Absolute normalisation in microscopic DWBA analysis of the three-nucleon transfer-reaction cross sections

Nuclear Physics 0

North-Holland

A398 (1983) 8492 Publishing

Company

ABSOLUTE NORMALISATION IN MICROSCOPIC OF THE THREE-NUCLEON TRANSFER-REACTION F. BRUNNER

DWBA ANALYSIS CROSS SECTIONS

and H. H. MULLER

Institut ,fCr Physik,

Uniwrsity

Ziiric~h, Switzerland

and C. DORNINGER Institut

ftir Kerphysik,

and H. OBERHUMMER Technical

Received

University

25 October

Vienna, Austria

1982

Abstract: Theoretical and experimental normalisation constants for the three-nucleon transfer reactions are determined. Zero-range and finite-range analyses have been performed. Theoretical cross sections obtained from microscopic DWBA calculations are two to three orders of magnitude smaller than experimental values.

1. Introduction The direct three-particle transfer reaction attention in the last few years (i) as a tool

(p, a) has received considerable for nuclear spectroscopy, and (ii)

regarding the formulation of a theory to analyse these processes in the framework of the distorted-wave Born approximation (DWBA). There has been a considerable improvement in the theoretical DWBA description for (p. a) reactions over the simple triton cluster pick-up approximation by using either semimicroscopic ‘) or microscopic ‘) models. These descriptions have been fairly successful in the analyses of the data for cross sections and analysing power as well as for the relative magnitudes of the transitions to different final states in the same residual nucleus. However little is known about the absolute magnitude of the theoretical cross section. In principle, the absolute normalisation can either be determined empirically from the comparison of DWBA calculations with experimentally determined cross sections or can be calculated using the microscopic model. In the two-nucleon pick-up reactions induced by light ions, i.e. (p, t), (p, T) and (d, a), the empirical determined values agree roughly with the theoretical calculations for the normalising constants 3). However there are still some discrepancies in the normalising constant which should be investigated further 4). In sect. 2 we calculate the zero-range normalisation constant 0;” for (p, a) 84

F. Brunner et al. / Absolute normalisation

reactions

using the microscopic

85

model. This value is then compared

in sect. 3 with

the empirical determined values D,. In sect. 4 we discuss the normalisation constant determined by including finite-range effects of the interaction potential in the DWBA calculations and compare these values with the normalisation constants from zero-range DWBA calculations. Finally in the last section we investigate how the normalisation constants depend upon the ambiguities of the optical potentials of composite particles.

2. The zero-range normalisation

constant

We denote the three transferred nucleons in the (p, c() reaction with 1, 2 and 3, and the incoming proton with 4. The outgoing m-particle consists then of particles 1, 2, 3 and 4 retaining only the direct and neglecting the exchange terms of the pick-up reaction. We choose Y for the relative coordinate between particle 4 and the centre of mass (cm.) of particles 1, 2 and 3, the coordinate r12 for the coordinate between particles 1 and 2 and r1 23 for the relative coordinate between the c.m. of particles 1 and 2 and particle 3 (fig. 1). In order to define the zero-range normalisation constant 0: we have to write down first the expressions for the internal a-particle wave function and the interaction potential in the microscopic model. For the a-particle wave function we assume a gaussian form ‘) (PaPi, r2, r3, rq) = N,exp

Fig.

i
- ‘1’ 1

Iri-rjl'

,

(2.1)

1. Representation of the relevant coordinates in the description of the A(p, a)B reaction. Particles 1, 2 and 3 refer to the transferred nucleons and particle 4 to the projectile proton.

86

where

F. Brunner et al. 1 Absolute

q = 0.233 fm-’

is the

size parameter

normalisarion

of the

cc-particle

normalisation constant of the m-particle wave function. Transforming to the coordinates Y, r12 and rr13 given above, q.(r,

and

N,

we obtain

is the 6, (2.2)

r12, rtZ3) = cpY’(rk~~~‘(r,~, r12A

where q?‘(r) = (6q2/7c)”exp [ - 3q2r2],

(2.3)

cph2)(r) = (64q4/3n2)*exp [ - q2(2rf2 +$tzJ).

