- Email: [email protected]
Investigation of the Sn(p , p0) and (p, n) reaction at sub-coulomb energies
2.E : 2.L
Nuclear Physics A339 (1980) 13-22 ; © NortAfXoJloed Pwblirhinp Co., Arrvterrlorrr Na to be reproduced by photoprlnt or mian®1m whhout writtsn permission tiom the publishsr
INVESTIGATION OF THE Su(~, po) AND (p, u) REACTION AT SIJS-COULOMB ENERGIES W . DRENCKHAHN, A . FEIGEL, E FINCKH, G. GADEMANN, K . R(7SKAMP, M . WANGLER and L. ZEMLO f Physikalisches Institut der Uniuersü6t Erlangen-Nfs nberg, Erwin-Romure!-Str . 1, D-8S10 Erlangen, Germmry Rxeived 10 December 1979 Abstract : Hy analyzing the absorption cross-sxtion and polarization data of the even Sn isotopes, opticalmodel parameters for energies below the Coulomb barrier are deduced . In this way the depth and the diffuseneas of the imaginary potential can be determined separately, the diffuseness has to be reduced to a p = 0.4 fm, a value which is much smaller than usual . The volume integrals of the real and imaginary potential and a slight different .!-dependence of the nuclear radius are discussed .
E
NUCLEAR REACTIONS "6 " rrs . rso, r"" '~`Sn(p, n), E = 4-10 MeV ; measured rre, r~o, ruS~ Pob E = 7.75, 8 .80 MeV, 'sosn~ Po ), E = 6.77 MeV ; measured A(~ ; deduced optical-model parameters Enriched targets.
1. Introduction Optical-model parameters are deduced from many experimental data') at proton energies above the Coulomb barrier. Global fits to these data reproduce very well the differential cross section, the analyzing power, and the absorption cross section. For energies below the Coulomb barrier, only the absorption gross section and the analyzing power still contain information on the nuclear potential, the differential cross section is predominantly Rutil~erford scattering . In an investigation of the (p, n) cross section of the tin isotopes, Johnson and I{ernell Z) found a resonance-like behaviour of the cross section after dividing by the Coulomb transmission . It was interpreted as a p-wave resonance of the optical model. Further investigations 3-s) confirmed this resonance for all tin isotopes . From these data, optical-model parameters were obtained, which differ from the conventional values . The diffuseness of the imaginary potential had to be reduced to aD = 0.4 fm and the vohune integral of the imaginary potential was smaller than reported in the literature 6). Furthermore, the depth of the real potential and its energy dependence was rather large contradicting theoretical predictions'). The unusual values of optical-model parameters induced us, to investigate the f Visitor from Institute of Nuclear Reaearrh, Warsaw, Poland.
l3
14
W. DRENCKHAHN et al.
(p, n)
cross section and in addition the analyzing power at sub-Coulomb energies in order to use both values for determining optical-model parameters. In preliminary reports 3 . s) we still applied the large energy dependence of the real potential. In this analysis, we use the accepted energy dependence and discuss other possibilities to explain the results. 2. Experim~tal method For the determination of the (~i, po) analyzing power, the Erlangen EN tandem accelerator was used in conjunction with the Erlangen Lamb-shift ion source. The. beam was momentum analyzed and focused through a 2 mm diameter aperture. The intensity on the target was 20 nA. The polarization of the bèam, Ps = 0.79, was continuously monitored with a "He polarimeter. The targets were selfsupporting foils of tin isotopes t with thieknesses between 0.84 and 1 .82 mg/cmZ and enriched to at least92.5 %. The thickness was measured by the energy loss of 5.8 MeV a-particles with an accuracy of ~ 3 ~. The angular distribution of the analyzing power was measured in the angular range from 35° to 150° at 5° intervals . On each side of the target, four movable solid-state detectors tt were mounted in 25 cm distance and one detector as an additional monitor at 20°. The counting rates of the monitor detectors had a constant ratio to the beam integrator of the Faraday cup. The signals from the 10 detectors in the scattering chamber and the 2 detectors of the polarimeter were analyzed each in 512 channels by using a deadtime corrected routing system and a PDP 11/40 with a Camac interface . The analyzing power at energies below the Coulomb barrier is rather small and false asymmetries would strongly influence the results. Therefore, the detectors were always located symmetrically to the beam and the position of the beam was stabilized. In addition, the polarization was switched on and off with a frequency of 1 or 10 Hz by a magnetic field near the Sona transition iri the ion source. The field acts on the neutral beam so that the position does not change . During the switching time, the measurement was stopped for 150 ~s. For the calculation.of the analyzing power, the counting ratios spin up to unpolarized and spin down to unpolarized were used. With this set up, the systematic error in the analyzing power deduced from the scattering .of the data was about f 1 .6 x 10 -3 , which is slightly above the counting statistics . The (p, n) experiment was performed by using the Erlangen tandem accelerator with the conventional ion source . The beam was momentum analyzed and focused on an aperture with a 3 mm diameter. A second quadrupole lens refocused the beam on the target without any further beam defining apertures to roduce the neutron background. The beam position on the target was frequently checked r Delivered by Oak Ridge National Laboratory, Oak Ridge, Tennessoe. rr Manufactured by Ortec aad by Siemeae.
