On the separation of the direct reaction contribution in fluctuating cross sections by the use of polarized particles

Nuclear Physics A281 (1977) 261-266 ; © North-Hol/and PublLrhing Co., Amsterdam Not to be reproduced by photoprint or microSlm wiWout written parmlsaion from tha publisher

ON THE SEPARATION OF THE DIRECT REACTION CONTRIBUTION 1N FLUCTUATING CROSS SECTIONS BY THE USE OF POLARIZED PARTICLES t R . HENNECK

it

Tandemlahor der Uniuersitüt Erlangen-NWrnherg, Erlangen, Germany and G . GRAW Sektion Physik der Universität Mûnchen, Garching, Germany Received I1 November 1976 Abstract : A method to determine the direct reaction contribution yo to the differential aoss section in the presence of Ericson fluctuations is proposed . For the scattering of polarized spin-} particles from spmless nuclei the statistical analysis of the differential cross section do/dß and of the differential analy~ng power (da/dß)A allows an unambiguous and model independent extraction of yn. The method is applied to the elastic scattering of polarized protons from ~bMg, ss Sr and'°Zr ; the results arc in agretment with Hauser-Feshbach calculations .

1. Introduction The study of cross-section fluctuations in the excitation functions of nuclear reactions by means of the Ericson model' -4) has contributed considerably to our knowledge of average nuclear parameters at high excitation energies in the compound nucleus. It offers a simple method to determine the coherence width T, which is related to the average compound nucleus level width. If both direct (DI) and compound nuclear (CN) reaction'mechanisms are present, the relative amount of the direct reaction contribution to the cross section yn is also obtained, provided additional assumptions, e.g. those concerning the number Nert of effectively contributing channels are introduced. Usually, N~.t is obtained from Hauser-Feshbach (HF) calculations. In this contribution we show that polarization experiments are able to remove this model dependence. We discuss this explicitly for the case oftwo spin channels, using elastic proton scattering from a spinles target as an example . To illustrate the method we analyze fluctuation data, measured with polarized protons by Glashauser et al.') and Paetz gen. Schieck et al . 6~ and compare the results with HF calculations. f rT

Supported in part by BMFT and DFG. Present address : ETH Zurich, Isbor f. Kernphyai~, Zurich, Switzerland . 261

262

R. RENNECK AND G . GRAW

2. Theory

The fluctuation theory is discussed in detail by Ericson i .z .<) and Brink and Stephen 3). Here we limit ourselves to the problems concerning the determination of yD. Iffor the scattering amplitudes a separation in a direct and a fluctuating part f° '+fwith
If the particles have spin, the reaction amplitudes ß = 1, . . . Nm,= contribute incoherently to Q :

fo

of the different spin channels

Q=~Qa=~~fa~2 .

s

a

The number N~ of spin channels is related to the spins of the projectile (i), target nucleus (I), ejectile (i') and residual nucleus (I~ in the following way °): for q even for q odd, where q = (2i+1x21+1x2i'+1x21'+1). For i = i' = i, 1 = l' = 0 we have a two-channel case. For N~ > 1 the normalized variance is no longer sufficient to determine yD since' " s) C(~) _ (1- Yn)2 ~ Pâ + 2Yn(1 - Yn) ~ Pßgp 0

depends also on pa = the assumption °" ' " e)

Qé

/QCri

and qo =

B

a~/Q°' .

A simple expression is obtained if

is introduced . This corresponds to defining via the CN contributions a number of effectively contributing channels Nor = (~pâ) - '. Then eq. (6) yields the widely used relation C(v) _ (1 - Yô)/Nerr

(8)

DIRECT REACTION CONTRIBUTION

26 3

The validity of eq. (7) has been studied by Ernst') and Dearnley e) by means of HF and DWBA calculations. Deviations from equality as large as 10 ~ have been found. Hence this approximation may be considered as critical ifa detailed discussion is attempted. The number N~rr is a function of the scattering angle B, symmetric to B = 90° (cf. also fig. 1) and by definition N~rr 5 Nox. Its angular dependence has to be calculated in the Hauser-Feshbach model, although around 90° the approximation N~rr x N~ x seems practicable. [In principle yn and N~rr may be determined separately via the cross section probability distribution function a . °) without HF calculations. In practice, however, this turns out to be rather impossible .] In polarization experiments the above-mentioned model dependence can be avoided, since for a reaction between particles with spin there are at least as many independent polarization observables as contributing spin channels No= [ref. 9 )] . This shall be demonstrated for a two-channel case, e.g. proton scattering (elastic or inelastic) from a spin-zero nucleus. The differential cross section Q and the differential analyzing power QA (A is the vector-analyzing power) are functions of two amplitudes, fi = a = a°r + ar` and f2 = 6 = b°` + b~ [ref.' °)] Q = Ia1 2 +161 2 = Qa+Q~, QA

= 2 Im (ab*).

