v = v S T Π S T S,T different correlations in the respective channels g = g S T Π S T S,T correlated interaction in two-body space ˆv = ( igs T e v S T e ig ) S T Π S T S,T

6 p r Central and Tensor Correlations r p C = C Ω C r p r = { r r ( r r p) + ( p r r p = p r + p Ω ) } r r, pω = r { l r r r r l} p Ω Central Correlations g r = { } pr s(r) + s(r)p r probability density shifted out of the repulsive core Tensor Correlations g Ω = ϑ(r) { 3 (σ p Ω )(σ r) + 3 (σ r)(σ p Ω ) } tensor force admixes other angular momenta S =, T = AV8 S =, T = AV8 replacements [fm 3 ] ρ () (r) ˆρ () (r) V [MeV] - AV8 v c (r) r [fm] [fm 3 ] ρ () (r) ˆρ () (r) V [MeV] ˆρ () L= () (r)ˆρ L= (r) - AV8 v c (r) v t (r) r [fm] r [fm] r [fm]

7 Central Correlations g replacements realistic interactions have short-range repulsion probability density of nucleons in the repulsive core strongly suppressed [fm 3 ] ρ () (r) ˆρ () (r) V [MeV] AV r [fm] - v c (r) AV r [fm] Radial Shift correlator shifts nucleons out of core radial shift generated by radial momentum p r r g r { } pr s(r) + s(r)p r, pr = ( i r + ) r Correlation Function use correlation function R ± (r) instead of shift function s(r) ± = R± (r) r dξ s(ξ), R ±(r) r ± s(r) Correlated Wave function X, r c r Φ = R (r) r R (r) X, R (r)ˆr Φ

8 Centrally Correlated Interaction Kinetic Energy [] ˆt = t ˆt [] = c r (t + t)c r (t + t) r + [ p r ˆµ Ω (r) + µ ˆµ r (r) + l r ˆµ r (r) p r ( 7R + (r) 4R +(r) R 4 ] + (r) ) R +(r) 3 [MeV ] 4 4 ˆµ Ω (r) ˆµ r (r) r [fm] ˆµ r (r) = µ ( R +(r) ), [MeV] û(r) r [fm] ˆµ Ω (r) = µ ( r R + (r) ) Potential c ˆv b ˆv t ˆv r v c (R + (r)) r v b (R + (r)) l s r v t (R + (r)) s (ˆr, ˆr) V [MeV] ˆv c (r) v c (r) r [fm]

9 Tensor Correlations replacements tensor force admixes higher angular momenta [fm 3 ] S =, T = ρ () (r) ˆρ () (r) V [MeV] ˆρ () L= () (r)ˆρ L= (r) AV r [fm] - ( + ) v c (r) v t (r) AV r [fm] Perpendicular Shift correlator aligns density with spin perpendicular shift generated by s (r, p Ω ) g Ω r ϑ(r)s (r, p Ω ), p = p r + p Ω s (r, p Ω ) = 3 (σ p Ω )(σ r) + 3 (σ r)(σ p Ω ) s (r, p Ω ) (J, )J (J)J (J +, )J (J, )J 3i J(J + ) (J, )J (J +, )J 3i J(J + ) Correlated Wave function r c Ω ϕ; (J, )J = ϕ(r) (J, )J r c Ω ϕ; (J, )J = cos ( θ (J) (r) ) ϕ(r) (J, )J ± sin ( θ (J) (r) ) ϕ(r) (J±, )J θ (J) (r) = 3 J(J + )ϑ(r)

10 Tensor Correlated Interaction Kinetic Energy (J, )J c r c Ω t r c Ω c r t r (J, )J c r cω t Ω cω r c tω ( cos ( θ (J) (R + (r)) )(J )(J +) µ R + (r) (J, )J r [ p r (J, )J r ˆµ r (r) + ] ˆµ r (r) p r + ( θ (J) (R + (r)) ) + û(r) µ + sin ( θ (J) (R + (r)) )(J±)(J±+) ) R + (r) µ ( (J )(J +) ) r (J, )J c r c Ω v c c Ω c r (J, )J r v c (R + (r)) Potential (J, )J c r c Ω v b c Ω c r (J, )J c r c Ω v t c Ω c r (J, )J r v b (R + (r)) ( cos ( θ (J) (R + (r)) ) (J, )J l s (J, )J (J, )J r + sin ( θ (J) (R + (r)) ) (J±, )J l s (J±, )J ) v t (R + (r)) ( cos ( θ (J) (R + (r)) ) (J, )J s (ˆr, ˆr) (J, )J ± cos ( θ (J) (R + (r)) ) sin ( θ (J) (R + (r)) ) (J, )J s (ˆr, ˆr) (J±, )J + sin ( θ (J) (R + (r)) ) (J±, )J s (ˆr, ˆr) (J±, )J )