S~. Po)+ (P, n)
15
using a quartz beam-viewer. The beam was stopped in a Faraday cup which was shielded with boron-loaded parafï'm . The targets were the same as in the (p, po) experiment . The proton energies were chosen in such a way that isobaric analogue resonances are mainly excluded The neutrons were detected in three long counters at B = 55°, 90° and 125°. The counting rates at the forward and backward angles are proportional to the total neutron yield if its angular distribution is of the form A+B sine B. The counting rates of the throe counters were the same within the experimental errors. That means the neutron emission was nearly isotropic and the contributions from direct reactions negligible. The neutron background, determined by using an empty frame at the target position, was always leas than 5 %. The long counters were calibrated with an Am13e neutron source at the target position. The error in the absolute neutron yield is about f 6 ~ mainly due to the source calibration. The change of the detection efficiency by increasing neutron energies is in our case less than f2 ~. 3. Analysls The analyzing power of the Sn(~, po) reaction, A y(~ was calculated from the intensities measured on theleft and right side, L and R, with spin up and unpolarized, e.g. L + and Lo+ , and spin down and unpolarized, L_ and Lo _ and the beam polarization PB, A~9)
_ 1 PH
L + R_ Lo _ Ro+ _ 1 Lo+ Ro- L_ 1 R +
_ L+ R_ Lo_ Ro+ +1 Lo+ Ro_ L_ R+
The errors are calculated using the above formula and the counting statistics, uncertainties from background subtraction are negligible. The measured analyzing power or the product QA could be fitted with the JIP3 optical-model program. Usually only two or three parameters are varied simultaneously. The (p, n) data are compared with thecalculated absorption cross section because at energies below the Coulomb barrier and about 1 MeV above the neutron threshold the competing (p, p), (p, x), and (P, Y) cross sections are small i. The cross section rises exponentially in this energy range and the nuclear effxts are hidden . Therefore, a reduced cross section is introduced ~ by dividing the measured and calculated cross section by the transmission through Coulomb and angularf The ak;ulated (p, y) iras sectiom of Johmion et al. a) aie rather Iarge oampared to measured velues at neighbouring elements s) .
w. DRENCKHAHN
16
er al.
momentum barrier:
This reduced cross section corresponds to the neutron strength function, but here all partial waves are added. 4. Results The (p, n) cross section was measured from Ep = 4 MeV or from the threshold up to 10 MeV. The data are in good agreement with the results of Johnson et al. The analyzing power was determined at Ep = 7.75 and 8.8 MeV for the tin isotopes 116, 120, and 124 and at Ep = 6.77 MeV for tzzSn. Fig. 1 shows our measured (p, n) cross sections reduced for the Coulomb barrier penetration. The p-wave resonance, clearly shown for all isotopes, is characterized by three quantities : the energy of the maximum which changes from isotope to isotope systematically, the absolute value of the reduced cross section andthewidth of the resonance. The position of the maximummainly depends on the real potential, the absolute cross section and the width on the imaginary potential. Urad
0.40
0.35 F-
0.35 0.40 0.35 0.30 0. 35 0.30 0.25
o.zu Fig . 1 . (p, n) cross section for eveö tin isotopes. The data and the calculated curves are divided by the traasmission through the Coulomb barrier alone to show the nuclear effect, the p-wave sine resonance of the optical model. The curves are calculated with the parameters of table 1, last column. The arrows indicate the threshold energies for the (p, n) and the (p, 2n) reactions .
s~, Po), (P, n)
17
Including the data of Johnson et al. 1 °) for ' t 'Sn and t '9Sn, a systematic shift ofthe maximum of the reduced cross section is observed . The increase of the nl~lear radius proportional to A} does not fully rproduce this effect . Therefore, the energy dependence of the real potential was enlarged from b = 0.32 MeV- ' to about 0.9 MeV- ' [refs. Z' a. s)]. Since such an increase at low proton energies contradicts theoretical calculations') it was searched for other possibilities to explain these shifts . The maxima belong to different isotopes and another strength in the isospin dependence could also reproduce the data . But the value in front of the (N-Z)/A term has to go up from 24 MeV to 36 MeV. This value is much larger than the one obtained from optical-model analysis " " ' 2). We, therefore, propose the following explanation The usual A dependence of the radius R = roAs with x = i is extracted from a global analysis over a wide range of elements. From the analysis of muonic X-rays [ref. 's)], electron scattering 14) and isotopes shifts t s . t e) and of particle scattering for different tin isotopes t' " ta) it is known that neither the electromagnetic mean square radius, nor a mass dependent radius follows a strict roA }law in a sequence of isotopes . The electromagnetic radius of different tin isotopes has an exponent x = 0.18, from particle scattering x ~ 0.44 is deduced. It is expected, that in a (p, n) reaction the A dependence of radii bends to mass dependent radii and not to electromagnetic ones. The maximum of the reduced (p, n) cross section of the even and odd tin isotopes is reproduced by an A dependence RA = R, 2o(A/120)04' and R12o = 1 .17(120)x. Teat~ 1 Optical-model parameters
9
~
`)
Vo ro o, Wp rp
54.0 1 .17 0.75 11 .8-0.25E+ 12(N-Z)/ .! 1 .32
57.38 1 .17 0.75 8.207 1 .32
57.0 1 .17 0.75 10.4 1 .36
V,,, . r,A, g, ., . rc
6 .2 1 .01 0.75 1 .21
6.2 1 .01 0.75 1 .21
6.2 1 .01 0.75 1 .21
(I/A) j Vdr (1 /A) f Wdr
463 125
499 95
498 79
The u~ notation m rued, all potential depths are in MeV, all lettgtór in fm. The volume inte8ra4 are calculated for Eo ~ 6 MeV. The tall pooential is ;iven by the formula V = Vo -bE+0.4 ZlA'~'+ 24(N-Z)/.! . The energy dependence was chosen in all ara b ~ 0.32 MeV'' Ret "). h see text. 7 see text.
