On modifications of the CSC GM code (R364n)

Transcription

1 O modfcatos of the CSC GM code (R364) Y. Satou Jue 3, 202 Abstract The code CSC GM was modfed to hadle proto ad eutro destes separately for both target ad core ucle so as to allow drect use of proto-proto (eutroeutro) ad proto-eutro profle fuctos for the elemetary ucleo-ucleo scatterg ampltudes. Some uderlg aalytc treatmets of parameters ad fuctos the CSC GM code are metoed together wth extesos ewly troduced the calculato of the phase-shft fuctos, χ CT ad χ NT. Itroducto The code CSC GM s a Fortra program to calculate cross sectos for varous reactos betwee composte partcles the framework of Glauber theory []. Ths code may readly be used for terpretato of oe-ucleo kockout reacto data take at the RIKEN faclty. I ts orgal form the code accepts a sgle profle fucto Γ, whch takes care of the ucleo-ucleo scatterg ampltudes, approprately averaged over the sosp of the system. Sce the code s to be used for the descrpto of oe-ucleo kockout processes ot oly o auclear target butalso o aproto target wheretheproflefucto descrbg the target-(valece) ucleo ampltudes ca dffer apprecably depedg o the kockedout ucleo speces (eutro or proto), t was felt to be beefcal to hadle sosp degrees of freedom explctly. Based o the above cosderato, the code was modfed to treat proto ad eutro desty dstrbutos separately whch allows drect use of the protoproto (eutro-eutro) ad proto-eutro profle fuctos avalable the lterature (see, e.g., Ref. [2]). 2 Phase-shft fuctos ad mpact parameters Wth the optcal-lmt approxmato (OLA) the projectle-target phase-shft fucto, χ PT, ca be expressed usg the core-target phase-shft fucto, χ CT, ad the ucleotarget phase-shft fucto, χ NT as follows []: e χ PT(b) ψ 0 e χ CT(b C )+χ NT (b C +s) ψ 0. () Here b refers to the mpact parameter betwee the projectle ad the target ad b C that betwee the core ad the target. The vector s s assocated wth the coordate of the ysatou/r364/csc GM SUNJI.pdf

2 valece ucleo measured from the ceter-of-mass of the core: r = (s,z). There s a relatoshp b C = b A p s, (2) where A p s the projectle mass. The phase-shft fuctos, χ CT ad χ NT, are expressed by usg the relevat uclear destes as follows: χ CT (b) = dr dr ρ C (r)ρ T (r )Γ(b+s s ), (3) χ NT (b) = drρ T (r)γ(b s). (4) Here the mpact parameter b wll be detfed, the actual calculatos, to that betwee the core ad the target, b C, ad that betwee the valece ucleo ad the target, b C +s, for χ CT ad χ NT, respectvely. The profle fucto Γ has the form Γ(b) = α 4πβ σ NNe b 2. (5) I the code the desty s expaded terms of a combato of Gaussas: ρ(r) = c e a r 2. (6) Ths allows a aalytcal evaluato of the tegrals volvg the uclear coordate. For a proto target, ρ T (r) s replaced by the delta fucto δ(r) ad the phase-shft fuctos reduce to the followg forms: χ CT (b) = drρ C (r)γ(b+s), (7) χ NT (b) = Γ(b). (8) I the above expresso the OLA s ot ay more appled to the teracto betwee the valece ucleo ad the target proto. 3 Aalytc expressos for the mpact parameters I the subroute rcs, where the reacto ad oe-ucleo removal cross sectos are calculated, the code chooses a coordate system whch the mpact parameter b betwee the projectle ad the target oly has the x-compoet b = (b,0) as show Fg.. I ths choce the absolute values for b C +s ad b C, whch are respectvely the argumets of subroutes zcht ad zchct, are gve as follows: b2 = b C +s = b3 = b C = b 2 +( Ac A p b 2 +( A p Here b = b, s = (x,x 2 ), ad A c s the core mass. ) 2 ( ) (x 2 +x2 2 )+2 Ac x b, (9) A p ) 2 (x 2 +x2 2 ) 2 ( A p 2 ) x b. (0)

