Three- and four-particle scattering

Transcription

1 Three- and four-particle scattering A. Deltuva Centro de Física Nuclear da Universidade de Lisboa

2 Few-particle scattering Three-particle scattering equations Three-nucleon system Three-body direct nuclear reactions Four-particle scattering equations Four-nucleon system Four-boson universal physics

3 Hamiltonian H 0 + α Three-particle system 1 v 2 3 v 1 v 3 v α 2

4 Hamiltonian H 0 + α Three-particle system 1 v 2 3 v 1 v 3 v α 2 Faddeev equations (E H 0 v α ) ψ α = v α σ Ψ = α ψ α δ ασ ψ σ

5 Alt, Grassberger, and Sandhas equations U βα = δ βα G σ δ βσ T σ G 0 U σα U 0α = G 1 + T 0 σ G 0 U σα σ T σ = v σ + v σ G 0 T σ G 0 = (E + i0 H 0 ) 1 channel states (E H 0 v α ) φ α = 0...

6 AGS equations: numerical solution U βα = δ βα G σ δ βσ T σ G 0 U σα 3 sets of Jacobi momenta momentum-space partial wave basis set of coupled 2-variable integral equations integrable singularities in kernel α q α p α Coulomb interaction: screening and renormalization

7 Proton-deuteron elastic scattering at E p = 135 MeV dσ/dω (mb/sr) 10 1 AV18 AV18 + UIX CD Bonn CD Bonn + A y (N) A y (d) A yy 0.4 A xx A xz [PRC 80, ]

8 Coulomb vs 3NF: 1 H(d,pp)n at E d = 130 MeV d 5 σ/ds dω 1 dω 2 (mb MeV -1 sr -2 ) (15 o,15 o,160 o ) (30 o,15 o,160 o ) 0.1 (20 o,15 o,160 o ) (20 o,20 o,160 o ) 0.1 (25 o,15 o,160 o ) (25 o,20 o,160 o ) (13 o,13 o,20 o ) 0.1 AV18(nd) AV18(pd) AV18+UIX(pd) (30 o,20 o,160 o ) S (MeV) (25 o,25 o,160 o ) S (MeV) S (MeV)

9 Application to 3-body nuclear reactions } p+(na) d + A n+(pa) p+(na) d + A p+n+a with A = 4 He, 10 Be, 12 C, 14 C, 16 O, 28 Si, 40 Ca, 48 Ca, 58 Ni,... Validity test of approximate nuclear reaction methods: DWBA, ADWA, CDCC,... Novel dynamic input: nonlocal potentials,...

10 CDCC test: 12 C(d, pn) 12 C & 58 Ni(d,d) 58 Ni 10 3 θ n =15 o Matsuoka (1982) Faddeev CDCC-BU 10 0 d 4 σ/dω n dω p (mb/sr 2 ) (dσ/dω)/(dσ R /dω) Exp. (8 MeV) Exp. (79.0 MeV) Faddeev CDCC-BU C(d,pn) 12 Ed =56 MeV θ p (deg) θ (deg) c.m. CDCC: A. M. Moro & F. M. Nunes [PRC 76, (2007)]

11 CDCC test: 12 C(d, p) 13 C and 12 C(d, pn) 12 C dσ / dω (mb/sr) MeV CDCC AGS ADWA 56 MeV θ (degrees) dσ / dω (mb/sr) MeV 56 MeV dσ / de 5 dσ / de E pn (MeV) CDCC AGS E pn (MeV) θ (deg) CDCC/ADWA: F. M. Nunes, N. Upadhyay [PRC 85, ]

12 Comparison with r-space FM results 1 (dσ/dω)/(dσ R /dω) AGS FM p+ 13 C p+ 13 C (dσ/dω)/(dσ R /dω) AGS FM d+ 12 C d+ 12 C 1 d+ 12 C p+ 13 C(1/2 - ) A y 0.5 dσ/dω (mb/sr) FM: R. Lazauskas [arxiv: ]

13 Nonlocal OP: transfer reactions dσ/dω (mb/sr) local OP nonlocal OP d+ 16 O p+ 17 O(5/2 + ) dσ/dω (mb/sr) E d = 14 MeV d+ 14 C p+ 15 C(1/2+) local OP nonlocal OP 1 d+ 16 O p+ 17 O(1/2 + ) 100 d+ 14 C p+ 15 C(5/2+) dσ/dω (mb/sr) E d = 36 MeV dσ/dω (mb/sr) [PRC 79, , PRC 79, ]

14 Four-particle scattering Hamiltonian H 0 + i> j v i j 1 v Wave function: Schrödinger equation Wave function components: Faddeev-Yakubovsky equations Transition operators: Alt-Grassberger-Sandhas equations