W. DRENCKHAHN et a1.
18 A(8) 0.008 0.006 0.004 0.002 0.000 -0 .002 -0 .004 -0 .006 -0 .008 -0 .010 -0 .012 -0 .014 -0 .016 -0 .018
120° 150° 90 00° 60° ° b
e q~
Fig. 2a. Anaiyzing power of the reaction `23 Sn(p, pu) measwed at Ep = 6.77 MeV. The curve is calculated with the parameters of Bexhetti et al., only the real and imaginary potential depths V and W are adjusted to give a good fit (table 1, column 2). u. .a 0.50 0.45 0.40 0.35 0.30 0.25 0 .20
Fig. 2b . Reduced cross saloon of the reaction 'x=Sn(p, n)'='Sb. The curve is calculated with the same parameters as in fig. 2a.
The imaginary potential not only influences the absolute value and the shape of the reaction cross section but also the analyzing power. With the global parameter set of Becchetti and Greenlees t 1) and by varying the potential depth of the real and imaginary potential (table 1, first and second rows), a good fit to the analyzing power is obtained (fig . Za). However, the calculated (p, n) cross section is too large (fig . 2b). By reducing the imaginary potential depth the shape of the resonance becomes smaller and the maximum value rises. The (p, n) cross section cannot be
Sn(p, Po)" (P. a)
19
AC6] 0. C106 0.006 0.004 0.002 0.000 -0 .002 -0 .004 -0 .008 -O .OOB -0 .010 -0 .012 -O.Ot4 -0 .018
-o.ole Fig. 3a . Analyzing power of the reaction '~'Sa(p, po) at E~ ~ 6.77 MeV. The curves are calculated with a constant product WDap, but various unlace of aa. [C~rve (a) as = 0.3 fm, (b) ap = 0.4 fm, (c) ap v 0.6 fm .] The other parameters are given in table 1, last toloma.
Fíg. 3b . Reduced cross auction of the reaction'='Sn(p, n)' ~'Sb. Thecurves arecalculated with a oonrtant product Waag, but various values of as [Ciuwe (a) ap = 0.6 fm, (b) as = 0.5 fm, (c) as e 0.4 fm, (d) aD ~ 0.3 fm .] The same parameters as in fig. 3a ene used.
fitted by a change of the depth WD alone, the diffuseness aD has to be reduced. Good agreement with all (p, n) data is obtained with the value ap = 0.4 fm. . The analyzing power determines only the product Wpap, but the (P, n) data are very sensitive whether ap (absolute cross suction) or Wp (shape of the ltsonance) is changed. This is shown explicitly in fig. 3a where the analyzing power is plotted for a constant value of W~ but different values of as For this example, all other
20
W. DRENCKHAHN et al. A(6] 0.06 0.04 0.02 0.00 -O.OZ -0 .04 -0 .06 30°
60° 80° 120° 150° 8cy iisSn~, Fig. 4, Analyzing power of the reaction puJ at Epb ~ 7.75 MeV. The curve is calculated with the parameters of table l, last column . AC6) 0.06 0.04 0.02 0 .00 -0 .02 -0 .04 -0 .06 00°
80°
90°
120°
160°
e q~
Fig. 5. Analyzing power of the reaction i 1sSn(~, pot at E~ = 8.8 MeV. The curve is calculated with the parameter of table 1, last column .