The a-particle wave function the motion between particle

(2.4)

has been split into parts, where the first part describes 4 and the c.m. of particles 1, 2 and 3, and the second

part describes the internal motion of particles 1, 2 and 3. For the interaction potential between particle 4 and the system particles 1, 2 and 3 we can write

W,, and we assume

r4) = i

i=l

for the effective nucleon-nucleon ~j(ri,

with a strength Transforming obtain 6,

r2, r3,

rj)

=

U,

consisting

(2.5)

VZi(Yq-ri), potential

exp [ - /?‘lri - rjl]

again

a gaussian

form ‘)

(i, j = l-4),

U, = 70 MeV and an inverse range /? = 0.632 fm-‘. the interaction potential to the above given coordinates

Ur,

r12, r123) = f?r)~‘2Yr,2,

of

r12,),

(2.6)

we

(2.7)

where V(')(r) = 3U, V2'(r12, r123) =

exp[-fi2r2],

(2.8)

~exp[-~2(~r~2+~r~2,)]+~exp[-P2(~r~23)].

(2.9)

Here the interaction potential is split into two parts in the same way as for the LXparticle wave function in (2.2). We can now easily formulate the different definitions for the zero-range normalisation constants. The first definition is given by “) V(r, r12, r123)(pr)(r) z

V"(r)cp~'(r)

z 0$5(r).

(2.10)

This approximation implies that the incoming nucleon interacts with the threenucleon system only through its centre of mass. Other possible definitions for the zero-range normalisation constants are 2, V(r, r12, r123) z

V')(r) z @S(r),

(2.11)

and “)

(2.12)

In the last approximation the cl-particle is assumed to be point-like, therefore this approximation should be called the point-alpha approximation analogous to the (p, t) reaction, where this approximation is called the point-triton approximation. We calculate now the theoretical values of the zero-range normalisation constant D:: using the microscopic description of the a-particle wave function (2.2)(2.4) and the interaction potential (2.7)-(2.9). Integrating eq. (2.10) we obtain 0: = ~V”)(r)yl~i)(r)dr = 37/, ((b::rrZ)h):

+

= 506 MeV. fm*.

Similarly the value of the other zero-range normalisation @’ =

s

6:: =

(2.13)

constants are obtained:

= 4620 MeV . fm*,

(2.14)

V”)(r)cp!‘)(r)G(r,2)6(r,23)drdr12dr123 s

= Dth 0 = 506 MeV. fm*.

(2.15)

In the following we use the definition (2.10) and the value (2.13) for the zero-range normalisation constant. The values of the different zero-range normalisation constants can be related with the help of relations (2.13~(2.15).

3. Empirical determination of the zero-range normalisation constant The zero-range normalisation constant can also be determined empirically by comparing the experimental data with the DWBA calculations using the microscopic model. In this method the zero-range normalisation constant D, is not calculated like in the last section, but is considered as an open parameter which can be determined from the comparison of the experimental with the theoretical cross section. The DWBA cross section is given by cDWBA

=

pp

0

3

(3.1)

where crZR is the zero-range cross section and the zero-range normalisation constant D, has been factored out. In the calculations performed by our group the DWBA cross section is calculated using the expression *oWBA_ 2s,+ 1 L); cFDWUCK4 -2s,+liZF 2Jfl ’

(3.2)

where s, is the spin of the incoming proton, sr, is the spin of the outgoing a-particle and J is the total angular momentum of the three transferred nucleons. The cross section is calculated with the code DWUCK49) with an external form factor

F. Brunner et al. 1 Absolute normalisation

88

calculated

with the code

Talmi-Moshinsky systems

after

functions

(j).