15 Symmetrized AGS equations t = v+vg 0 t G 0 = (E + iε H 0 ) 1 u j = P j G P j tg 0 u j 3+1 : P 1 = P 12 P 23 + P 13 P : P 2 = P 13 P 24 U 11 = (G 0 tg 0 ) 1 ζp 34 + ζp 34 u 1 G 0 tg 0 U 11 + u 2 G 0 tg 0 U 21 U 21 = (G 0 tg 0 ) 1 (1+ζP 34 )+(1+ζP 34 )u 1 G 0 tg 0 U 11 U 12 = (G 0 tg 0 ) 1 + ζp 34 u 1 G 0 tg 0 U 12 + u 2 G 0 tg 0 U 22 U 22 = (1+ζP 34 )u 1 G 0 tg 0 U 12 ζ = 1 (+1) for fermions (bosons) basis states partially symmetrized

16 Scattering amplitudes: E + iε E + i0 2-cluster reactions: T f i = s f i φ f U f i φ i φ j = G 0 tp j φ j Φ j = (1+P j ) φ j 3-cluster breakup/recombination: T 3i = s 3i φ 3 [(1+ζP 34 )u 1 G 0 tg 0 U 1i + u 2 G 0 tg 0 U 2i ] φ i 4-cluster breakup/recombination: T 4i = s 4i { φ 4 [1+(1+P 1 )ζp 34 ](1+P 1 )tg 0 u 1 G 0 tg 0 U 1i φ i + φ 4 (1+P 1 )(1+P 2 )tg 0 u 2 G 0 tg 0 U 2i φ i }

17 Wave function Ψ i = s i {[1+(1+P 1 )ζp 34 ](1+P 1 ) ψ 1,i +(1+P 1 )(1+P 2 ) ψ 2,i } with Faddeev-Yakubovsky components ψ j,i = δ ji φ i +G 0 tg 0 u j G 0 tg 0 U ji φ i

18 Solution of 4N AGS equations U 11 φ 1 = G 1 0 P 34P 1 φ 1 P 34 u 1 G 0 tg 0 U 11 φ 1 +u 2 G 0 tg 0 U 21 φ 1 1 lx l lx l l y 2 z y l z momentum-space partial-wave basis k x k y k z [l z ({l y [(l x S x ) j x s y ]S y }J y s z )S z ]JM,[(T x t y )T y t z ]T M T 1 k x k y k z [l z {(l x S x ) j x [l y (s y s z )S y ] j y }S z ]JM,[T x (t y t z )T y ]T M T 2 large system (up to 30000) of coupled 3-variable integral equations with integrable singularities Coulomb interaction: screening and renormalization [PRC 75, , PRL 98, ]

19 Singularities of 4N AGS equations 3 H, 3 He, or d+d bound state poles G 0 u j G 0 P j φ j s j j φ j P j E + iε E b j k2 z/2µ j treated by subtraction below 3-cluster threshold Z q p kz 2 F(k z ) dk z k0 2 k2 z + i0 =P = Z q Z q p k 2 z dk z F(k z ) k 2 0 k2 z 1 2 iπk 0F(k 0 ) kz 2 F(k z ) k0 2 dk F(k 0) z p k0 2 k2 z 1 [ 2 k 0F(k 0 ) iπ+ln (k 0 + p)(q k 0 ) ] (k 0 p)(k 0 + q)

20 p- 3 He scattering dσ/dω [mb/sr] MeV Famularo 1954 Fisher 2006 I-N3LO AV18 low-k MeV Mcdonald 1964 Fisher MeV Mcdonald Fisher 2006 George 2001 Fisher 2006 Alley 1993 A y A 0y 0.1 Daniels 2010 Daniels 2010 Alley 1993 Daniels θ c.m. [deg] θ c.m. [deg] θ c.m. [deg] AGS/HH/FY (Lisbon/Pisa/Strasbourg, PRC 84, )

21 -isobar excitation: effective 3N and 4N forces Fujita-Miyazawa higher order 3N force 4N force [PLB 660, 471]

22 n- 3 He elastic scattering dσ/dω (mb/sr) AV18 N3LO CD Bonn CD Bonn + INOY04 E n = 1 MeV E n = 1 MeV E n = 3.7 MeV E n = 3.5 MeV A y [PRC 76, ]

23 p- 3 H elastic scattering dσ/dω (mb/sr) N3LO INOY04 AV18 CD Bonn A y E p = 4.15 MeV p- 3 H max(a y ) p- 3 He 0 CD Bonn E p (MeV)

24 Charge exchange reaction 3 H(p,n) 3 He dσ/dω (mb/sr) AV18 N3LO CD Bonn CD Bonn + INOY04 E p = 2.48 MeV E p = 6 MeV A y [PRC 76, ]

25 d-d elastic scattering at E d = 3 MeV dσ/dω (mb/sr) 10 3 CD Bonn CD Bonn + T [PLB 660, 471]

26 2 H(d,p) 3 H and 2 H(d,n) 3 He dσ/dω (mb/sr) 20 d+d p+ 3 H d+d n+ 3 He E d = 1.5 MeV dσ/dω (mb/sr) AV18 N3LO CD Bonn CD Bonn + INOY04 E d = 3 MeV dσ/dω (mb/sr) 20 E d = 4 MeV 0 [PRC 81, ]