S~P. Pá.1P. n)
21
parameters are kept constant, the values are given in the third row of table 1 . With small additional changes in V and W, all curves would agree with the data points equally well. The (P, n) cross section (fig. 3b) does not show this ambiguity and from both measurements ap and Wp are determined In the energy range analyzed, EP = 4-10 MeV, no energy dependence of aD and WD was found. So it is not clear how the small value of ap and of the volume integral of the imaginary potential smoothly turns into the known values at higher. energies. Further, the radius of the imaginary potential, rp, was increased by 3 ~, shifting the reduced cross section to somewhat higher values without influencing the shape of the cross suction . Our absolute calibration of the (P, n) cross section is about 4 % higher than the values of Johnson et al. That is, within the experimental errors and limits the accuracy for determining rp. Using these parameters ; all Sn(p, n) cross section (fig . 1) and the analyzing power measured at Ep = 6.77 MeV (fig . 3a), 7.75 MeV (fig. 4) and 8.8 MeV (fig . 5) are well reproduced . The measurement at 9.8 MeV [ref. '~] is in good agreement with the calculation, too. In a recent investigation s°) of the 'i6 . izoSn(p. po) scattering an energy dependence of the real potential, b = 1 .13 (MeV) - ' was deduced and the geometry parameters of Perey Zi) gave better fits to the data than the ones of Beochetti and Greenlees. Probably the most important difference in the two sets is the small düiuseness an = 0.47 in the set of Perey. Using our parameters (third column in table 1) the calculated dif%rential cross sections are in good agreement with these data . S. Condoeion
This investigation has shown that measurements of (p, n) cross suction or analyzing power alone are not sufficient to extract uniquely optical-model parameters in the sub-Coulomb region. Hut the good agreement between experiment and calculation found in our analysis by using both measurements together, still does not seem satisfying . Thechanges in the depth of the real and imaginary potential are not unusual, the reduction in the diffuseness of the imaginary potential is rather unexpected . The strong energy dependence of the real potential in older analyses can be avoidal. by introducing a slightly different A-dependence of the radius in a row of isotopes, which is consistent with other measurements. The volume integral of the real potential is higher than the values otherwise used. That seems to appear also for other elemènts at sub-Coulomb energies zs-z4). For the imaginary potential, the volume integral is strongly reduced due to the small diffuseness. This is probably an exception for the magic tin nucleus. Since for silver s3) and rhodium 2') the volume integral is even above the average value in spite of the small diffuseness au = 0.4 fm. More systematic studies are necessary bore defining conclusions about the volume integrals and the imaginary diffuseness can be drawn. We thank H. M. Koenner, R Kumpf, and K.. H. Uebel for their help in the measurements and Miss H. Rowedder for therpatience during the many calculations .
22
W. DRENCKHAHN er al.
References 1) C. M. Percy and F. G. Perey, Atomic Data and Nucl . Data Tables 13 (1974) 293 2) C. H. Johnson and R. L. Kernen, Phys . Rev. C2 (1970) 639 3) W. Drenckhahn, A. Feigel, E Finckh, R. Kempf, M. Koenner, P. Krümmer and K. H. Uebel, Proc. Fourth Int. Symp . on polarization phenomena in nuclear reactions, Zûrich 1975, al . W. Graebkr and V. König (Birkhâuaer, Basel, 1976) p. 613 4) C. H. Johnson, J. K. Bair, C. M. Jones, S. K. Penny and D. W. Smith, Phys. Rev. C15 (1977) I% 5) W. Drenckhabn, A. Fcigel, E. Finckh, G. Gademann, K. Raskamp and M. Wangkr, in Microscopic optical potentials, ed. H. V. ven Geramb, Lectures Noten in Physics, vol. 89 (Springer, Berlin, 1979) p. 225 ~ S. Kaden and S. K. Gupta, Phys . Left . 71B (1977) 271 7) C. Mahaux, in Microscopic optical potentials, ed . H. V. ven Geramb, Lecture Notes in Physics, vol. 89 (Springer, Berlin, 1979) p. 1 8) P. J. Daly and P. F. D. Shaw, Nucl . Phys. 56 (1964) 322 9) J. P. Schif%r and L. L. Lee Jr., Phys. Rev. 109 (1958) 2098 10) C. H. Johnson, A Galonaky, R. L. Kernen, Phys. Rev. Lett. 39 (1977) 1604 II) F. D. Bachetti and G. W. Greenlees, Phys. Rev. 18Z (1969) 1190 12) S. Kailas and S. K. Gupta, Phys. Rev. C17 (1978) 2236 13) R. Enger, H. Schneuwly, J. L. Vuilleumier, H. K. Walter and A. Zehnder, Atomic Data and Nucl . Data Tabka 14 (1974) 509 14) C. W. de Jager, H. de Vries and C. de Vries, Atomic Data and Nucl . Data Tabla 14 (1974) 479 15) F. Boehm and P. L. Lee, Atomic Data and Nucl . Data Tables 14 (1974) 605 16) K. Heilig and A. Steudel, Atomic Data and Nucl . Data Tables 14 (1974) 613 17) R. N. Boyd, J. Fenton, M. Williams and T. Kruse, Nucl . Phys . A162 (1971) 497 18) S. L: Tabor, B. A. Watson and S. S. Hanna, Phys . Rev. Cll (197 198 19) G. W. Greenkes, C. H. Poppe, J. A. Sievers and D. L. Watson, Phys . Rev. C3 (1971) 1231 20) B. Gyatmati, T. Vertse, L. Zolnai, A. I. Baryshnikov, A. F. Gurbich, N. N. Titararko end E. L. Yadrovsky, J. of Phys. C9 (1979) 1225 21) F. G. Perey, Phys . Rev. 131 (1%3) 745 22) B. Badum, Zulassungsarbeit, Univ . Erlangen-Narnbcrg (1979) 23) A. Feigel, E. Finckh, K. Raskamp and U. Weise, to be published 24) W. Kretschmer, H. Lóh, K. Spitzer and W. Stack, Phys. Lett. B (1979) to be published 25) H. Rowdder, Zulass~mgsarbeít, Unív. Erlangen-NÛmbag (1979)
Nuclear Physics A339 (1980) 13-22 ; © NortAfXoJloed Pwblirhinp Co., Arrvterrlorrr Na to be reproduced by photoprlnt or mian®1m whhout writtsn permission tiom the publishsr
INVESTIGATION OF THE Su(~, po) AND (p, u) REACTION AT SIJS-COULOMB ENERGIES W . DRENCKHAHN, A . FEIGEL, E FINCKH, G. GADEMANN, K . R(7SKAMP, M . WANGLER and L. ZEMLO f Physikalisches Institut der Uniuersü6t Erlangen-Nfs nberg, Erwin-Romure!-Str . 1, D-8S10 Erlangen, Germmry Rxeived 10 December 1979 Abstract : Hy analyzing the absorption cross-sxtion and polarization data of the even Sn isotopes, opticalmodel parameters for energies below the Coulomb barrier are deduced . In this way the depth and the diffuseneas of the imaginary potential can be determined separately, the diffuseness has to be reduced to a p = 0.4 fm, a value which is much smaller than usual . The volume integrals of the real and imaginary potential and a slight different .!-dependence of the nuclear radius are discussed .
