HAUSER-FESHBACH CROSS-SECTION CALCULATIONS FOR ELASTIC AND INELASTIC SCATTERING OF ALPHA PARTICLES PROGRAM CORA

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RAPORT No 865/PL

HAUSER-FESHBACH CROSS-SECTION

CALCULATIONS FOR ELASTIC AND INELASTIC

SCATTERING OF ALPHA PARTICLES

PROGRAM CORA

A.HARTMAN» M.SIEMASZKO. W. ZIPPER

HAUSEE-PESHBACH CROSS-SECTION CALCULATIONS

FOR ELASTIC AND IHELASTIC SCATTERING

OF ALPHA PARTICLES . PROGRAM СОВА

OBLICZENIA PRZEKROJU CZYNNEGO NA ELASTYCZNE

I NIEELASTYCZNE ROZPRASZANIE CZĄSTEK ALFA

NA GRUNCIE FORMALIZMU HAUSERA-PESHBACHA, PROGRAM CORA

ВЫЧИСЛЕНИЕ ДИФФЕРЕНЦИАЛЬНЫХ СЕЧЕНИЙ

ДЛЯ УПРУГОГО И НЕУПРУГОГО РАССЕЯНИЯ АЛЬФА ЧАСТИЦ

НА БАЗЕ ФОРМАЛИЗМА ХАУЗЕРА-ФЕШЕШ

ПРОГРАММА СОКА

A. Hartman М. Siemaszko W. Zipper

Institute of Physics, Sileeian University, Katowice

Poland

and

Institute of Nuclear Physics, Cracow, Poland

C R A C O W

II

-The program CORA «as prepared on the basis of Hauser

and Feshbaoh compound reaction formal lent. It allows the

differential cross-section distributions for the elastio

and inelastic scattering of alpha particles / via

com-pound nucleus state/ to be calculated. The transmission

coefficients are calculated on the basis of a four

para-meter optical model. The search prooedure / for ę and

G* parameters / is also included. , '

1

-The program CORA ia written in ALGOL 1204- for the

QDBA 1204 computer.

Section 2 gives the basic formalism of compound

reaction oroaa-seotion calculations.

In section 3 the simplified block scheme of the

program is presented and the data input is shown* The

optical model procedure comes from reference Л

Section 4 gives the data description*

Section 5 and б contain the results of the test

run and listing of the program, respectively.

The energy averaged differential cross-section for

compound reactions can be calculated following Hauser

1/

and Feschbach » by the use of optical model

transmis-sion coefficients*

г

-spins or the nucleus and incoming particle, respectively!

S is the channel spin» which is given by

I Is the orbital angular momentum of the Incoming

particle; Л

is the channel parity, defined by

The compound nucleus etete has the spin 7 » according

to the relation

The final state la denoted Ъу cXn',l

,Vfi%K) . Therefore

the energy averaged differential cross-section is

given by

*Z(lJlJ;sL} Z d ' J t ' J - , s ' L ) { - ! ) * ' * ' ,

«here Цц is the wave number in the entrance channel;

Г(. and IV are the transmission coefficients, which

are connected with phase shifts by the relation

r

i*)=i-ie

l

. /5/

3

-P

(cos6) is the Legendre polynomial of order L • The

Z coefficient» are defined aa foliowa

«here W ie the ВасаЬ coefficient. The Z. coefficients

are equal zero, «ben the sum L

Hi

L is odd, becauae

of vanlahixs of the Clebah-Gordan coefficient (L^OOlLO).

The aain difficulty connected with the application

of the formtla A / ie evaluation of the term

This sun involves all open exit channels into which the

compound nucleus can decay* Eberhard et al. " proposed

the calculation of this sum on the basis of the atatia

-tical nodel

'. This procedure gives the following result

3 1 Л

4

-is a correction factor ^' , approzlznated by W ^ j * ! ;A-->

Here A

is the masa of the emitted particle and Ares

is the mass of the corresponding residual nucleus»

Substituting /8/ intc /4/ we obtain

where

5

-1. W.Hawer and H.7eshbach, Pbys.Bev. §2 /1952/ 366,

H.Feahbaeh, in Nuclear Speetrosoopy, fart В

/Academic Press, Hew Xdrk, 1960/ p. 625*

2. P.A.Moldauer, Phye.Bev. Д22 /1961/ 968,

Phya.Lettera 8 /1964/ 249,

Phye.Lettere 19. /1967/ 1047.