11 Tensor Correlated Interaction Tensor Correlated Operators Baker-Campbell-Hausdorff cω ac Ω = eig Ω ae ig Ω = e L Ω a, L Ω = [ g Ω, ] s (l, l) = 3(σ l)(σ l) (σ σ )l. s (p Ω, p Ω ) = r s (p Ω, p Ω ) + s (l, l) s (ˆr, ˆr) [ gω, p r] = i ( p r ϑ (r) + ϑ ) (r)p r s (r, p Ω ) [ gω, [ [ g Ω, p ] ] r = ϑ (r) (8 + 6 l )Π + 45 l s + 3 ] s (l, l) [ gω, [ g Ω, [ g Ω, p ] ] ] r = [ gω, Π ] = [ gω, s (ˆr, ˆr) ] = iϑ(r)[ 4 Π 8 l s + 3 s (ˆr, ˆr) ] [ gω, l s ] = iϑ(r)[ s (p Ω, p Ω ) ] [ gω, l ] = iϑ(r)[ s (p Ω, p Ω ) ] [ gω, s (l, l) ] = iϑ(r)[ 7 s (p Ω, p Ω ) ] [ gω, s (p Ω, p Ω ) ] = iϑ(r)[ (96 l + 8)Π + (36 l + 53) l s + 5 s (l, l) ] evaluate e L Ω in truncated operator space use in HF or FMD calculations

12 Determination of Correlator determination of R + (r) und ϑ(r) by variational principle φ S T trial C r C Ω HC Ω C r φ S T trial min R + (r),ϑ(r) Central Correlations Tensor Correlations.5 AV8.8 AV8. replacements R+(r) r [fm].5..5 const 4 He R + (r) r.6.4. α β γ ϑ d (r) ϑ(r) S r [fm] r [fm] correlator depends only weakly on the trial state limit the range of the correlator

13 Many-Body Calculations Many-Body Trial States Slater determinant of harmonic oscillator states 4 He = (s) 4 6 O = (s) 4 (p) 4 Ca = (s) 4 (p) (s) 4 (d) Talmi Transformation harmonic oscillator allows separation of relative and cm motion n l m χ ξ ; n l m χ ξ [] ĥ n l m χ ξ ; n l m χ ξ = n l m NLM n l m NLM n l m nlm l n l m C( n l m l ξ ξ nn jmll s tm t m l m l m s m s NLM C( χ χ s m s ) C( χ χ s m s ) ( l s C m l m s j m t m t ) ( l s C m l m s ) C( ξ ξ t m t ) j n(ls) j, t ĥ [] m n (l s) j, t )

14 Many-Body Calculations two-body densities z [fm] C r C Ω x [fm] x [fm] x [fm] ρ () S,T (r r ) S =, M S =, T = central correlator shifts density out of the repulsive core tensor correlator aligns density with spin orientation both central and tensor correlations are essential for binding energies [A MeV] He 6 O 4 Ca C r C Ω T V H 6

15 Many-Body Calculations Benchmark AV8 Binding Energy Contributions to the Binding Energy -5 4 He 4 He placements α β γ Eb [MeV] α β ref γ r rms [fm] [MeV] - T T cm V b V t Vc r rms [fm] extremely simple trial state, only one Slater determinant big cancelations between kinetic and potential energy, therefore binding energy very sensitive ref: PRC64 () 44

16 Many-Body Calculations Argonne V8 and Bonn-A Argonne V8 Eb [MeV] β γ α VMC r charge [fm] CC Exp 6 O 4 Ca Eb [MeV] α β γ r charge [fm] Exp VMC: PRC46 (99) 46 CC: PRC59 (999) 44 BHF: Prog. Part. Nucl. Phys 45 () 43 Bonn-A Eb [MeV] BHF r charge [fm] Exp 6 O 4 Ca Eb [MeV] r charge [fm] Exp simulation of threebody contributions by a three-body delta force δ(r i r k )δ(r k r j )

17 Interaction in Momentum Space klm Ĥ [] k l m = i l l M S channel d 3 x Y lm (ˆx) j l(kx) x Ĥ [] x j l (k x)y l m (ˆx) 3 S channel k Ĥ [] k, k V k [fm] replacements uncorrelated correlated, T =, ; L= - uncorrelated α - β Bonn A γ AV8 uncorrelated -3 fully correlated k [fm ] correlated unique effective potential identical to V lowk Kuo, Schwenk, nucl-th/84 k Ĥ [] k, k V k [fm] S, T =, ; L= Bonn A AV k [fm ] V lowk Cutoff Λ =.. fm

18 Bonn A Interaction in Momentum Space S : k V k replacements k [fm ] k [fm ] 3 S : k V k PSfrag 4 replacements k [fm ] 3 k [fm ] 4 5 bare potential S : k Ĥ [] k replacements k [fm ] k [fm ] 3 S : k Ĥ [] k PSfrag 4 replacements k [fm ] 3 k [fm ] 4 5 correlated interaction

19 Nucleon Momentum Distributions placements 4 He Bonn A AV8 α β γ radial VMC LDA ˆn(k)/A [fm 3 ] Bonn A 6 O 3 4 k [fm ] central+ tensor central 4 He Bonn A AV8 α β γ radial ˆn(k)/A [fm 3 ] Argonne V8 VMC LDA 6 O 3 4 k [fm ] correlations induce high-momentum components contributions of tensor correlations very big different correlator ranges relevant especially at the fermi surface