E
NUCLEAR REACTIONS "6 " rrs . rso, r"" '~`Sn(p, n), E = 4-10 MeV ; measured rre, r~o, ruS~ Pob E = 7.75, 8 .80 MeV, 'sosn~ Po ), E = 6.77 MeV ; measured A(~ ; deduced optical-model parameters Enriched targets.
1. Introduction Optical-model parameters are deduced from many experimental data') at proton energies above the Coulomb barrier. Global fits to these data reproduce very well the differential cross section, the analyzing power, and the absorption cross section. For energies below the Coulomb barrier, only the absorption gross section and the analyzing power still contain information on the nuclear potential, the differential cross section is predominantly Rutil~erford scattering . In an investigation of the (p, n) cross section of the tin isotopes, Johnson and I{ernell Z) found a resonance-like behaviour of the cross section after dividing by the Coulomb transmission . It was interpreted as a p-wave resonance of the optical model. Further investigations 3-s) confirmed this resonance for all tin isotopes . From these data, optical-model parameters were obtained, which differ from the conventional values . The diffuseness of the imaginary potential had to be reduced to aD = 0.4 fm and the vohune integral of the imaginary potential was smaller than reported in the literature 6). Furthermore, the depth of the real potential and its energy dependence was rather large contradicting theoretical predictions'). The unusual values of optical-model parameters induced us, to investigate the f Visitor from Institute of Nuclear Reaearrh, Warsaw, Poland.
l3
14
W. DRENCKHAHN et al.
(p, n)
cross section and in addition the analyzing power at sub-Coulomb energies in order to use both values for determining optical-model parameters. In preliminary reports 3 . s) we still applied the large energy dependence of the real potential. In this analysis, we use the accepted energy dependence and discuss other possibilities to explain the results. 2. Experim~tal method For the determination of the (~i, po) analyzing power, the Erlangen EN tandem accelerator was used in conjunction with the Erlangen Lamb-shift ion source. The. beam was momentum analyzed and focused through a 2 mm diameter aperture. The intensity on the target was 20 nA. The polarization of the bèam, Ps = 0.79, was continuously monitored with a "He polarimeter. The targets were selfsupporting foils of tin isotopes t with thieknesses between 0.84 and 1 .82 mg/cmZ and enriched to at least92.5 %. The thickness was measured by the energy loss of 5.8 MeV a-particles with an accuracy of ~ 3 ~. The angular distribution of the analyzing power was measured in the angular range from 35° to 150° at 5° intervals . On each side of the target, four movable solid-state detectors tt were mounted in 25 cm distance and one detector as an additional monitor at 20°. The counting rates of the monitor detectors had a constant ratio to the beam integrator of the Faraday cup. The signals from the 10 detectors in the scattering chamber and the 2 detectors of the polarimeter were analyzed each in 512 channels by using a deadtime corrected routing system and a PDP 11/40 with a Camac interface . The analyzing power at energies below the Coulomb barrier is rather small and false asymmetries would strongly influence the results. Therefore, the detectors were always located symmetrically to the beam and the position of the beam was stabilized. In addition, the polarization was switched on and off with a frequency of 1 or 10 Hz by a magnetic field near the Sona transition iri the ion source. The field acts on the neutral beam so that the position does not change . During the switching time, the measurement was stopped for 150 ~s. For the calculation.of the analyzing power, the counting ratios spin up to unpolarized and spin down to unpolarized were used. With this set up, the systematic error in the analyzing power deduced from the scattering .of the data was about f 1 .6 x 10 -3 , which is slightly above the counting statistics . The (p, n) experiment was performed by using the Erlangen tandem accelerator with the conventional ion source . The beam was momentum analyzed and focused on an aperture with a 3 mm diameter. A second quadrupole lens refocused the beam on the target without any further beam defining apertures to roduce the neutron background. The beam position on the target was frequently checked r Delivered by Oak Ridge National Laboratory, Oak Ridge, Tennessoe. rr Manufactured by Ortec aad by Siemeae.