3 z T y x b=(b,0) bc C } N s=(x,x2) s/ap Fgure : Vectors used the subroute rcs to calculate absolute values of the mpact parameters b C +s ad b C. 4 Aalytc expressos for the phase-shft fuctos The phase-shft fuctos χ NT ad χ CT are aalytcally evaluated the subroutes zcht ad zchct, respectvely. Here t s to be oted that sde these subroutes the coordate systems used for the tegrato over the uclear destes are re-defed so that the relevat mpact parameters, b C +s ad b C, respectvely, have oly the x-compoet. For a uclear target χ NT s expaded as follows: χ NT (b) = drρ T (r)γ(b s), () = = α 4πβ σ NN = α σ NN dxdydz c c e a r 2 α 4πβ σ NNe b s 2, (2) c π π a dxdydze a (x 2 +y 2 +z 2) e (b2 2bx+x 2 +y 2), (3) b 2 a e +a. (4) +a For tegrato over the target uclear volume followg aalytc relatoshps were used: dxe a x 2 ( 2bx+x2 b ) 2 (+a = e ) π, (5) a + dye a y 2 π y2 =, (6) a + 3

4 dze a z 2 = π a. (7) For a proto target χ NT s gve Eq.(8). For a uclear target χ CT s expaded as follows: χ CT (b) = dr dr ρ C (r)ρ T (r )Γ(b+s s ), (8) = dxdydz dx dy dz c j e a jr 2 c e a r 2 j α 4πβ σ NNe b+s s 2, (9) = α c j c 4πβ σ NN dxdydze a jr 2 dx dy dz e a r 2 j e {(b+x x ) 2 +(y y ) 2}, (20) = α c j c σ π 3 b 2 a j a a NN e j a +a j +a. (2) aj a a j j a +a j +a Itegratos over the target ad core uclear volumes were performed aalytcally usg the followg relatoshps: dx e a x 2 e (x 2 2xx 2x b) = e (x+b) 2 (a +) π, (22) a + dy e a y 2 e ( 2yy +y 2 y ) 2 (a = e +) π, (23) a + dz e a z 2 π =, (24) a dxe a jx 2 e (x2 +2bx) e x2 +2bx (a +) dye a jy 2 e y2 y 2 (a e +) = dze a jz 2 = a ( b a = exp + )2 π (a j + a a + ), (25) aj + a a + π, (26) aj + a a + π aj. (27) For a proto target χ CT s expaded as follows: χ CT (b) = drρ C (r)γ(b+s), (28) = = α 4πβ σ NN = α σ NN dxdydz c c e a r 2 α 4πβ σ NNe b+s 2, (29) c π π a dxdydze a (x 2 +y 2 +z 2) e (b2 +2bx+x 2 +y 2), (30) b 2 a a + e +a. (3) 4

5 Fgure 2: The eergy depedece of parameters for the Γ pp ad Γ p profle fuctos. The les are the gude for the eye. Red pots represet values at 70 MeV cdet eergy per ucleo obtaed by a terpolato procedure. 5 proto-proto ad proto-eutro profle fuctos The profle fuctos, Γ pn, for proto-proto (eutro-eutro) ad proto-eutro scattergs, are separately parametrzed the forms Γ pp (b) = α pp σ pp e 4πβ pp Γ p (b) = α p σ p e 4πβ p b pp, 2 (32) b p. 2 (33) A lst of parameters, σ pn, α pn, ad β pn, a eergy rage from 40 MeV to 000 MeV ca be foud Ref. [2]. They are depcted Fg. 2. I order to allow explct use of these profle fuctos desty dstrbutos for the core ad the target ucle eed to be decomposed to proto ad eutro dstrbutos as follows: ρ C = ρ p C +ρ C, (34) ρ T = ρ p T +ρ T. (35) Wth those decompostos takg to accout the phase-shft fuctos are re-formulated. The results are summarzed the followg. It s to be oted that for χ NT a dstcto betwee the proto ad eutro kockout reactos eeds to be troduced. 5

