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					Standard four-body CDCC
       calculations
               ı
        M. Rodr´guez-Gallardo


                  ı
       Centro de F´sica Nuclear
       Universidade de Lisboa




                                  Standard four-body CDCC calculations – p. 1/3
Outline
³ Motivation
  ³ Borromean nuclei
³ Discretization methods based on Hyperspherical
  Harmonics (HH)
  ³ Bin method extended to 3-body systems
³ Application to 6 He
  ³ Structure: B(Eλ)
  ³ Reactions(6 He+64 Zn@13.6MeV): Four-body
    Contiuum Discretized Coupled Channels
    (CDCC)
³ Summary and conclusions

                                       Standard four-body CDCC calculations – p. 2/3
Motivation: General scheme
                   quantum system
          ւ                             ց
   unbound states                      bound states
       ⇓                                      ⇓
continuum spectrum                  discrete spectrum
non normalizable                        normalizable
                                           strongly
      w                         w¯          bound
      weakly bound system                   system

       discretization method
                                           Standard four-body CDCC calculations – p. 3/3
Three-body discretization methods
³ HH method: The states of the system are
  expanded in Hyperspherical Harmonics

   Ψjµn (ρ, Ω) =        β Rβjn (ρ)    νι    jab νIι|jµ κι
                                                        I
                                              lx ly
                        mσ   lmSx σ|jab ν    ΥKlm (Ω)χσ x
                                                      S
                                                                       n

   Ω ≡ {α, x, y}                                 4             y              x
                                                 He
   β ≡ {K, lx , ly , l, Sx , jab }
                                                                                   n




                                                     Standard four-body CDCC calculations – p. 4/3
Three-body discretization methods
³ HH method: The states of the system are
  expanded in Hyperspherical Harmonics

   Ψjµn (ρ, Ω) =        β Rβjn (ρ)    νι    jab νIι|jµ κι
                                                        I
                                              lx ly
                        mσ   lmSx σ|jab ν    ΥKlm (Ω)χσ x
                                                      S
                                                                       n

   Ω ≡ {α, x, y}                                 4             y              x
                                                 He
   β ≡ {K, lx , ly , l, Sx , jab }
                                                                                   n




³ The hyperradial functions {Rβjn } can be
  constructed by different discretization methods
                                                     Standard four-body CDCC calculations – p. 4/3
THO method
³ Introduction: M. Rodríguez-Gallardo et al.,
  PRC 72 024007 (2005)
³ 4-body CDCC: M. Rodríguez-Gallardo et al.,
  arXiv:0710.0769, submitted to PRC




                                       Standard four-body CDCC calculations – p. 5/3
Bin method (I)
³ Continuum states can be expanded in HH as
  Ψκjµ (ρ, Ω, Ωκ ) = ββ ′ Rββ ′ j (κρ)Yβjµ (Ω)
                                               ′ ′
                                              lx ly
                    ×   m′ σ ′ l′ m′ Sx σ ′ |jµ
                                      ′
                                             ΥK ′ l′ m′ (Ωκ )
     ′
                                          √
   {β } incoming; {β} outgoing; κ = 2mε/




                                          Standard four-body CDCC calculations – p. 6/3
Bin method (I)
³ Continuum states can be expanded in HH as
  Ψκjµ (ρ, Ω, Ωκ ) = ββ ′ Rββ ′ j (κρ)Yβjµ (Ω)
                                                         ′ ′
                                                        lx ly
                         ×
                       m′ σ ′ l′ m′ Sx σ ′ |jµ
                                     ′
                                                       ΥK ′ l′ m′ (Ωκ )
    ′
                                         √
  {β } incoming; {β} outgoing; κ = 2mε/
³ Bins for each incoming channel are calcuated as
    bin                    2        κ+∆κ
   Rjβ{β ′ εav } (ρ)   = √πN        κ    dκfβ ′ j (κ)Rββ ′ j (κρ)
                             β′ j

    n−r
   fβ ′ j (κ) = e−iδβ′ j (κ) fβ ′ j (κ) = sin δβ ′ j (κ)e−iδβ′j (κ)
                              r