FORM

transformation expanding

lo). This

form

to the centre

the

functions

in

factor

is determined

of mass

terms

and

by using

internal

of harmonic

oscillator

By equating the DWBA cross section gDWBAand the experimental aexp, for instance with the help of a least-square fit

wave

cross section

anwB* Z ,yxp we can determine

the empirical

zero-range

a

coordinate

(3.3)

normalisation

constant

with

0; = lo4

(3.4)

In table 1 the results of these empiricail determined zero-range normalisation constants are shown for (p, a) reactions on different nuclei. All of these values have been determined by using the microscopic DWBA calculations described in ref.(j) and then extracting the zero-range normalisation constant with the help of (3.4). In

TABLE 1 Zero-range Target nucleus

normalisation

Residual nucleus

constants

Wave function of target nucleus

94Zr, + 92Zr,, “Zr, + 902r, + 89Y,,2_ ‘*Nio + 56Fe, a

refs. refs. refs. ref. ref.

‘I) 13)

;z;

1,“) ‘)

55Mn5,2 2’AI,,,~

ref. “1 ref. 23)

“) as in b, ‘) d,

for (p, G()reactions

“.”

(our results)

Wave function of residual nucleus

G (MeV’ 0.14x 0.23 x 0.17 x 0.53 x 0.66 x 0.17 x 0.70 x 0.84 x 0.26 x

ref. 13) “)

)

’ ‘. I2) 16. ” )

ref.13)

ref. 13) refs ‘7.18) b, ref. ‘17) zf. 20) ref. 20) ref. 24)

For 9’Y,,2_ the same proton configuration as in “Y, i2 and the same 92Zro, has been assumed. For “Y, ,z the same neutron configuration as in %r, + has been assumed. For 58Nio+ the same neutron configuration as in 56Feo * has been assumed. For 55Co,,,_ a proton If,/, hole configuration has been assumed.

neutron

IO* [ref. IO* [ref. 10’ [ref. 10s [ref. IO* [ref. IO’ [ref. 10’ [ref. IO8 [ref. 10’ [ref.

normalisation Reaction

26Mgo+(p. cWNa,,,+ “Zr,+(p, a)*V,,,. “sSn,+(p, a)“%,,,+

constants

for (p, a) reactions

taken 0;

from the literature

(MeV’

fm3)

0.87 x IO8 [ref. ‘“)I 0.36 x 10s [ref. “)I 4.12 x lo8 [ref. ‘)I

r4)] IS)] IS)] 14)] “)I *‘)I *‘)I 22)] 25)]

configuration

TABLE 2 Zero-range

fm3)

89

F. Brunner et al. / Absolute normalisation

table

2 we show empirical

from the literature. Comparing the empirically

values

of the zero-range

determined

normalisation

values of the squares

constant

taken

of the normalisation

constants 0; with the theoretical value (#,h)2 one sees that the theoretical value is too low by a factor of approximately 50 to 500, except for the last reaction in table 2 where this value is even 1600. The observed fluctuations of the 0; factor may be attributed to incomplete shell-model wave functions of the target and the residual nucleus, to different forms of the optical potentials and to different treatments of the asymptotic form of the form factor. However, the theory generally underestimates the experimental cross section by about two to three orders of magnitude. Therefore we conclude that there must be some fundamental deficiency in the microscopic, zero-range DWBA theory.

4. Results for finite-range calculations In order to see how the zero-range approximation influences the value of the normalisation constants of the microscopic DWBA calculation, we also performed calculations in finite-range using the formalism of ref. 6). We define a normalisation constant 0 exp

=

Crgth,

(4.1)

where Y is the absolute normalisation factor either for zero-range or finite-range microscopic calculations. A value of V = 1 would mean perfect agreement between theory and experiment. The normalisation constant +’ is determined by equating the product of this constant times the theoretical cross section ath to the experimental cross section fYxP with the help of a least square fit. In table 3 we show the results for the normalisation constants in zero-range and finite-range

using our experimental

data for the cross sections.