6 6 Exteto to the phase-shft fuctos, χ NT For a uclear target ad for the kockout of a valece eutro χ NT s expaded as follows: χ NT (b) = drρ p T (r)γ p(b s) drρ T (r)γ pp(b s), (36) = α p α pp σ p σ pp c p c π π a p + p a p e b 2 p a +pa p π π a + pp a e b 2 a +ppa. (37) Here (c p,ap ) ad (c,a ) are coeffcets descrbg respectvely the proto ad eutro desty dstrbutos of the target. For a uclear target ad for the kockout of a valece proto χ NT s expaded as follows: χ NT (b) = = α pp drρ p T (r)γ pp(b s) α p σ pp σ p c p c drρ T (r)γ p(b s), (38) π π a p + pp a p e b 2 p a +ppa p π π a + p a e b 2 a +pa. (39) For a proto target ad for the kockout of a valece eutro χ NT s gve by χ NT (b) = Γ p (b), (40) = α p σ p e b 2 p. (4) p For a proto target ad for the kockout of a valece proto χ NT s gve by χ NT (b) = Γ pp (b), (42) = α pp σ pp e b 2 pp. (43) pp 7 Exteto to the phase-shft fuctos, χ CT For a uclear target χ CT s expaded as follows: χ CT (b) = dr dr ρ p C (r)ρp T (r )Γ pp (b+s s ) dr dr ρ p C (r)ρ T(r )Γ p (b+s s ) dr dr ρ C (r)ρp T (r )Γ p (b+s s ) dr dr ρ C (r)ρ T (r )Γ pp (b+s s ), (44) 6

7 = j c p jc cp α pp π 3 σ pp a p jc a p pp a p e jc ap +ap jc +ap b 2 a p jc ap ppa p jc ap +ap jc +ap j j j c p jc c c jcc p c jcc α p π 3 σ p a p a p e jc a +ap jc +a a p jc α p π 3 σ p a jc a p p a e jc ap +a jc +ap α pp π 3 σ pp a jc a pp a e jc a +a jc +a b 2 a p jc a pa p jc a +ap jc +a b 2 a jc ap pa jc ap +a jc +ap b 2 a jc a ppa jc a +a jc +a. Here (c p C,ap C ) ad (c C,a C ) are coeffcets descrbg respectvely the proto ad eutro desty dstrbutos of the core. For a proto target χ CT s expaded as follows: χ CT (b) = drρ p C (r)γ pp(b+s) drρ C (r)γ p(b+s), (46) = α pp α p 8 A sample calculato σ pp σ p c p C c C π π a p pp a p C C +e π π a C p a C +e b 2 a p C +ppa p C (45) b 2 a C +pa C. (47) TheewerversooftheCSC GMcode(amedCSC GM SUNJI)takestoaccout proto ad eutro degrees of freedom a maer as descrbed Sectos 5, 6, ad 7. I the followg, put fles (csc.p ad wf.p) ad a output fle (reac.out) of CSC GM SUNJI for the calculato of the oe-eutro removal cross secto the 3 C 2 C reacto o a carbo 2 C target at 800 MeV/ucleo are show. 8. csc.p 2.d0 3.d0 2.d0! At Ap Ac 6.d0 6.d0 6.d0! Zt Zp Zc 800.d0! cdet eergy per ucleo ( MeV) ! pp profle fucto(sgma_pp,alpha_pp,beta_pp) ! p profle fucto(sgma_p,alpha_p,beta_p)! l (orbtal agular mometum) d0-8765! Mote Carlo parameters (Ns,delta,rad) 0! cod(: avalable, 0: ot avalable) 0! cod2, cod3 20.d0 0! max.ag (or mom) ad umer of pots 7

8 6! o. of Gaussas to ft the target proto desty ! coeffcet,rage ( fm^-2) ! o. of Gaussas to ft the target eutro desty ! coeffcet,rage ( fm^-2) ! o. of Gaussos used to ft the core proto desty ! coeffcet,rage ( fm^-2) ! o. of Gaussos used to ft the core eutro desty ! coeffcet,rage ( fm^-2) wf.p 70.0! tal depth of the uclear potetal ( MeV) 0.6! dffuseess ( fm).2! radus ( fm) ! eergy ege value for the s.p. state ( MeV) 0.5! j-value of the valece-ucleo orbt 0! ode umber of the valece-ucleo orbt 8.3 reac.out Projectle:mass= charge= Target:mass= charge= Eergy (MeV/)= Proj. reacto cross secto= [mb] Core reacto cross secto= [mb] N-removal cross secto= [mb] N (elastc)= [mb] N (elastc)= [mb] 8

9 Refereces [] B. Abu-Ibrahm et al., Comp. Phys. Comm. 5 (2003) 369. [2] B. Abu-Ibrahm et al., Phys. Rev. C 77 (2008)