                                                    Standard four-body CDCC calculations – p. 6/3
Bin method (I)
³ Continuum states can be expanded in HH as
  Ψκjµ (ρ, Ω, Ωκ ) = ββ ′ Rββ ′ j (κρ)Yβjµ (Ω)
                                                          ′ ′
                                                         lx ly
                          ×
                       m′ σ ′ l′ m′ Sx σ ′ |jµ
                                     ′
                                                        ΥK ′ l′ m′ (Ωκ )
    ′
                                         √
  {β } incoming; {β} outgoing; κ = 2mε/
³ Bins for each incoming channel are calcuated as
     bin                    2        κ+∆κ
    Rjβ{β ′ εav } (ρ)   = √πN        κ    dκfβ ′ j (κ)Rββ ′ j (κρ)
                              β′ j

     n−r
    fβ ′ j (κ) = e−iδβ′ j (κ) fβ ′ j (κ) = sin δβ ′ j (κ)e−iδβ′j (κ)
                               r


 The vectorial space for 3-body case is huge!!

                                                     Standard four-body CDCC calculations – p. 6/3
Bin method (II)
 How establish a hierarchy among the states for
each energy ε and total angular momentum j?




                                         Standard four-body CDCC calculations – p. 7/3
Bin method (II)
 How establish a hierarchy among the states for
each energy ε and total angular momentum j?
 Œ Truncate in the number of oc’s β ′ as K increases




                                          Standard four-body CDCC calculations – p. 7/3
Bin method (II)
 How establish a hierarchy among the states for
each energy ε and total angular momentum j?
 Œ Truncate in the number of oc’s β ′ as K increases
  Define ec’s γ of S-matrix for each ε & truncate in
   the number of ec’s as the eigenphase decreases




                                         Standard four-body CDCC calculations – p. 7/3
Bin method (II)
 How establish a hierarchy among the states for
each energy ε and total angular momentum j?
    Œ Truncate in the number of oc’s β ′ as K increases
     Define ec’s γ of S-matrix for each ε & truncate in
      the number of ec’s as the eigenphase decreases
            4                                                       4

                                  OC        oc 1
                                            oc 2
                                            oc 3
                                                                                          EC             ec 1
                                                                                                         ec 2
                                                                                                         ec 3
                                                                                                                            j = 0+
                                            oc 4                                                         ec 4
            3                               oc 5                    3                                    ec 5

                                                                                                                        ε=3MeV
2




                                                        2
Σβ|Rβ(ρ)|




                                                        Σγ|Rγ(ρ)|


            2                                                       2




            1                                                       1




            0                                                       0
                0   10   20            30    40    50                   0   10   20            30         40           50
                              ρ (fm)                                                  ρ (fm)



                                                                                                    Standard four-body CDCC calculations – p. 7/3
6
    He Hamiltonian

              H(ρ, Ω) = T (ρ, Ω) + V (ρ, Ω)
              V = Vnα + Vnα + Vnn + Vnnα

    ³ n + α Vnα = Vc + VSO
      Vc , VSO : Woods-Saxon
    ³ GPT n + n Vnn = Vc + VSO + Vt
      Vc , Vt , VSO : Gaussian
                                    a
    ³ n + n + α: power Vpow =   [1+(r/b)c ]

 Pauli forbidden states: repulsive Vc for s-waves
U
                                              Standard four-body CDCC calculations – p. 8/3
Structure: B(E1)&B(E2)
                                             Total                                                          total
                                             oc 1                                                           ec 1
            0.25                             oc 2                     0.25                                  ec 2
                                             oc 3                                                           ec 3
                                             oc 4                                                           ec 4
                                             oc 5                                                           ec 5
             0.2                                                       0.2
dB(E1)/dε