Phys.Bev. 121 /4968/ 1164,

ibye.Hev. 122 /1969/ 1841.

3. J.M.Blatt and L.C.Biedenharn, Bev.M0d.fb7a. 24 /1952/

258.

4. W. you Witach, P. von Brent ano, Т.Мауег-Kuckuk and

A.Hichter, Huol.Pbars. 80 /1966/ 394.

5* K.A.Eberbard, P. von Brentano, ll.BOhring and

R.O.Stephen, Nucl.Pnye. A125 /1969/ 673.

6

-Ъ. Data Irrout лпл Ргоегав Scheme.

ТВ Ai, Z l , A t , Z t , Elab, S e x , J , CQN3, СШ4

СОЯ4=1

tain, tmax, dt

ТВ delgd, erusigd

I

wcc

optparami, optparamf M

ВО, BOIttX, DRO, SIGMA, BISUiMX, DSIGU

CALL OPTICAL MODEL РВОСБШГОЕ

L

8 -JNTSt N И nrparaa ' хш ^ ^

CALL SEABCH HIOCKIDURK

^ 1

CALL COMPOOTD EROCEDUBB

DSIGM

9

-. Input Pata Description-.

гЛ grid procedure is ueed

com

-2 search procedure ie ueed

3 one run for f and <r

1 transnission coefficients are calculated

О0И2

2 transmission coefficients are read from tape

С

i:

1 final state parity is odd

COHJ

-2 final state parity is even

-0 regular step of the angles

COB*

irregular step of the angles

nn - number of parameters used in the search

proce-dure / nns 1 or 2 /

nrparam- the parameters, which are searched / if

cxparam«1, then СГ

if nrparaa=2

then P /

Ai - mass

of the incident particle

Zi - charge -»

At - mass

of the target nucleus

Zt - charge J

10

-Вех - excitation energy of tbe final state

/ In OH eyatem / [lleV]

j - epin of the final at&be

tain - »<r>5»»T -j p value of the angle for which

the fl^gfi *т* dietribtttione are

teaz

— calculated

dt - atep of the angle

И - number of angles for which the experimental

croae-eeotion waa aeaaured

ТЕГА - table of angles / TBTA=TBTA[iJ, where

i=1,2, ... ,H

dalgd - table of experimental differential

cross-aeotion data / in CM ayatea /

erdsigd - table of errors of the experimental data

optparami - optical aodel parametera: U(, Wj, Г[ ,(3i

for the entrance channel

optparamf - optical model parametera t Uf, Wp, Tf,Qf

for the exit channel

1 1

-rep - table of real parte of n

coefficients for

the entrance channel

isp - table of imaginary parts of ^ coefficients

for the entrance channel

Rsp - table of real parts of n

coefficients for

the exit channel

lap - table of imaginary parts of n

coefficients

for the exit channel

WCC - "width correlation factor"

HO - initial value of the p parameter

EOUAX - final value of the 0 parameter

Ш 0 - step of p in the grid procedure

SIGMA - initial value of the 0* parameter

SIGMAMAX - final value of the 0" parameter

DSIGMA. - step of СГ in the grid procedure

12

-Description ot the keyв» key down

XL 2 must be used during the search

Kb 10 graph «111 be printed Kb 11 experimental and

cal-culated oross-eeotione «111 be printed

1 3

The following input data were used in the test run

of the GOBI programs

5,1. *

14

-0.095,0.09,0.048,0.03,0.032,0.059,0.041,

0.046,0.044,0.045,0.085,0.095,0.1,

0.H,0.08,0.085,0.085,0.072,0.08,0.07,

0.08,0.115,0.122,0.13,0.11,0.11,0.101,

50.875,10.501,1.699,0.505,

50.875,10.501,1.699,0.505,

1750,3.1,

The program requires one unit of magnetic drum

sto-rage. With the above input data the test run takes about

25 minutes of ODRA. 1204 computer time. It depends

CORA-7.3 PROGRAM

IHBLASTIC SCATTERING OF ALPHA PARTICLES WITH HAUSER-FESHBACH FORMALISM

REACTION* 2 8 S i ( a l p h a , a l p h a * ) 2 8 S i BINGHAM EXPERIMENTAL DAXA

INCIDENT PARTICLE ENERGY E» 2 4 . 0 0 0 MEV (LAB) ( FINAL STATE ENERGY E e z - 6 . 2 7 0 MEV (CM) •£

SPIN AND PARITY J« 3 ( + ) *

OPTICAL MODEL PARAMETERS

ENTRANCE CHANNEL № 5 0 . 8 7 5 W-10.501 R-1.699 A- . 5 0 5

COMPODHD NUCLEUS PARAMETERS RO-.750,03 SIGMA» 3..00

THETA DCS THEGR DOS EZ? ERR DCS EZP

20

-Zl,Stf 0OH1,С0Н2,С0ИЗ,С0К4, J,lmax,1,хш,H,mi1 j A±,At»Elab,tmlnttmai,dt,Eex,EO,RCMAI,DRO,SIGBSAt SIQ11AMAI,DSIOII, WCCtrOE J

