The statistical multistep direct emission theory revisited

Volume 112B, number 3

PI1YSICS LI-TTF RS

13 May 1982

THE STATISTICAL MULTISTEP DIRECT EMISSION THEORY REVISITED ¢z Mahlr S HUSSEIN Instttuto de l-tstca cla Umverstdade de S~o Paulo, Sao Paulo S P, Brastl and

Roberto BONETTI Istttuto dt Ftstca dell Untverstta, Mgan Italy

Received 3 December 1981

A recurslon equatmn has been derwed for the multbtep direct enussmn probability The solution of the equation resembles the one gwen by I'eshbach, Kerman and Koonm except for an overall multtphcatwe factor which depends on the number ol steps considered, and the compound nucleus transmlsston coctficmnts for the grazing parnal waves Fstlmates of these factors are given and the consequences of thetr presence are discussed for the reaction 12°Sn(p n) at 45 MeV

With increasing bombardmg energies, nuclear direct reactmns become more complex as mulustep processes become tmportant Examples o f these are preequfltbrmm reacttons and heavy-ion deep melasttc colhslons (DIC) In either case many complex channels are mvolved Exact, quantal many-coupledchannels calculatmns o f these processes are certatrdy not viable Recently several alternatwe theories have been developed having m common the stat~sucal treatment of the couphngs along the channels In pamcular, we mentmn the theory o f Agasst et al [1] for DIC, and the staUstlcal mulhstep direct emlssmn (SMDE) theory o f Feshbach et al [2] One dtstmctwe feature of the quantal SMDE developed m ref [2] Is its smaphclty Several calculauons have been made [3] and the overall SMDE account of the data on preequdlbnum spectra at different angles were found to be good Recently, it has been observed [4] * ~ that the individual one-step cross sectmns that appear m the expressrun for the mulustep cross sectmn o f SMDE of ref [2] are not o f the DWBA type The point raised m refs [4,5] refers to the fact that the matrix elements * Supported m part by INFN, Sezlone dl Mflano, Italy, and CNPq-Brasfl ~1 This problem has been discussed prevtously m ret [5] 0 031-9 ! 6 3 / 8 2 / 0 0 0 0 - 0 0 0 0 / $ 0 2 75 © 1982 North-Holland

/ ~ , ( + ) l tV I .i Xv+l, (+) \ for successive transitions are of the type "xv and therefore not of the DWBA form (Y(-)lVIY(+),) "'V "~/)t 1 Ttus point is qmte relevant as the calculatxons that have been performed so far use DWBA cross sections for the mtermedlate smgle-steps Clearly one may calculate the former matrtx element from the DWBA by inserting, m the partial wave expanstun o f the latter, an appropriate p a m a l wave elasuc Smatrtx exp(-2tS~'+l), where 6~'+1 is complex [6] An mamedtate consequence ts that

(~(+)Av V Av+l'(+) X 2 i> i<~(-)lVix~+)l)12 smce the modulus o f exp(-216~'+l), t e exp(+27?y +l) is larger than umty due to the presence of absorption (compound nucleus, other open channels) m the system The above rrnphes that the extracted values of the restdual mteractmn strength, V0, found m ref [3] should be smaller than reported In the present note, we demonstrate that by a proper treatment o f the distorted waves and, further, by recogmzmg the non-umtary nature o f the underlying average S-matrLx, the above questmn discussed m refs [4,5] can be fully, albeit approximately, accounted for by the mtroductmn, into the SMDE expression used m the calculatmn o f ref [3], o f overall multlphcatwe 189

Volume I12B, number 3

PIIYSI(S LEFIF RS

factors that depend on the number of steps included and the incident channel compound nucleus transmlsstun coefficients for the grazing partml waves Clearly had the authors of ref 13] used non-DWBA cross sectmns as the FKK theory m&cates (see our comments preceedlng fig 1), there would be no need for using eq (14) below whMa is an approximate version of the FKK results Howe,er, wc believe that our eq (14) is useful as It clearly exhabxts the role of compound processes on the multlstep direct ones Further, all mdwldual single-step cross sections present m eq (14) are of the DWBA type which may be calculated in the usual way The transition matrix element is as given m ref [2] Ttl = tft +

(~:~-)lV£~)pt(E)Vl~ol+)),

(1)

where tfi is the usual DWBA amplitude I~#I-)) and I~pl+)) are the final and mmal channel distorted waves respectively The couphng potentml (that couples among the different channels) is ln&cated by V F1nally £(+p)(E) is the opttcal Greens function given by £(o~)t(L) = (E(+) - / / o p t )

1,

(2)

where Hop t is the usual optical bamlltonlan with an imaginary part that simulates absorption due to compound nuclear processes The mulustep processes are describable by the second term in eq (1) In ref [2], the optical Greens function was expanded in terms of dressed channel Greens function that were eventually replaced by their on-energy-shell parts The contributions of the principal parts of these Greens functions average out to zero [2] We shall adapt the same procedure later We now write eq (1) as

