# On-Line RGM calculation for the α+α, α+n, and α+p, scattering phase shifts

(last update 15'th Jul. 2004.)

Chose the system ,
α+α elastic scattering ( 8Be ) ,
α+n elastic scattering ( 5He ),
α+p elastic scattering ( 5Li ) .
Give the size parameter of (0s)4 function for the alpha cluster internal w.f. , ( να = mω/2ħ).
να = fm-2
or
chose automatically the size parameter to minimize the binding energy.
Chose the central nucleon-nucleon force.
Minnesota Pot.     ( D. R. Thompson et al. Nucl. Phys. A286(1977)p53.) ,
Volkov No.1    ( A. B. Volkov, Nucl. Phys. A74(1965)p.33.) ,
Volkov No.2    ( A. B. Volkov, Nucl. Phys. A74(1965)p.33.) ,
* Give the exchange mixture parameter u in "Minnesota pot."
or majorana parameter m in "Volkov pot." if you chose either.
u   or   m = ,
others
If you chose "others", give potentail parameters (up to 3-range),
V(r) = Σi=1,3 Vi Exp[- μi r2] (Wi + Mi Pr + Bi Pb + Hi Ph ) ,
(where Pr, Pb, and Ph are the Majorana, Bartlett, and Heisenberg operator, respectively.)
Strength
V1= ,   V2= ,   V3= MeV.
Range
μ1= ,   μ2= ,   μ3= fm-2 .
Wigner, Majorana, Bartlett, and Heisenberg parameter
for 1'st range
W1= , M1= , B1= , H1= ,
for 2'nd range
W2= , M2= , B2= , H2= ,
for 3'rd range
W3= , M3= , B3= , H3= .
Put the parameter of spin-orbit force (up to 2-range),
V(r) = Σi=4,5 Vi Exp[- μi r2] (1 - mls - mls Pr) ħ -1 L⋅S ,
(where majorana parameter mls is same for i=4,5) .
Strength
V4= ,   V5= , MeV.
Range
μ4= ,   μ5= ,
Majorana parameter
mls= ,

Put 'channel radius' in R-matrix calculation,

# Note

• This code can calculate the α+α(L=0, 2, 4), α+p, and α+n(L=0, 1),
elastic scattering phase shifts in the RGM (Resonating Group Method)[1,2].

• The internal wave function of the α particle is approximated by the single (0s)4 h.o. wave function.
να (= mω/2ħ) = 0.26 fm-2 reproduce nearly the experimental charge radius of the α particle 1.47 fm.

• The cluster relative wave function inside the channel radius is approximated
by a superposition of the tempered Gaussian basis, see Eqs. (4) and (5) in Ref.,
and is then connected with the exact Coulomb function at the channel radius.
The microscopic R-method [4,5] is employed in order to obtain the scattering S-matrix.

• The basis set used to approximate the cluster relative w.f. are following,

   i             νi                                  bi
1.    0.6122448979591778E+01    0.4041451884327400E+00
2.    0.3876910717810504E+01    0.5078753097033172E+00
3.    0.2454971329932769E+01    0.6382293729798237E+00
4.    0.1554558427952545E+01    0.8020408252808562E+00
5.    0.9843910910293477E+00    0.1007897023626226E+01
6.    0.6233447406503856E+00    0.1266589403200112E+01
7.    0.3947198112999908E+00    0.1591679188144666E+01
8.    0.2499479329369778E+00    0.2000208300789485E+01
9.    0.1582742172826667E+00    0.2513592736744086E+01
10.    0.1002237848582533E+00    0.3158745238542828E+01
11.    0.6346458206376170E-01    0.3969486120866696E+01
12.    0.4018759800604615E-01    0.4988316205906549E+01
13.    0.2544794247400798E-01    0.6268644810043293E+01
14.    0.1611436881753891E-01    0.7877589577812519E+01
15.    0.1020408163264653E-01    0.9899494936614833E+01

where the νi and bi are parameters in the tempered Gaissian basis
such as Exp[-νi * r2] = Exp[-( r / bi )2] .

• As for the effective nucleon-nucleon force Vij(rij),
the Minnesota potential has a follwing form,
Vij(rij) = {U1 + 1/2*(1+Pσ) U2 + 1/2*(1-Pσ) U3} { 1/2*u + 1/2*(2 - u Pr)} ,
where u is the exchange mixture-parameter.
The Volokov force has,
Vij(rij) = (U1 + U2) ( 1 - m + m Pr)} ,
where m is the majorana parameter.
The Ui ( i=1,2,3 ) is definced as
Ui = Vi Exp[- μi r2] ,
and Pσ and Pr are the Bartlett(spin exhange) and Majorana(space exchange) operator, respectively.

• The Coulomb force is included in this calculation.

• As for the spin-orbit force, the definitions of L and S are
L = - (1/2) i ħ rij (∇j - ∇i)
S = (σi+ σj)
See Refs [6, 7].

References

. K. Wildermuth and Y.C. Tang, A Unified Theory of the Nucleus
(Vieweg, Braunschweig, 1977).

. K. Langanke, in Advances in Nuclear Physics, vol. {\bf 21},
editors J.W. Negele and E. Vogt (Plenum, New York, 1994), p. 85.

. K. Arai, P. Descouvemont, D. Baye, and W. N. Catford, Phys. Rev. C 68, 014310(2003).

. D. Baye, P.-H. Heenen, and M. Libert-Heinemann, Nucl. Phys. A291, 230 (1977).

. H. Kanada, T. Kaneko, S. Saito and Y. C. Tang, Nucl. Phys. A444, 209 (1985).

. A. Csoto, Phys. Rev. C 48, 165 (1993).

. L. Reichstein and Y. C. Tang, Nucl. Phys. A158, 529 (1970).