The wave functions

of the target and residual nuclei and the parameters used in the DWBA calculations can be found already in the corresponding references of table 1. The values of the normalisation constants of table 3 show that the zero-range as well as

TABLE 3 Normalisation

Reaction

*‘Al(p, a)=Mgs... ‘*Ni(p, a)55Co,,,, “Wp, @9Y,.,.

constants

JV” for microscopic Zero-range normalisation

102 66 66

DWBA

calculations Finite-range normalisation

545 1268 341

90

F. Brunner et al. / Absolute

normalisation

the finite-range result is too small by about two to three orders of magnitude. However, the theoretical cross section in finite range is still reduced compared to the zero-range cross section. This means that the discrepancy between theoretical and experimental cross section is even enlarged for the considered nuclei when finite-range effects of the interaction potential are included. In summary, the inclusion of the finite-range of the interaction potential cannot explain the large discrepancy between the absolute value of the theoretical and experimental cross sections.

5. Ambiguities of the optical potentials It was well known that the optical potentials for the elastic scattering of composite particles have ambiguities. In this section we investigate how these ambiguities affect the absolute value of the (p, CC)cross section. In particular we consider the reaction 27Al(p, C()24Mgp,s,.Table 4 shows four different optical potentials for the channel a+ 24Mg and a optical potential for the channel ~i-~‘Al. All of these potentials are of the Saxon-Woods type 28) with parameters given in ref. 29) and reproduce the elastic scattering data. The differences in the a-potentials are: the potentials c~i, a,, CQ differ mainly in the depth and range of the real part; there is a deep, a medium and a shallow potential. The potential ~1~is also a deep potential, but the volume-absorption term is replaced by a surface-absorption term. Using the optical potentials given in table 4 we made microscopic unite-range DWBA calculations for the reaction 27Al(p, a)Z4Mgp.s,. We determined the differential cross section and the analysing power and made a least-square fit to the experimental data given in ref. 24). The results for the least squares of the differential cross section and the analysing power are shown in the last two columns of table 5. The normalising constants differ as much as a factor of 30 for

TABLE 4 Parameter p-potential

for optical

for 27Al(p, p)“AI

and 24Mg(~, @Mg

V, 39.7

1.18

V0

lo

aI a2

199.7 132.6

1.35 1.42

a3 a4

33.3 190.0

1.74 1.40

a-potential

potentials

0.70

7.0

1.40

0.70

7.5

W

rw

a,

wD

0.67 0.67

30.4 36.2

2.70 1.58

0.47 0.55

0.59 0.52

17.2

1.56

0.87 60.0

1.40

0.70

1.20

25)

‘D

aD

rc

ref.

1.40 1.40

29) 29)

1.40 1.40

29) 29)

1.35

0.37

F. Brunner

et al. /

Absolute

normalisation

91

TABLE5 Normalising

constants

Potential

a, a3 a3 a4

Jlr for differential cross sections and analysing powers for different optical potentials Normalising constants

x2 for differential cross section

x2 for analysing power

545 432 20 170

16.0 31.4 43.8 16.9

16.5 17.3 98.6 17.2

the different LYoptical potentials. One sees also from table 4 that only the deep potentials CC,and ~1~give an acceptable lit to the experimental data. The shallower a-potentials produce a low normalising constant but make the fit of the data worse. From the point of view of the normalising constant shallower potentials seem to be preferable. However, to obtain optimal fits to the differential cross sections and analysing power deep LXoptical potentials are required.

6. Summary Summarising our results regarding the absolute normalisation of microscopic DWBA calculations for (p, a) reactions we can say the following: (i) The value of the empirical determined zero-range normalisation constants D, is between about 0.4 x lo4 and 2 x lo4 MeV * fmt. These fluctuations may be attributed to incomplete shell-model wave functions for the target and residual nucleus, to different forms of the optical potentials and to different treatments of the asymptotic form of the form factor. Also it has been shown that in some cases multistep processes can give important contributions to the cross section for the (p, 01)reactions. (ii) The theory generally underestimates the absolute values of the cross section by about three orders of magnitude. These discrepancy cannot be explained by finite-range effects or by the use of different c( optical potentials of the SaxonWoods type. It is obvious that there are some fundamental deficiencies in the present DWBA treatment of the (p, a) reaction. Therefore further theoretical studies are required.

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92 4) 5) 6) 7) 8) 9) IO) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 23) 24) 25) 26) 27) 28) 29)

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