S~. Po)+ (P, n)
15
using a quartz beam-viewer. The beam was stopped in a Faraday cup which was shielded with boron-loaded parafï'm . The targets were the same as in the (p, po) experiment . The proton energies were chosen in such a way that isobaric analogue resonances are mainly excluded The neutrons were detected in three long counters at B = 55°, 90° and 125°. The counting rates at the forward and backward angles are proportional to the total neutron yield if its angular distribution is of the form A+B sine B. The counting rates of the throe counters were the same within the experimental errors. That means the neutron emission was nearly isotropic and the contributions from direct reactions negligible. The neutron background, determined by using an empty frame at the target position, was always leas than 5 %. The long counters were calibrated with an Am13e neutron source at the target position. The error in the absolute neutron yield is about f 6 ~ mainly due to the source calibration. The change of the detection efficiency by increasing neutron energies is in our case less than f2 ~. 3. Analysls The analyzing power of the Sn(~, po) reaction, A y(~ was calculated from the intensities measured on theleft and right side, L and R, with spin up and unpolarized, e.g. L + and Lo+ , and spin down and unpolarized, L_ and Lo _ and the beam polarization PB, A~9)
_ 1 PH
L + R_ Lo _ Ro+ _ 1 Lo+ Ro- L_ 1 R +
_ L+ R_ Lo_ Ro+ +1 Lo+ Ro_ L_ R+
The errors are calculated using the above formula and the counting statistics, uncertainties from background subtraction are negligible. The measured analyzing power or the product QA could be fitted with the JIP3 optical-model program. Usually only two or three parameters are varied simultaneously. The (p, n) data are compared with thecalculated absorption cross section because at energies below the Coulomb barrier and about 1 MeV above the neutron threshold the competing (p, p), (p, x), and (P, Y) cross sections are small i. The cross section rises exponentially in this energy range and the nuclear effxts are hidden . Therefore, a reduced cross section is introduced ~ by dividing the measured and calculated cross section by the transmission through Coulomb and angularf The ak;ulated (p, y) iras sectiom of Johmion et al. a) aie rather Iarge oampared to measured velues at neighbouring elements s) .
w. DRENCKHAHN
16
er al.
momentum barrier:
This reduced cross section corresponds to the neutron strength function, but here all partial waves are added. 4. Results The (p, n) cross section was measured from Ep = 4 MeV or from the threshold up to 10 MeV. The data are in good agreement with the results of Johnson et al. The analyzing power was determined at Ep = 7.75 and 8.8 MeV for the tin isotopes 116, 120, and 124 and at Ep = 6.77 MeV for tzzSn. Fig. 1 shows our measured (p, n) cross sections reduced for the Coulomb barrier penetration. The p-wave resonance, clearly shown for all isotopes, is characterized by three quantities : the energy of the maximum which changes from isotope to isotope systematically, the absolute value of the reduced cross section andthewidth of the resonance. The position of the maximummainly depends on the real potential, the absolute cross section and the width on the imaginary potential. Urad
0.40
0.35 F-
0.35 0.40 0.35 0.30 0. 35 0.30 0.25
o.zu Fig . 1 . (p, n) cross section for eveö tin isotopes. The data and the calculated curves are divided by the traasmission through the Coulomb barrier alone to show the nuclear effect, the p-wave sine resonance of the optical model. The curves are calculated with the parameters of table 1, last column. The arrows indicate the threshold energies for the (p, n) and the (p, 2n) reactions .
s~, Po), (P, n)
17
Including the data of Johnson et al. 1 °) for ' t 'Sn and t '9Sn, a systematic shift ofthe maximum of the reduced cross section is observed . The increase of the nl~lear radius proportional to A} does not fully rproduce this effect . Therefore, the energy dependence of the real potential was enlarged from b = 0.32 MeV- ' to about 0.9 MeV- ' [refs. Z' a. s)]. Since such an increase at low proton energies contradicts theoretical calculations') it was searched for other possibilities to explain these shifts . The maxima belong to different isotopes and another strength in the isospin dependence could also reproduce the data . But the value in front of the (N-Z)/A term has to go up from 24 MeV to 36 MeV. This value is much larger than the one obtained from optical-model analysis " " ' 2). We, therefore, propose the following explanation The usual A dependence of the radius R = roAs with x = i is extracted from a global analysis over a wide range of elements. From the analysis of muonic X-rays [ref. 's)], electron scattering 14) and isotopes shifts t s . t e) and of particle scattering for different tin isotopes t' " ta) it is known that neither the electromagnetic mean square radius, nor a mass dependent radius follows a strict roA }law in a sequence of isotopes . The electromagnetic radius of different tin isotopes has an exponent x = 0.18, from particle scattering x ~ 0.44 is deduced. It is expected, that in a (p, n) reaction the A dependence of radii bends to mass dependent radii and not to electromagnetic ones. The maximum of the reduced (p, n) cross section of the even and odd tin isotopes is reproduced by an A dependence RA = R, 2o(A/120)04' and R12o = 1 .17(120)x. Teat~ 1 Optical-model parameters
9
~
`)
Vo ro o, Wp rp
54.0 1 .17 0.75 11 .8-0.25E+ 12(N-Z)/ .! 1 .32
57.38 1 .17 0.75 8.207 1 .32
57.0 1 .17 0.75 10.4 1 .36
V,,, . r,A, g, ., . rc
6 .2 1 .01 0.75 1 .21
6.2 1 .01 0.75 1 .21
6.2 1 .01 0.75 1 .21
(I/A) j Vdr (1 /A) f Wdr
463 125
499 95
498 79
The u~ notation m rued, all potential depths are in MeV, all lettgtór in fm. The volume inte8ra4 are calculated for Eo ~ 6 MeV. The tall pooential is ;iven by the formula V = Vo -bE+0.4 ZlA'~'+ 24(N-Z)/.! . The energy dependence was chosen in all ara b ~ 0.32 MeV'' Ret "). h see text. 7 see text.