fttn»flv paranC 1 s2 ] , optparami, optparaaf [ 114 J f lap»rep,

2 5 -q«=2 • 5/0c*k><param[2 ]) xfTCC; pomo». 5/(param[1 ]»q>aram[1 ] ) ;

aten 2 ntit41 2xlmax

1&Ю «tfłti 1 ]щЦЛ Злах doeaioe+RA[l,L+2]

дай i;

B[lH-2>=dcex(-i)tJ;

дай Ł;

a2esin(a)t

»=arctan(ganmaxa2/8qrt (i-ganmaxganmaxa2xa2) ) 4a;

a2eein(a) j

2 7 -space(pp2); print(«o»)t space(pp1-pp2-i)| print(«x»); and dlaV; dlaWł l i n e d ) ; format (' itMMJuCELSQmi. 123456,12»); print(onleq); line(2); wykres; cLrukAkey(12) print ( * tnnmiiimmiimii AUSIII3SIOIM3OEPPICBiraS») l i n e (2)t print ( * ?i • • • »t« ч «HE *

p r i n t (* ??• * "If * * " 'HP1 TETA* "If" * " — Д iETA1 -if --- •

l i n e d ) ;

2 9

-read(optparaaf);

read Омах)»

eat Input О);

- 30

fMeaat(*111«11i*)t

print (• ?i • • i f i • • • • i tIHOIDEHaViPARgICŁEuEMEHGY"E- ' , Blab, «tJMMET«(LAB) •);

print (* ?•» ••• tPHAi^jS IAIEXIEHEBGYI • >«• t ««Bex» *,

format(*11»); print («?>« ••••••••• t t i . t i . f ••itSPH^AHDuPARIHutJ»', J) i £ C0N3-2 print (•(+)») print(«(-)»); l i n e ( 2 ) ; print (' ?••«•»• • •• I I •OPTICAI^ODBlMPARAMEgERS») t lined); format (* ?!»«»«•••ttiEHTRAKCBuCHAiniElM^U«111.111 W-11.11 1"JJE»1 .11 1UMJA«1 .111» ) {

print (optparami [ 1 ] , optparami [2 ] , optparaml [3 ] , optparami [ 4 ] ) ;

31 -print (optparamf[1],optparamf[2],optparamf[3], optparamf[4]); line (2); 1£ COH1-2 V СОЫ1-3 print('?i«iintiini»C0MP0UH3>->RUCLEUS^PARAMETERS») ; lined); f o r m a t ( ' ^ t u n n i i i i R 0 » 1 » 1 1 1 •»! 1«-»-»-«-»SIGyA»11.111') ; p r i n t (BO, SIGMA) | 1 £ C0H2-1 OPTICAbMODEL(lmax,ZifZtfAi,At,Elab,.O,rsp,isp, optparami,el); 0PTICAlM0DEL(lmax+J,2i,ZttAl,At,Elab,-Eei,Rsp,Iflp, optparamftel); apA C0N2; Elab-Elab+Eexx(Ai+At) / A t ; fry ]>=0 atftn 2 i i a i i l 2xlaax do

•POT» leO qtan 1 jjnJij^ lmaz ЦЦ

3 3 -Ило(2)|

'Rfr -ЗИМА*);

param[1 JaSIGMA j t a a D3IQH u n t i l SKBUMAZ JUUEifl

" " " " " » ^ ' » » ' ! ••11.111»);

рахявС2 ЗчИО д £ 0 Д DRO JHQ^UL BOKAZ

da

OOHPOUBD(ohlaq);

prlafe(ebleq,p«raaC2]) i

U 2)

print ( '••» ••••••••• .MBBDupfiOCEDORDuu. . .uuCORA-»);

COF1-3

55

-param[ll>=aaxi ,0011

COHPOOHD(chleq)t

param[ll>=aa;

aaP.OOIxaa;