                                                          dB(E1)/dε
            0.15                                                      0.15


             0.1                                                       0.1


            0.05                                                      0.05


              0                                                         0
                   0   2   4             6     8     10                      0   2   4             6          8           10
                               ε (MeV)                                                   ε (MeV)
             3                                                         3

                                             Total                                                         total
                                             oc 1                                                          ec 1
            2.5                              oc 2                     2.5                                  ec 2
                                             oc 3                                                          ec 3
                                             oc 4                                                          ec 4
                                             oc 5                                                          ec 5
             2                                                         2
dB(E2)/dε




                                                          dB(E2)/dε

            1.5                                                       1.5


             1                                                         1


            0.5                                                       0.5


             0                                                         0
                  0    2   4             6    8      10                    0     2   4             6         8            10
                               ε (MeV)                                                   ε (MeV)


*
                                                                                                       Standard four-body CDCC calculations – p. 9/3
Structure: B(E0)

       B(E0) ≡ C| gs|ρ2 |cont |2
            0.4                                                      0.4

                                            Total                                                          total
                                            oc 1                                                           ec 1
                                            oc 2                                                           ec 2
                                            oc 3                                                           ec 3
            0.3                             oc 4                     0.3                                   ec 4
                                            oc 5                                                           ec 5
                                                                                                           ec 14
                                                                                                           ec 15
dB(E0)/dε




                                                         dB(E0)/dε
            0.2                                                      0.2




            0.1                                                      0.1




             0                                                        0
                  0   2   4             6    8      10                     0   2   4             6           8            10
                              ε (MeV)                                                  ε (MeV)




                                                                                                     Standard four-body CDCC calculations – p. 10/3
Structure: B(E0)

       B(E0) ≡ C| gs|ρ2 |cont |2
            0.4                                                      0.4

                                            Total                                                          total
                                            oc 1                                                           ec 1
                                            oc 2                                                           ec 2
                                            oc 3                                                           ec 3
            0.3                             oc 4                     0.3                                   ec 4
                                            oc 5                                                           ec 5
                                                                                                           ec 14
                                                                                                           ec 15
dB(E0)/dε




                                                         dB(E0)/dε
            0.2                                                      0.2




            0.1                                                      0.1




             0                                                        0
                  0   2   4             6    8      10                     0   2   4             6           8            10
                              ε (MeV)                                                  ε (MeV)




 Eigenchannels with negative eigenphase,
Pauli forbidden states??

                                                                                                     Standard four-body CDCC calculations – p. 10/3
Structure: B(Eλ) vs eigenphases
                 0.4
                                                           total
                                                           ec 1
                  0.3                                      ec 2
dB(E0)/dε



                                                           ec 3
                                                           ec 4
                 0.2                                       ec 5
                                                           ec 14
                  0.1                                      ec 15


                      0
                           0   2   4             6   8                   10
                      2
   eigenphase (rad)




                       1

                      0

                      -1

                      -2
                           0   2   4             6   8                   10
                                       ε (MeV)

                                                         Standard four-body CDCC calculations – p. 11/3
Structure: B(Eλ) vs eigenphases

                0.25                                          total
                                                              ec 1
dB(E1)/dε



                       0.2                                    ec 2
                                                              ec 3
                0.15                                          ec 4
                                                              ec 5
                       0.1
                0.05
                        0
                             0   2   4             6   8                   10
                        2
    eigenphase (rad)




                       1.5

                        1

                       0.5

                        0
                             0   2   4             6   8                   10
                                         ε (MeV)

                                                           Standard four-body CDCC calculations – p. 11/3
Structure: B(Eλ) vs eigenphases
                        3
              2.5                                     total
                                                      ec 1
dB(E2)/dε



                        2                             ec 2
                                                      ec 3
              1.5                                     ec 4
                        1                             ec 5

              0.5
                        0
                            0   2   4             6           8                    10
                        4
     eigenphase (rad)