W. DRENCKHAHN et a1.
18 A(8) 0.008 0.006 0.004 0.002 0.000 -0 .002 -0 .004 -0 .006 -0 .008 -0 .010 -0 .012 -0 .014 -0 .016 -0 .018
120° 150° 90 00° 60° ° b
e q~
Fig. 2a. Anaiyzing power of the reaction `23 Sn(p, pu) measwed at Ep = 6.77 MeV. The curve is calculated with the parameters of Bexhetti et al., only the real and imaginary potential depths V and W are adjusted to give a good fit (table 1, column 2). u. .a 0.50 0.45 0.40 0.35 0.30 0.25 0 .20
Fig. 2b . Reduced cross saloon of the reaction 'x=Sn(p, n)'='Sb. The curve is calculated with the same parameters as in fig. 2a.
The imaginary potential not only influences the absolute value and the shape of the reaction cross section but also the analyzing power. With the global parameter set of Becchetti and Greenlees t 1) and by varying the potential depth of the real and imaginary potential (table 1, first and second rows), a good fit to the analyzing power is obtained (fig . Za). However, the calculated (p, n) cross section is too large (fig . 2b). By reducing the imaginary potential depth the shape of the resonance becomes smaller and the maximum value rises. The (p, n) cross section cannot be
Sn(p, Po)" (P. a)
19
AC6] 0. C106 0.006 0.004 0.002 0.000 -0 .002 -0 .004 -0 .008 -O .OOB -0 .010 -0 .012 -O.Ot4 -0 .018
-o.ole Fig. 3a . Analyzing power of the reaction '~'Sa(p, po) at E~ ~ 6.77 MeV. The curves are calculated with a constant product WDap, but various unlace of aa. [C~rve (a) as = 0.3 fm, (b) ap = 0.4 fm, (c) ap v 0.6 fm .] The other parameters are given in table 1, last toloma.
Fíg. 3b . Reduced cross auction of the reaction'='Sn(p, n)' ~'Sb. Thecurves arecalculated with a oonrtant product Waag, but various values of as [Ciuwe (a) ap = 0.6 fm, (b) as = 0.5 fm, (c) as e 0.4 fm, (d) aD ~ 0.3 fm .] The same parameters as in fig. 3a ene used.
fitted by a change of the depth WD alone, the diffuseness aD has to be reduced. Good agreement with all (p, n) data is obtained with the value ap = 0.4 fm. . The analyzing power determines only the product Wpap, but the (P, n) data are very sensitive whether ap (absolute cross suction) or Wp (shape of the ltsonance) is changed. This is shown explicitly in fig. 3a where the analyzing power is plotted for a constant value of W~ but different values of as For this example, all other
20
W. DRENCKHAHN et al. A(6] 0.06 0.04 0.02 0.00 -O.OZ -0 .04 -0 .06 30°
60° 80° 120° 150° 8cy iisSn~, Fig. 4, Analyzing power of the reaction puJ at Epb ~ 7.75 MeV. The curve is calculated with the parameters of table l, last column . AC6) 0.06 0.04 0.02 0 .00 -0 .02 -0 .04 -0 .06 00°
80°
90°
120°
160°
e q~
Fig. 5. Analyzing power of the reaction i 1sSn(~, pot at E~ = 8.8 MeV. The curve is calculated with the parameter of table 1, last column .