EIA[ Л, zapis MEZA [j , zapla 3-EXA [ j , 0 ] ) /as;

nn-fc< 1

, zapls X )

dbfifi 1

aaPElA[l,i]><EXA[l,j]/ertielgd[l]t2+aa;

HAC[,J,i3=MAC[l

j3=aa

lei fl^fiii i ш х ш H Да

37

-j*=ll aten 1 j m J ^ nn1

JCfiC

MACEi.pj >iIAC[i,pj ]-MAC[i,ph]xb

Alflft

fQJJ jtsll Д^ДД 1 дщЦ^ |щ1 MAC[l,p[j]]nO;

рорг[р[1]]ив

3 8 -№0; ±e1 Ht«p 1 until an da IUUCLA b»abe(popr[i]/b)i i i b>a iłjea 1 *1м 1С a<.3 t^**?8 «5 e l a f t 1С .2 »1яа i £ a<1 Ниш • 1 e^ee »05/a; ±tal «tan 1 jjjjtil ХШ dft

param[nrparam[i ] ]>7aramzm[i ]+popr [ i ] xZ; C0HFOUHD(F2) ;

a-fcftT>

ССМРОШП)(РЗ) i i Р1Ж

39

-for jU=1 gtgn 1 jHjJii Ж do

EXA[i,O]=EIA[i,-i];

£2KU £& Jift PARABOLA

alaa

iti?

<& 1& PARABOLA

40

-print('parab.nienoz*);

XL

i

PARABOLIl

xm fa

param[axparamCi 3 >=paramsan[i 3-4?opr [ i ]»*вт{

41

-1 a a t i l ПП dfl

paramzmCl ]&2>aram[nrparam[i ] ]t=paraozm[i ]+popr [1 ]><ZL; foznat (• ?it «11»***jch±eq-1 • 12345«12») }

read(RD,R0HAX,DRO);

read (SIGMA, SIOBUHAX,DSIGM) j

com-2

com-3

com-2 v сош-з

format (• ?k*A*******»jROei. 111 „1 U***JSIGMA«1 1.111');

OPTICALMODEL(lmax,Zl,Zt,Al,At,Blab,Q,rflp,i'jp,

parallel);

lmax,Zl

Zt;

yL Q,Blab,Ai,At;

rep,lep,param;

Aoolean e l ;

begin

Integer l,m,i,

uu,lmax1,p1,p2,pp1,pp2,lmax3;

real k,gamma,cbl8q,zboinaz,BE,Un,Wn,r1,a1,k2,a,b,c,d,e,

f

pom,xo,rl,pofflo,ro,z«k,gamna2,r3,r4,Wne;

a»*Att. 333333333;

Blab^EE/b;

k«=sqrt( AixElab)xbx. 2167;

gamnapsqrt ( Al/Elab) xZixZtx. 15745;

U»=param[ 1 ]/EB; r1«=param[3]*<a;

ГОВ1.34;

44

-(kx(e+f*ln(e)))>(k*(e+f*ln(d)))

then k»<(e+fxin(a))+8.059047825

else kx(e+fxln(d))+8.059047825;

If el

then lmasPentier(itiemax>+2;

1тах1е1лшх+1;

pp1=lmax3+lmax3; "

pp2«=lmax3+1;

begin

array RS,IS[1tpp1];

procedure CDIV(a,b,c,d,e,f);

value o

d

e

f ;

reap. a

b,c,d,e,f;

begin

b=e*e+fxf;

i f b»0

4 5

-procedure CMDXi(a,b,c,d,e,f);

value c,d,e,f;

real a,b

c,d,e,f;

begin

a?=cxe-d»<f;

b«=dxe+cxf

begin

reel BYC,IVC,e1

e2,ee1,ee2,eh1.eh2,r,phA,pIA,hr,hc,

hc1,rl2

rik,rl12,aa,bb,e,chiBq2,co,grck;

array pRS,pIS[1ipp1];

procedure FUH;

begin

hc=Łc+hc1;

chi*q=chlaq+rlk;

chlaq2i=chieqxchleq;

eMRVC+1-ЦХ r>rc then gamna2/ohieq elae

(3-hc>4ic)xgrok)

ri12sri2/12{

grckt=gamma/rok;

eh1«=exp(rl/a1);

e1>=ee1«=exp(-r1/a1);

real Ra2,Ia2;

for xerl,rl+rl do

d«12;

e>=20;

fbohieq;

• 7

ДНЯХ 7S

48

-(у [n»,a2]-y [m,n43 )*.75)/rlk»