Using the random phase approximation for the fluctuating scattering amplitudes Thompson 1 °) could show that the CN part of the differential analyzing power vanishes in the energy average : a° and b r`,

((6A)cN)

= 2(Im(a°br`7) = 0.

(10)

This has been applied in ref.' 1) to separate DI and CN parts in cases, where due to poor experimental energy resolution the fluctuations are averaged out completely . A straightforward calculation with Ericson's statistical assumptions and eq . (10) yields for the variances of Q and QA R(Q) _ (Q Z ) - (Q) 2 = (Qn)2-(~r)2+(Qb)2-(~r)2,

(lla)

Then the sum of both variances, normalized to the square of the energy-averaged cross section yields 1 z) ~~~ QA)

=

R(Q)+R(QA) (Q)2

z 1- yo~

(12)

Using polarization data hence reduces the two-channel case to a one-channel situation. This allows determining yo independently of N~rr and of the validity of eq. (~. The coherence width may also be obtained independently from the auto-

264

R. RENNECK AND G. GRAW

correlation function R(QA,e) of QA [ref.' Z)] : (13) This may ease the determination of r in certain cases since it has been shown theoretically 13,14) that the strength of the fluctuations should be larger for QA than for Q. 3. Analysis and discasion In order to illustrate the proposed procedure, we reanalyzed elastic scattering experiments with polarized protons in the fluctuation region which have recently been performed for 26 Mg(Ii, p°) by Glashausser et al . s) and for 88 Sr(~i, p°) and 9° Zr(~, p°) by Paetz gen. Schieck et al . 6). In table 1 the incident proton energy E~ Teel .e 1

Energy range EP, scattering angle B, step width of data taking s and energy resolution d of the experiments of Glashausser et al.') and Paetz gen. Schixk et al. °) Eo (MeV)

Realion :b

B,,e

s (keV)

d (keV)

50°, 60°, 70', 80°, 100° 120', 140', 160°

50

2030

Mg($ . Po)

5.4- 9.4 5.17.5

eesr(~ Po)

11 .9-13 .0

140', 160'

10

10

9oZli()3 .

12 .7-13.5

140'', 160'

10

10

Po)

') Ref.').

°) Ref.6).

the scattering angle B,,b, the step width of data taking s, and the experimental energy resolution d of these experiments are listed. The analysis of Q was performed in the conventional way. Because of gross structures in Q the original data were trend-reduced. This was accomplished by dividing each experimental cross-section point by a local, sliding energy average. The length of the averaging interval was determined according to the method suggested by Pappalardo ls), For the analysis of QA we applied a modified method of trend reduction which has also been used for the analysis of fluctuations in Za Si(~ d) and 28 Si(~, p) 29 Si [ref. 'Z)] : instead of dividing by a local energy average
DIRECT REACTION CONTRIBUTION

265

"unresolved" fluctuations. Since the experimental energy resolutioq d x 10 keV, is much larger than the estimated coherence width of about 200-300 eV [ref.' 8 )], the fluctuations get smeared out to structures of width z d . In this case the tight hand sides of formulas (8) and (12) have to be multiplied' e ) by a correction factor j(d/r) with j(x) = 2x - t arctan (x)-x -z ln (1 +xZ). (14)

Thus yo cannot be evaluated without a knowledge of r, whereas N~f f is obtained independently of this correction by dividing Q(a, QA) from eq. (12) by C(Q) from eq. (8). Therefore we used N°ff as the quantity to compare with HF calculations. The HF predictions for N~ff were calculated by Berg t 9 ) and are more or less identical for e8 Stü3, p0) and 9° Zr(13, p°): NiÉ(140°) = 2.00 and N"f(160°) = 1 .79 . These values have to be compared to the average values extracted from both experiments: N~~(140°) = 2.47 f0.5 and N~ff (160°) = 1 .62 f0.38 . The errors are rather large, mostly due to the finite range of the data; however, N~ff shows the correct trend and agrees within the error with NHf. In the experiment Z6 Mg((i, p°) [ref. s)] the fluctuations are resolved ; they have widths r between 30 to 80 keV depending on the energy region t 2, z°). [According to Roeders' 6 ), the values of C(Q) remain independent of the step width of datataking s, as long as s/I' < 2.] Therefore, N~ff and yp could be determined unambiguously from eqs. (8) and (12). We have plotted Neff in fig. 1 together with HF predictions for N°ff by Häusser et al. s°) at EP = 9.8 MeV (dots) and at EP = 12.6 MeV (open circles). These calculations include a correction for the finite range of the data eE= 5.G- 9.4MeY eE= 9.0-14.OMeV eE =13.0-17.SMeV

yD LO.90

e

0.8 0.7 -

a 30°

60°

90°

BLAB

120°

150°

180°

Fig . 1 . Comparison of experimentally extracted values of N,ff (triangles) to HF calculations =q at Ep = 9 .8 MeV (dots) and at EP = 12.6 MeV (open circles) .