S~P. Pá.1P. n)
21
parameters are kept constant, the values are given in the third row of table 1 . With small additional changes in V and W, all curves would agree with the data points equally well. The (P, n) cross section (fig. 3b) does not show this ambiguity and from both measurements ap and Wp are determined In the energy range analyzed, EP = 4-10 MeV, no energy dependence of aD and WD was found. So it is not clear how the small value of ap and of the volume integral of the imaginary potential smoothly turns into the known values at higher. energies. Further, the radius of the imaginary potential, rp, was increased by 3 ~, shifting the reduced cross section to somewhat higher values without influencing the shape of the cross suction . Our absolute calibration of the (P, n) cross section is about 4 % higher than the values of Johnson et al. That is, within the experimental errors and limits the accuracy for determining rp. Using these parameters ; all Sn(p, n) cross section (fig . 1) and the analyzing power measured at Ep = 6.77 MeV (fig . 3a), 7.75 MeV (fig. 4) and 8.8 MeV (fig . 5) are well reproduced . The measurement at 9.8 MeV [ref. '~] is in good agreement with the calculation, too. In a recent investigation s°) of the 'i6 . izoSn(p. po) scattering an energy dependence of the real potential, b = 1 .13 (MeV) - ' was deduced and the geometry parameters of Perey Zi) gave better fits to the data than the ones of Beochetti and Greenlees. Probably the most important difference in the two sets is the small düiuseness an = 0.47 in the set of Perey. Using our parameters (third column in table 1) the calculated dif%rential cross sections are in good agreement with these data . S. Condoeion
This investigation has shown that measurements of (p, n) cross suction or analyzing power alone are not sufficient to extract uniquely optical-model parameters in the sub-Coulomb region. Hut the good agreement between experiment and calculation found in our analysis by using both measurements together, still does not seem satisfying . Thechanges in the depth of the real and imaginary potential are not unusual, the reduction in the diffuseness of the imaginary potential is rather unexpected . The strong energy dependence of the real potential in older analyses can be avoidal. by introducing a slightly different A-dependence of the radius in a row of isotopes, which is consistent with other measurements. The volume integral of the real potential is higher than the values otherwise used. That seems to appear also for other elemènts at sub-Coulomb energies zs-z4). For the imaginary potential, the volume integral is strongly reduced due to the small diffuseness. This is probably an exception for the magic tin nucleus. Since for silver s3) and rhodium 2') the volume integral is even above the average value in spite of the small diffuseness au = 0.4 fm. More systematic studies are necessary bore defining conclusions about the volume integrals and the imaginary diffuseness can be drawn. We thank H. M. Koenner, R Kumpf, and K.. H. Uebel for their help in the measurements and Miss H. Rowedder for therpatience during the many calculations .
22
W. DRENCKHAHN er al.
References 1) C. M. Percy and F. G. Perey, Atomic Data and Nucl . Data Tables 13 (1974) 293 2) C. H. Johnson and R. L. Kernen, Phys . Rev. C2 (1970) 639 3) W. Drenckhahn, A. Feigel, E Finckh, R. Kempf, M. Koenner, P. Krümmer and K. H. Uebel, Proc. Fourth Int. Symp . on polarization phenomena in nuclear reactions, Zûrich 1975, al . W. Graebkr and V. König (Birkhâuaer, Basel, 1976) p. 613 4) C. H. Johnson, J. K. Bair, C. M. Jones, S. K. Penny and D. W. Smith, Phys. Rev. C15 (1977) I% 5) W. Drenckhabn, A. Fcigel, E. Finckh, G. Gademann, K. Raskamp and M. Wangkr, in Microscopic optical potentials, ed. H. V. ven Geramb, Lectures Noten in Physics, vol. 89 (Springer, Berlin, 1979) p. 225 ~ S. Kaden and S. K. Gupta, Phys . Left . 71B (1977) 271 7) C. Mahaux, in Microscopic optical potentials, ed . H. V. ven Geramb, Lecture Notes in Physics, vol. 89 (Springer, Berlin, 1979) p. 1 8) P. J. Daly and P. F. D. Shaw, Nucl . Phys. 56 (1964) 322 9) J. P. Schif%r and L. L. Lee Jr., Phys. Rev. 109 (1958) 2098 10) C. H. Johnson, A Galonaky, R. L. Kernen, Phys. Rev. Lett. 39 (1977) 1604 II) F. D. Bachetti and G. W. Greenlees, Phys. Rev. 18Z (1969) 1190 12) S. Kailas and S. K. Gupta, Phys. Rev. C17 (1978) 2236 13) R. Enger, H. Schneuwly, J. L. Vuilleumier, H. K. Walter and A. Zehnder, Atomic Data and Nucl . Data Tabka 14 (1974) 509 14) C. W. de Jager, H. de Vries and C. de Vries, Atomic Data and Nucl . Data Tabla 14 (1974) 479 15) F. Boehm and P. L. Lee, Atomic Data and Nucl . Data Tables 14 (1974) 605 16) K. Heilig and A. Steudel, Atomic Data and Nucl . Data Tables 14 (1974) 613 17) R. N. Boyd, J. Fenton, M. Williams and T. Kruse, Nucl . Phys . A162 (1971) 497 18) S. L: Tabor, B. A. Watson and S. S. Hanna, Phys . Rev. Cll (197 198 19) G. W. Greenkes, C. H. Poppe, J. A. Sievers and D. L. Watson, Phys . Rev. C3 (1971) 1231 20) B. Gyatmati, T. Vertse, L. Zolnai, A. I. Baryshnikov, A. F. Gurbich, N. N. Titararko end E. L. Yadrovsky, J. of Phys. C9 (1979) 1225 21) F. G. Perey, Phys . Rev. 131 (1%3) 745 22) B. Badum, Zulassungsarbeit, Univ . Erlangen-Narnbcrg (1979) 23) A. Feigel, E. Finckh, K. Raskamp and U. Weise, to be published 24) W. Kretschmer, H. Lóh, K. Spitzer and W. Stack, Phys. Lett. B (1979) to be published 25) H. Rowdder, Zulass~mgsarbeít, Unív. Erlangen-NÛmbag (1979)