13 May 1982

Tfi=tfi

n'l ~ (~Pl-)[ v ~v'(+)./7 . . , . ,(+) W vv ~p,(+)',

(5a)

v

= tfi - 7rl ~ v

Ttv(~g~(+) v Iris, (+) >

(Sb)

The amplitude ( ~ + ) l Vltpl +)) can be further reduced to a manageable expressmn as follows F~rst we notice that (~+)1 may be formally related to (~(-)1 through the relation '7(-), = ~

Suu,(~(u+,)I

(6)

#

where S ~ , is the/l/f-element of the average (optical) S-matrix Upon multiplying eq (6) from the left by SSu and summing over # we obtain

Svu(gg. ~(-) =~, (S~S)vu,(~d (+,)] t.t

(7)

t~

From umtanty, we know that

(S i S)~., = 8~u, - / % , , ,

(8)

where P is Satchler's penetration matrix t2 Accordmgly we can recast eq (7) as

~S~u(~(-)l:(~b~+)l - ~ Pvu,(~(+)l

(9)

At this point we employ the approximation

Pvu' "~ 6 vu'Pvv - 6 vu'Pv ,

(10)

which states that the effects of directly coupled open channels on the compound nucleus processes are small Thus (~+)[~(1-Pv)

-1 ~_jSvu(Ou -(-) I

(1l)

# ,' ( + ) \ / ~ ( + ) 1

TI =tf '+~J~dev(~l-){ v

v , ~ v ,,_~v ,V[~0}+)), E(+) - ev

(3)

where (E -

Hopt)[~ (+)) = 0

(4)

Notice that e v contains a kinetic-energy part and an intrinsic excitation energy related to the excited target nucleus m stage v We now make the on-energy-shell approximation for £(+)t(E), anteclpatlng that the offshell part will generate terms m the series (3) that eventually average to zero upon taking the modulus squared and averaging eq (3) Thus 190

where we understand Pv above as the transmission coefficients for the grazing parhal waves ,2 Using the familiar relation between S and T, we can rewrite eq (11) in the following simple form • 2 N o t i c e t h a t P is n o t t h e usual, partial wave t r a n s m i s s i o n m a t r i x as it d e p e n d s on the angle f o r m e d b y the m o m e n t u m vectors k v and k/~ I l o w e v e r , as a c o n s e q u e n c e o f the surface n a t u r e o f the m u l t t s t e p d i r e c t process, 1 e its b e i n g l o r w a r d peakt.d, we e x p e c t t h a t e l all partzal waves ~.ontnb u t m g to Pvt2, o n l y the grazing one to be i m p o r t a n t [ h r o u g h o u t the p a p e r w e shall t h e r e f o r e c o n s i d e r Pvv or Pv to be a p p r o x i m a t e l y gxven b y Pig, w h e r e lg _~ kR, w i t h R being the r a d m s o f the s y s t e m

Volume 112B, number 3

-1

PHYSICS LE 1 FERS

~(-

+2hi

* ~(-) (12)

which, upon insertion into eq (5b), gives our fundamental "on-shell nuclear Low-equation" T l , = / f l - 7 / ' 1 ~ v Try ~ + 2n"2 ~

1

(13)

To calculate the inclusive preequlhbrlum cross section we have to take modulus squared o f e q (13) and sum over the relevant final excitation-energy Interval In doing this we shall, in keeping with the basic statistical hypothesis underlying the theory, retain only manifestly positive definite terms, thus iZf,12 = itf,12 +rr2 ~ iTf~12 _ _ 1 _ _ iZ.,i2 v (1 - Pu) 2 + 4n2 ~

ITf"12 (1 _1p . ) 2

Table 1 The correction factors listed for three different numbers, n, of the steps in the SMDF reaction The symbol X stands for (1 - 1'1)-2 (see text for details) Note that for each ~alue of n, the correspondmg power of Vo m the cro~ section is 2n Thus V0 is scaled by (An)l/2(n-l) n

Correction factor An

A n for l:~°Sn(p n) at b p = 45 MeV

2 3 4

X 2(2X + X 2 ) 20X 2 + X 3

1 235 7 98 32 388

Tvl

Tf~ i- _ L ~ ( T f T ) . i

[(T-T)v,]2

three-step, and four-step processes As one clearly sees, these corrections depend on two quantities the number of steps involved, and the incident channel, compound nucleus, transmission coefficients P1 One further approximation was employed to obtain these correction factors, namely P~ ~ P i