20 No-Core Shell Model MTV and ATS3M preliminary MTV - 4 He ATS3M - 4 He MTV ATS3M -5-5 Eb [MeV] ref 4 6 Eb [MeV] ref 4 He 3 a [fm ] 4 He 3 a [fm ] oscillator parameter Ω = Ma ref: PRC5 (995) 885 use OXBASH shell model code calculations from to 6 Ω possible

21 No-Core Shell Model AV8 preliminary Argonne V8-4 He α β γ ag replacements Eb [MeV] ref 4 6 Eb [MeV] Eb [MeV] He 3 a [fm ] 4 He 3 a [fm ] 4 He 3 a [fm ] oscillator parameter Ω = Ma use OXBASH shell-model code calculations from to 6 Ω possible energy benefits more from enlarged model-space for short-ranged correlator binding energy depends less on oscillator parameter for larger model spaces radii increase with enlarged model-spaces convergence should be tested with larger model-spaces

22 FMD Calculations Fermionic single Slater determinant Q = A ( q q A ) antisymmetrized A-body state Gaussian Wavepackets Molecular x q = c i exp { (x b i) } χ i ξ i a i i Dynamics Variational Principle ˆQ i d dt δ dt H ˆQ ˆQ ˆQ = Interaction Ĥ eff = Ĥ C + Ĥ correction Ĥ C = [ C r C Ω HC Ω C ] C r Ĥ C correlated interaction in two-body approximation Ĥ correction adjusted on doubly magic nuclei (compensates for missing three-body contributions of the correlated interaction and genuine three-body forces) Variation minimize Q Ĥ eff Q by variation of the parameters of the single-particle states

23 FMD Matrix Elements One-Body Matrix Elements Two-Body Matrix Elements o = ( q ) i q j ˆQ O [] ˆQ ˆQ [] O ˆQ ˆQ ˆQ = ˆQ ˆQ =: qk [] O q l olk k,l Gaussian Integrals k,l,m,n a qk, q l O [] q m, q n ao mk o nl G(x x ) = exp { (x x ) } κ ak b k, a l b l G ( κ ) 3/ ρ a m b m, a n b n = Rkm R ln exp{ } klmn α klmn + κ (α klmn + κ) α klmn = a k a m a k + a m + a l a n a, ρ l + a klmn = a mb k + a k b m n a k + a m a nb l + a l b n a l + a n, R km = a k b k a m b m

24 FMD Nuclear Chart Ti4 Sc4 Ca39 Ca4 Ca4 Ca4 Ca43 Ca44 Ca46 Ca (E E exp )/A [MeV] 8 Gaussian per single-particle state K38 K39 K4 K4 Ar36 Ar37 Ar38 Ar39 Ar4 Cl35 Cl36 Cl37 6 S3 S33 S34 S35 S36 4 Si4 P7 P3 Si6 Si7 Si8 Si9 Si3 Al4 Al5 Al6 Al7 Al8 Al9 Mg Mg3 Mg4 Mg5 Mg6 Mg7 Mg8 Na Na Na Na3 Na4 Na5 Na6 Na7 Ne8 Ne9 Ne Ne Ne Ne3 Ne4 Ne5 8 O3 O4 O5 O6 O7 O8 O9 O O O O3 O4 F6 F7 F8 F9 F F F F3 N N3 N4 N5 N6 N7 N8 N9 N N N 8 O3 O4 O5 O6 O7 O8 O9 O O O O3 O4 6 C9 C C C C3 C4 C5 C6 C7 C8 C9 C 6 N N3 N4 N5 N6 N7 N8 N9 N N N C9 C C C C3 C4 C5 C6 C7 C8 C9 C 4 B8 B9 B B B B3 B4 B5 B6 B7 Be7 Be8 Be9 Be Be Be Be3 Be4 4 B8 B9 B B B B3 B4 B5 B6 B7 Be7 Be8 Be9 Be Be Be Be3 Be4 Li5 Li6 Li7 Li8 Li9 Li Li He3 He4 He5 He6 He7 He8 H H3 He3 He4 He5 He6 He7 He8 H Li5 Li6 Li7 Li8 Li9 Li Li H3 Gaussians. per single-particle state

25 Selected Nuclei 4 He C 6 O ρ () (r) [ρ] Ne 8 Si 4 Ca spherical nuclei intrinsically deformed nuclei

26 Summary and Outlook Summary unitary correlation operator can describe short-range central and tensor correlations unitary correlator provides common low-momentum interaction correlated interaction can be used in many-body methods (HF, shell model, FMD) other observables can be correlated as well long range of tensor correlations has to be addressed M S = M S = Feldmeier, Neff, Roth, Schnack, NP A63 (998) 6 Neff, Feldmeier, NP A (accepted for publication), nucl-th/73 Outlook use correlated interaction with short-ranged tensor correlator in shell model calculations with configuration mixing use correlated interaction with long-ranged tensor correlator and three-body contributions (approximations) in simple manybody states (HF, FMD) genuine three-body forces are needed in addition for a successful description of nuclei