                        3

                        2

                        1

                        0
                            0   2   4             6           8                   10
                                        ε (MeV)
                                                                  Standard four-body CDCC calculations – p. 11/3
Structure: B(E1)&B(E2) with bins

                                             ec 1
                                             ec 2
            0.25                             ec 3


             0.2
dB(E1)/dε




            0.15


             0.1


            0.05


              0
                   0   2   4             6    8                     10
                               ε (MeV)


                                                    Standard four-body CDCC calculations – p. 12/3
Structure: B(E1)&B(E2) with bins
             3

                                            ec 1
                                            ec 2
            2.5                             ec 3


             2
dB(E2)/dε




            1.5


             1


            0.5


             0
                  0   2   4             6    8                     10
                              ε (MeV)

                                                   Standard four-body CDCC calculations – p. 12/3
4-body CDCC formalism

               Coupled channels system
         2
              d2      L(L+1)                         J
    − 2mr dR2 − R2               + εnj − E fLnj (R)
                 L′ −L J                 J
   + L′ n′ j ′ i      VLnj,L′ n′ j ′ (R)fL′ n′ j ′ (R) = 0
             projectile


                                         111111
                                         000000
                   x               r1
                                         000000
                                         111111
                                         111111
                                         000000
                          r2             000000
                                         111111
                  y            R         111111
                                         000000
                                         111111
                                         000000
                          r3             000000
                                         111111
                                              target




                                                  Standard four-body CDCC calculations – p. 13/3
Coupling potentials
  J                                       3
 VLnj,L′ n′ j ′ (R)   = LnjJM |           k=1 Vkt (rk )|L′ n′ j ′ JM

                                where

ΦJM (R, x, y) =
 Lnj                      µML   ψjµn (x, y) LML jµ|JM YLML (R

                       multipolar expansion

    J                                ˆˆ            L Q L′
   VLnj,L′ n′ j ′ (R) =    Q (−1)J−j LL′
                                                   0 0 0
                                 ′   ′         Q
                       × W (LL jj        , QJ)Fnj,n′ j ′ (R)

                                                        Standard four-body CDCC calculations – p. 14/3
Form factors
 Q                          Q+2j−j ′ ˆˆ′
Fnj,n′ j ′ (R)   = (−1)               j j (2Q + 1)
                  3
×       ββ ′      k=1       βk βk Nββk Nβ ′ βk
                                 ′             ′

                         ′
            lxk +Sxk +jabk −jabk −Ik
×   (−1)                              δlxk lxk δSxk Sxk
                                            ′        ′

                                                 ′
    ˆyk l′ yk lˆ lˆ ˆabk j ′ abk   lyk Q lyk
×   l ˆ k kj ˆ    ′
                            0 0 0
         ′      ′                   ′       ′
× W (lk lk lyk lyk ; Qlxk )W (jabk jabk lk lk ; QSxk )
                ′
×W (jj ′ jabk jabk ; QIk )     (sin αk )2 (cos αk )2 ρ5 dαk dρ
                                      ′
             lxk lyk  k          lxk lyk
×   Rβjn (ρ)ϕKk (αk )VQ (R, yk )ϕK ′ (αk )Rβ ′ j ′ n′ (ρ)
                                   k

W

                                                          Standard four-body CDCC calculations – p. 15/3
Energy spectrum
          8


          6


          4                                     Kmax = 8
ε (MeV)




          2                                                       1ec

          0

                   +       -       +
               0       1       2
          -2




                                       Standard four-body CDCC calculations – p. 16/3
6                   64
    He+ Zn@13.6MeV: elastic
                                            A. Di Pietro et al., PRC 69, 044613 (2004)
                                            one channel
           1                                4 ec (6,9,6) N=85


          0.8
σ/σRuth




          0.6
                                                                         εmax = 6 MeV
          0.4


          0.2


           0
                0        30   60       90           120         150         180
                                   θc.m. (deg)