(14)

which is the recurslon equation we seek An Important point which ts to be recognized IS that the pverall ~,thstep delta function would guarantee, upon taking the modulus squared of eq (13), the appearance of the correct momentum phase-space factors that are needed, m eq (14), to define the individual one-step DWBA cross-sectmns We have not indicated tins explicitly in eq (14) in order to simplify the notation Eq (14) exhibits the statistical multlstep nature o f the theory, as exemplified In the second and third terms One may easily generate the SMDE expression o f ref [2] by repeated application o f the recurslon equation above and by assuming the chaining hypothesis, namely u = t + 1, etc The only major difference IS the presence o f the factor (I - Pv) - 2 which IS generated In a successive way, and the appearance of genuine DWBA cross sections In the series The expressmn used in the calculation reported in ref [3] can be obtained from eq (14) by dropping the third term on the right-hand side and setting Pu = 0 0 e no compound absorption) The above approximations o f course amount to setting (~+)1 = (~9~-)1 lsee eq (9)] In table 1 we list the first three multlpllcatlve corrections to the results o f ref [3], namely for two-step,

13 May 1982

(15)

This, we believe, is reasonable in view of the fact that the energy loss encountered in each step is rather small, and the optical potential one uses in the distorted waves defining the basic DWBA cross sectlon,[tv,~+l [2, Is usually taken to be the same as the one that generates the elastic channel distorted wave, [¢}+)) We have calculated the correction factors shown In table 1, for the recently analysed reaction 12°Sn(p, n) at Fp = 45 MeV For this system the values o f PI, for the grazing p a m a l wave, namely lg ~ l 1, is about 0 1 and therefore the correction factors are not very large As a matter o f fact, by introducing these factors Into the calculations reported In ref [3], we obtain an almost Identical fit to the data with a smaller number of steps and, more importantly, with a smaller value of the residual lntelactlon strength V0 Fag 1 shows our results that were obtained with only four steps, compared to the corresponding results o f ref [3] obtained with six steps The adjusted value o f the strength V0 was found to be 35% less than that found in ref [3] Tins result, we believe, is reasonable as it goes in the direction of closing the gap between the values of V0 extracted from multlstep compound processes [8] and from multistep direct processes reported in ref [3] An Important feature o f our fundamental eq (14) is the appearance o f compound nucleus quantmes in the cross section for direct processes This is clearly a manifesta191

Volume 112B, number 3

PHYSICS LFTTERS

I0

[..

,2o

't

so,p.n,

""-'%...

: ":"%

~

O:12MeV

o

oo, .

0

.

.

.

I

50

,

,

,

,

Ocm

I

,

100

,

,

,

I

150

I lg 1 Calculated neutron angular distributions for the 120Sn(p, n),kp = 45 MeV at three final neutron energies Dashed curves are from ref [31 (see this reference for details) Full lines include the corrections factors of table 1 The data pmnts axe from Galonsky et al 17] tlon of u n l t a n t y Notice that when one increases the incident energy, the grazing lg Increases and accordingly the correction factor becomes less important as It should, since multlstep direct emission dominates over compound emission Inversely when the energy is reduced, PI becomes larger but [TviI 2 etc become smaller since they are roughly proportional to (1 - P~) Thus the second term in eq (14) becomes less tmportant [being proportional to (1 - P1)] The third term in eq (14) also becomes less significant since the fundamental DWBA cross scctmns that appear implicitly in It become small (since the average S is reduced in mag-

192

13 May 1982

mtude) This is certainly what one would expect In conclusion, we have shown that the recognition of the fact that the individual one-step cross sections that appear In the FKK expression for the statistical multlstep direct emission (SMDE) processes are not of the DWBA type, results in a better numerical convergence By converting all non-DWBA cross sections into DWBA ones, we have exphcltly exhibited, in an approximate way, the role of compound absorption m the SMDE cross section In this way, the Interplay between multlstep direct and compound processes becomes quite clear Of course our results and those of FKK are both based on the same statistical assumption namely Involving non-DWBA amplitudes m the linear momentum representation Presumably different statistical assumptions involving, e g , the DWBA amplitudes, would lead to different results Which assumptions should one use9 This and other related questions connected with the validity of the nearest-nmghbour (chaining) approximation are discussed fully in ref [9] This work was initiated during a visit by MSH to the Umverslty of Milan He expresses his thanks to Professor L Colh Mllazzo and to this coauthor for their generous hospltahty Discussions with Professor H Feshbach, Professor M Kawaa and Professor A Kerman are appreciated

References 11 ] D Agassl,C M Ko and H A Weldenmuller, Ann Phys (NY) 107 (1977) 140 [2] H Feshbach, A K Kerman and S Koonm, Ann Phys (NY) 125 (1980) 429 [3] L Avaldl, R Bonettl and L Colh Mflazzo, Phys Lett 94B (1980) 463, R Bonetti, M Camnaslo, L Colh MtlazToand P E llodgson, Phys Rev C, to be pubhshed [4] M Kawal, private commun,catlon to II Feshbach and A Kerman [5] N Austern et al, Phys Rev 128 (1962) 733, I) Robson, Phys Rev 7C(1973)1 [6] II Feshbach, private commumcatlon [7] A Galonsky et al, Phys Rev C14 (1976) 748 [8] R Bonettl et al, Phys Rev 21C (1980) 816 [9] R Bonettl, P E Hodgson M S Hussem and M Kawal, m preparation