°

0°

60°

0°

1 0°

1 0°

1 0° 9LAg

Fig . 2 . The direct reaction contribution yo for the ' 6 Mg(~, pu) reaction') as a function of the scattering angle for three different energy ranges dE.

266

R. HENNECK AND G. CRAW

effect of this experiment and are therefore larger than N,a= = 2 by about 10 ~. Hence for a comparison with N~ii~, these curves should be shifted down so that the value at B,,b = 90° is equal to Nm,x . Then our experimental values agree well with the HF curves. The value at B,,b = 160° was evaluated from an analysis over the whole energy range Ep = 5.17.5 MeV, whereas the values at the other angles represent an average N~~, extracted from analyses over subintervals of length x 4 MeV (see also fig. 2). Therefore for the value at 160° the error is much smaller and it clearly demonstrates the trend to lower N~rr values at angles close to 0° and 180°. It is especially in this angular region that the proposed method might be interesting for the extraction of yn : the approximation N~rr x NQx is definitely no longer valid and the reliability of NHf and the use of eq. (8) may be in question in special situations. Angular distributions of yo for the reaction Ze Mg(13, p°), extracted by means of this method, are plotted in fig. 2 for three different energy ranges dE. For the lowest energy range dE = 5.4-9 .4 MeV the direct reaction contribution decreases from x 0.95 at more forward angles to x 0.7 at the backward angles with a pronounced dip around 120°. The existence of a dip has been shown already by ref. Z°) and coincides with a minimum in the energy-averaged angular distribution of the differential cross section. With increasing energy the direct reaction contribution gets larger and the dip in yo gradually disappears. The authors are indebted to Drs . H. Paetz gen . Schieck and C. Glashausser for supplying us with their experimental data prior to publication. One of us (R .H .) acknowledges the hospitality of Duke University. Stimulating discussions with Drs. C. Glashauser, P. Moldauer, H. Paetz gen. Schieck and W. J. Thompson are appreciated. 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) IS) 16) 17) 18) 19) 20)

References

T. Ericson, Phys . Lett. 4 (1963) 258 T. Ericson, Ann. of Phys . 23 (1963) 390 D. M. Brink and R. O. Stephen, Phys . Left . S (1963) 77 T. Mayer-Kuckuck and T. Ericson, Ann. Rev. Nucl . Sci. 6B (1966) 183 C. Glashaussen, A. B. Robbins, E. Ventura, F. T. Baker, J. Eng and R. Kaita, Phys . Rev. Left . 35 (1975) 494; and private communication H. Paetz gen. Schieck, G. Berg, G. G. Ohlsen and C. E . Moss, Proc. 4th Symp. on polarization phenomena in nuclear reactions, Zürich 1975, p. 791 J. Ernst, H. L. Harney and K. Kotajima, Nucl . Phys . A136 (1969) 87 G. Dearnley, W. R. Gibbs, R. B. Leachman and P. V. Rogens, Phys . Rev. 139 (1965) 1170 G. C. Wick, E. P. Wigner and A. S. Wightman, Phys . Rev. 88 (1952) 101 W. J. Thompson, Phys . Lett . 25B (1967) 454 W. Kretschmer and G. Grave, Phys. Rev. Left . 27 (1971) 1294 R. Henpeck, G. Gnaw, T. B. Clegg, R. F. Haglund and E. J. Ludwig, Proc . 4th Symp . on polarization phenomena in nuclear reactions, Zürich 1975, p. 795; R. Henpeck, Ph .D. Thesis, Univ . of Erlangen-Nürnberg, 1976, to be published M . Lambent and G. Dumazet, Nucl . Phys . 83 (1966) 181 R. O. Stephen, Nucl . Phys . 70 (1965) 123 G. Pappalardo, Phys . Lett. 13 (1964) 320 J. D. A. Roeders, Ph .D . Thesis, Univ . of Groningen 1973 G. Berg, W. Kühn, H. Paetzgen. Schixk, K. Schulte and P. von Hrentano, Nucl . Phys. A2S4 (1975)169 P. Felsenden, W. R. Gibbs and R. B. Leachman, Phys . Rev. C3 (1971) 807 G. Berg, private communication O. Häuser, A. Richter, W. von Witsch and W. J. Thompeon, Nucl . Phys . A109 (1968) 329