                                                                              Standard four-body CDCC calculations – p. 17/3
6                   64
    He+ Zn@13.6MeV: convergence
                                            A. Di Pietro et al., PRC 69, 044613 (2004)
                                            one channel
           1                                1 ec (6,9,6) N=22
                                            2 ec (6,9,6) N=43
                                            3 ec (6,9,6) N=64
          0.8                               4 ec (6,9,6) N=85
                                            (6,9,4) (6,9,6) (3,6,6) (0,6,6) (0,0,4) N=72
σ/σRuth




                                            (6,9,4) (6,9,6) (3,6,6) (0,6,6) (0,4,4) N=76
          0.6
                                                                           εmax = 6 MeV
          0.4


          0.2


           0
                0        30   60       90            120         150          180
                                   θc.m. (deg)




                                                                                Standard four-body CDCC calculations – p. 18/3
6              64
    He+ Zn@13.6MeV: S-matrix


        0.2




         0.1
Im[S]




          0
                                        one channel
                                        1 ec
                                        2 ec
        -0.1                            3 ec
                                        4 ec
                                        (6,9,4) (6,9,6) (3,6,6) (0,6,6) (0,0,4)
                                        (6,9,4) (6,9,6) (3,6,6) (0,6,6) (0,4,4)
        -0.2
               0    0.2   0.4           0.6              0.8                  1
                                Re[S]




                                                                      Standard four-body CDCC calculations – p. 19/3
6                 64
    He+ Zn@13.6MeV: breakup
                  j=0+                          j=1-                       j=2+

      4                            4                           4
      3                            3                           3
                                                                                              EC 1
σbu




      2                            2                           2
      1                            1                           1
      0                            0                           0
      40    1 2    3 4   5 6   7   40    1 2    3 4    5 6 7   40    1 2    3 4   5 6 7
      3                            3                           3
                                                                                              EC 2
σbu




      2                            2                           2
      1                            1                           1
      0                            0                           0
      40    1 2    3 4   5 6   7   40    1 2    3 4    5 6 7   40    1 2    3 4   5 6 7
      3                            3                           3
                                                                                              EC 3
σbu




      2                            2                           2
      1                            1                           1
      0                            0                           0
      40    1 2    3 4   5 6   7   40    1 2    3 4    5 6 7   40    1 2    3 4   5 6 7
      3                            3                           3
                                                                                              EC 4
σbu




      2                            2                           2
      1                            1                           1
      0                            0                           0
        0   1 2     3 4 5 6    7     0   1 2     3 4 5 6 7       0   1 2     3 4 5 6 7
                  ε (MeV)                      ε (MeV)                     ε (MeV)

                                                                                  Standard four-body CDCC calculations – p. 20/3
Summary and conclusions
³ We have extended of the bin method for
  discretizing the continuum of three-body system,
  based on expansion in HH.
³ We have presented strucuture observables in order
  to undeestand the three-body continuum
³ We have used the bin method to perform standard
  four-body CDCC calculations.
³ The formalism has been applied to the Borromean
  nucleus 6 He scatterd by 64 Zn.
³ We have seen as standard CDCC calculations is an
  efficient procedure for the study of four-body
  reactions.
                                       Standard four-body CDCC calculations – p. 21/3
Open questions
³ Are the eigenphases ordered in the optimal way?
³ Are the eigenchannels the best option to truncate
  the basis?
³ Are eigenchannels with negative eigenphases
  Pauli forbidden states?




                                         Standard four-body CDCC calculations – p. 22/3
Collaborators
³ Depto. de Física Atómica, Molecular y Nuclear,
  Universidad de Sevilla
  J. M. Arias, J. Gómez-Camacho, and A. M. Moro
³ Livermore National Laboratory
  I. J. Thompson
³ Department of Physics, University of Surrey
  R. C. Johnson, and J. A. Tostevin




                                      Standard four-body CDCC calculations – p. 23/3

				
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