3 atea 1 until n Ja

POT;

p1ts1;

p2=pp2;

- 49

i-n3

begin

RS[p1]sy[p2,i]|

IS[p1]«=iy[p2,i]

end.

p2ep2+1;

end 1

i;

far ш=рр2 ętep 1 until pp1

RS[m]=dlaS(y);

IS[m]=dlaS(iy);

end

end

real sigmaO, sig2,alpha,FF, 14, QQ,Gd.RA,IJL.RB,IB.RC, 1С,

RD,ID,RE,IE;

астазг t[C:lmax+1O];

procedura SLJ(RA,IA);

50

-рЗер2+1;

CDIV(a,b,RS[p2]tIS[p2],RS[p1]»IS[p1]); CMUL(o,d,atb,PPt0O);

cMUL(e,f,a,b,pp,-aa);

CDI7(RA, IA,Fd-o, Gd-d, e-Fd, f+Od); end; real Aom,eig,rho,eta,sto; ' ato«i6+etaxeta; aigt=-eta+eta/2xln(eto)+3.5xarctan(eta/4)-(arotan(eta) +arctan(eta/2)+erctan(eta/3))-eta/(12xsto)x (1+1/30x(etat2-48)/(etoxsto)+1/105x(etat4-I60xetaxeta+1280)/(etoxstoxstoxsto))• i £ (ibo<175/12Aeta<4)Vrho>(etaxeta+4xeta+3}x5/12 then begin s x , l e , t x , l t , S 8 , e l , t t , t l , n s , n t , e n , t n , a n , b n , U rho>(eta*eta+4*eta+3)x5/12 then iiion»=rho

else rhoi»=(etaxeta+4><eta+3)x5/12;

51

-tt«=tn«=1-eta/rhom;

for 1«<I step 1 until 10,11,1+1

52

53 -nh?=entier(abe(rhom-rho)x10+1) • h«=( rho-rhom)/nh; hh=h/2; x»=rłiom; y«=GO; yd«=Gd; d=*hx(2xeta/x-1);

far i«1 step 1 qptll nh ^2

5*

-real fdO;

Integer w;

for ича-1 step -1 until 1 flp

t [w-1 ]«vr/eqrt ( etaxeta+wxw)x( (( 2xw+1 )xeta/(wx

+(2xw+1)/rho)xt[w]-eqrt(etaxeta+(«r+1)x(w+1))

faO«=(eta+1/rho)xt[0]-8qrt(etaxeta+1)xt[i3;

a«=(-Gd+( 1/rho+eta)xGG)/sqrt{ 1+etaxeta);

alphaPi/(eqrt(1+etaxeta)x(t[O]xa-PPt=alphaxt[o];

sigg22=sigmaOs=sigg+sigg;

end Coulomb;

SLJ(RA,IA);

rep[O]t=RA;

iep[O]«IA;

for 1«=1 step 1 until lmax djg

5 6

-a,b,o

d»*

f i

x,ISJS;

SJS(J1,J2

J3

Ł1,L2,L3)

J1,J2,J3»L1

L2,L3;

58

-IWBf=(J2+J3+L2+L>IW)/2+1;

ii (lW/4-entier(IW/4))*O

PB=1;

KAPPA«KAPPA+P№<SH(IW1) /SH(IW2) /SN(IW3) /SJT(IW4) / SN(IW5)/SH(ie6)/SH(IVf7) /SN(IW8); end Iffj SJbKAPPA*DELTA( Л f J2, J3)xDELTA( J1 ,L2,L3)x DELTA(L1, J2,Ł3)xDELTA(L1 ,L2, J3) { EHDjSJ&=SJ*10; SJS; DELTA(JI , J2, J3); J1,J2,J3; ^t^^^:Agft•r^ J 1tJ raal 0; Hn«(J1+J2-J3)/2*1; IW2B( J1-J2+J3) /2+1; IW3»(-J1+J2+J3)/2+1; IW4»(J1+J2+J3+2)/2+1;

<*=eqrt (SK(IW1 )xSN(IW2)xSH(IW3) /SH(IW4) )

DEŁZAPG/3. 16227765 *,

DELTA;

- 6 7 PCl S№=SL; PBOCEDUEB SLY;

<k/2-entier(k/2))łO

stop»

Z dostaroconego maszynopisu druk 1 opraw? wykonano w Zakładzie Graficznym Politechniki Krakowskiej w Krakowie. Hakład 120 egs.