27 2 H(d,p) 3 H and 2 H(d,n) 3 He 0.2 E d = 1.5 MeV 0.2 d+d p+ 3 H d+d n+ 3 He it 11 T 21 d+d p+ 3 H d+d n+ 3 He E d = 1.5 MeV it INOY04 CD Bonn + CD Bonn N3LO AV18 E d = 3 MeV E d = 4 MeV T AV18 N3LO CD Bonn CD Bonn + INOY04 E d = 3 MeV it 11 T 21 E d = 4 MeV

28 Above breakup: additional singularities in AGS equations deuteron bound state poles t v φ d φ d v E + iε e d k 2 y/2µ y j k2 z/2µ j free resolvent G 0 1 E + iε k 2 x/2µ x j k2 y/2µ y j k2 z/2µ j

29 Above breakup: additional singularities in AGS equations deuteron bound state poles t v φ d φ d v E + iε e d k 2 y/2µ y j k2 z/2µ j free resolvent G 0 1 E + iε k 2 x/2µ x j k2 y/2µ y j k2 z/2µ j treated by complex-energy method: 1. solve for U f i (E + iε) with finite ε = ε 1,...,ε n 2. extrapolate to ε 0 for physical amplitudes U f i (E + i0) [H. Kamada et al, Prog. Theor. Phys. 109, 869L (2003)]

30 Integration with special weights accuracy & efficiency of the complex-energy method is greatly improved by a special integration Z b a f(x) x n 0 + iy 0 x n dx N j=1 f(x j )w j (n,x 0,y 0,a,b) where the quasi-singular factor is absorbed into special weights w j (n,x 0,y 0,a,b) = Z b a S j (x) x n 0 + iy 0 x n dx that may be calculated using spline functions {S j (x)} for standard Gaussian grid {x j } [PRC 86, ]

31 Extrapolation ε 0: n+ 3 H at 22.1 MeV dσ/dω (mb/sr) [ε min,ε max ]/MeV: [1.0, 2.0] [1.2, 2.0] [1.4, 2.0] [1.2, 1.8] ε = 1.4 MeV A y

32 Extrapolation ε 0: n+ 3 H at 22.1 MeV [ε min,ε max ] δ( 1 S 0 ) η( 1 S 0 ) δ( 3 P 0 ) η( 3 P 0 ) δ( 3 P 2 ) η( 3 P 2 ) [1.0, 2.0] [1.2, 2.0] [1.4, 2.0] [1.2, 1.8] [PRC 86, ]

33 n+ 3 H elastic scattering E n = 14.1 MeV dσ/dω (mb/sr) INOY04 CD Bonn Frenje Debertin dσ/dω (mb/sr) dσ/dω (mb/sr) Debertin Seagrave E n = 18.0 MeV E n = 22.1 MeV Seagrave, 21 MeV Seagrave, 23 MeV A y E n = 22.1 MeV INOY04 CD Bonn Seagrave INOY04, 14.1 MeV INOY04, 18.0 MeV

34 n+ 3 H total and breakup cross sections 2.5 σ t (b) 2.0 AV18 N3LO CD Bonn CD Bonn + INOY04 σ t (b) 2 1 Phillips Battat INOY04 CD Bonn E n (MeV) E n (MeV) 60 σ b (mb) INOY E n (MeV)

35 Recombination reaction 2 H+n+n n+ 3 H N A 2 K3 (cm 6 s -1 mol -2 ) E 3 (MeV) dρ t dt = K γ 2 ρ dρ n + K 3 ρ d ρ 2 n +...

36 p+ 3 He elastic scattering dσ/dω (mb/sr) INOY04 Murdoch E p = 21.3 MeV 0.8 Birchall 0.4 A y

37 4-boson system with resonant interactions Energy 1/a a+t(n) IVS T(n,1) T(n,2) d+a+a d T(n,0) d+d a+t(n 1) T(n 1,1) T(n 1,2)

38 Four-atom recombination at threshold Energy 1/a d+a+a a+t(n) IVS T(n,2) d T(n,0) T(n,1) 10 9 a+t(n 1) d+d T(n 1,2) κ n 10 6 T(n 1,1) n = 1 n = 2 n = 3 n = 4 a 0 n : b n = 0 a 0 n,k : B n,k = 0 [PRA 85, ] a/a n 0 a 0 n,1 /a0 n = a 0 n,2 /a0 n =

39 Atom-trimer scattering length 200 Re A n /a n dd Im A n /a n dd Energy 1/a d+a+a A n /a n dd 0 a+t(n) IVS T(n,1) T(n,2) d T(n,0) -200 a+t(n 1) d+d T(n 1,1) T(n 1,2) a n dd /a a dd n : b n = 2b d [EPL 95, 43002, PRA 85, ]

40 Summary 3/4-body Faddev/AGS equations in momentum space complex-energy method with special integration weights

41 Summary 3/4-body Faddev/AGS equations in momentum space complex-energy method with special integration weights 3N scattering 3-body nuclear reactions 4N scattering universal 4-boson physics