The excitation functions for inelastic $\alpha$-scattering leading to the low lying excited states in $^{24}$Mg and $^{28}$Si were measured at $\Theta_{LAB}$ = 170°, 175° and 179° in the LAB energy range 22.75–28.40 MeV. Statistical analysis of these excitation functions and those previously measured for elastic scattering was performed. The direct interaction contribution $\mathit{y}_{D}$ to the reaction studied was obtained from probability distributions of cross sections and from correlation coefficients. Cross correlation coefficients between different reaction channels were calculated

Bobrowska, A.

Budzanowski, A.

Grotowski, K.

Jarczyk, L.

Kamys, Bogusław

Kapuścik, A.

Płoskonka, J.

Strzałkowski, A.

Wróbel, Z.

Zastawniak, L.

Zybert, R.

Acta Physica Polonica B

English

Jagiellonian Univeristy Repository

Vol. B5 (1974) A C T A  PHYSICA POLONICA No 1STATISTICAL ANALYSIS OF EXCITATION FUNCTIONS FOR ELASTIC AND INELASTIC SCATTERING OF a-PARTICLES ONMg AND Si NUCLEIB y  A. B o b r o w s k a ,  A. B u d z a n o w s k i ,  K .  G r o t o w s k i ,  L. J a r c z y k ,  B . K a m y s ,A. K a p u ś c i k ,  J. P ł o s k o n k a ,  A. S t r z a ł k o w s k i ,  Z. W r ó b e l ,  L. Z a s t a w n i a ka n d  R . Z y b e r tInstitute o f Physics, Jagellonian University, CracowInstitute o f Nuclear Physics, Cracow*(Received July 27, 1973; Revised paper received October 1, 1973)The excitation functions for inelastic ^-scattering leading to the low lying excited  states in 24Mg and 28Si were measured at © l a b  =  170°, 175° and 179° in the LAB energy range 22.75-28.40 MeV. Statistical analysis o f these excitation functions and those pre­viously measured for elastic scattering was performed. The direct interaction contribution y o  to the reaction studied was obtained from probability distributions of cross sections and from correlation coefficients. Cross correlation coefficients between different reaction channels were calculated.1. IntroductionOur recent data on elastic scattering o f a-particles o f Mg and Si nuclei show a strong energy dependence o f the cross section measured at extremely backward angles (179°) in the energy range 23-28 MeV [1 ]. A similar structure o f the excitation functions was observed at smaller angles by other authors [2],The reaction mechanism responsible for the appearance o f resonance-like structures in the excitation curves is not yet well known. It therefore seems interesting to examine the energy dependence o f the cross section in other reaction channels.2. ExperimentIn the present experiment the excitation functions for inelastic a-scattering leading to the low lying excited states o f 24Mg and 28Si were measured at 0 LAB =  170°, 175°, and 179°. The energy resolution was good enough to separate the first and the sum o f the second and third excited states in 24Mg and 28Si.* Address: Instytut Fizyki Jądrowej. Radzikowskiego 152. 31-342 Kraków'. Poland.(125)126The measurements were performed in the energy range from 22.75 to 28.40 MeV  in steps o f about 250 keV, using a-particles from the 120 cm cyclotron o f the Institute of Nuclear Physics, Cracow. The energy spread o f the beam was about 200 keV. The experimental arrangement and procedure were the same as those described in [I],The results together with elastic data are presented in Fig. 1 and 2 for 24Mg and 28Si respectively. The absolute errors o f all cross sections are around 10%.Fig. 1. Excitation functions for the 24Mg(a, a')24Mg reactions. Dotted line indicates the averaged crosssection1273. AnalysisA rich resonance-like structure has been found in all excitation functions. It has been suggested by various authors that this kind o f structure may be connected with some intermediate resonances [3 ].In order to understand the reaction mechanism responsible for these structures it seems interesting to examine the energy dependence o f  the cross sections in terms o f151001052©lab-170°( OC , 0Co )■ t ' il\ \  \ VV\Y8Si ( « , < * ’ ) 28Si® LAB = 175°( or , oc0 )- \ l  KV II \i , i .© LA B  =179°(oC . oC0) ,f tV  \ 1V . . . A  ' ̂ ( oC , oC , )J kjV \ a{'A  . V " '1 1 ....■ y* »- 2̂ co / M/ M H ( <* • “A 2*3)* I 1]My H■ V(ot.oc2.3)x  ■' \ 4 /23 24 25 26 27 28 23 24 25 26 27 28 23 24 25 26 27 28E LAB(MeV)Fig. 2. Excitation functions for the 28Si(ot, a')28Si reactions. Dotted line indicates the averaged crosssection128Ericson’s theory o f cross section fluctuations and to separate in this way possible contribu­tions from nonstatistical effects.The simple fluctuation theory was developed for excitation functions with constant averaged cross section. In our case, however, this condition seems not to be fulfilled. In order to eliminate the nonstatistical modulations from the excitation functions theseELAB(MeV)Fig. 3. Reduced excitation functions for the 28Si(a, a ')28Si reactions at 179°functions were reduced according to the procedure given by Rubbino [4]. The averaged cross sections were evaluated by fitting the polynomial of the order 8 in the case o f 24Mg and 4 in the case o f  28Si to the data points. The averaged cross sections and examples o f  reduced excitation functions are shown in Figs 1, 2 and Fig. 3 respectively.129Fig. 4. Probability distributions o f the cross sections for the24M g(a, a')24M g reactionsP(y)130F ig. 5. Probability distributions o f  the cross sections for the 28Si(a, a ')28Si reactions131The experimental probability distributions P  ( ----- 1 were obtained for the groundstate, first excited state, and the sum o f the second and third excited states o f  both nuclei. In order to determine the relative contribution o f direct interaction y D these histograms were fitted by the following distribution function [5]:,  °Dwhere: y D =  7 ~ ,  y< »(T 2 NJ  y y D > A - 1 (,z) is the cylindrical Bessel function1 — J bof order N —l, and N  is the number o f independent reaction channels.The number o f independent reaction channels N  in our case was equal to one, due to the extreme backward angles o f our measurements and zero spins o f  a-particles and target nuclei [6]. However, the correction for the finite energy resolution AE had to be taken into account. According to Fessenden et al. [7], this correction is equivalent toAEincreasing the number o f independent channels by a factor o f n =  h i, where F mdi-nEcates the coherence width. In our case, assuming AE =  250 keV and r  x  80 keV [8], n was equal to 2.The histograms o f P  ( ----- ] and fitted theoretical probability distributions are pres-\ < ar> /ented in Figs 4 and 5. The values o f direct interaction contributions y D obtained in this way are shown in Table I.TABLE IValues o f direct inteiaction contributionsTargetn Angle Reae- \  stionOor- 175° 179’(«, «o) 0.73 0.78 0.67Mg 0.95 0.91 0.74(a, <*o) 0.85 0.89 0.84Si (a, *i) 0.95 0.94 0.91(cc, a 2+3) 0.96 0.91 0.90It is also possible to find the direct interaction contribution y D from the formula for the autocon elation coefficient [9]:C(0)<<r2( £ ) > - < u >2<°>2i - y lN¿iVz  =  --------132These coefficients were calculated for all measured excitation functions except for the sum o f the second and third excited states in 24Mg. The values o f  y D obtained from these coefficients are presented in Table II.TABLE IIValues o f  direct interaction contributionsIt can be seen that these values are in good agreement with those found from prob­ability distributions.The cross correlation coefficients between different reaction channels may, in principle, give us information about possible intermediate resonances. The normalized cross correla­tion coefficients:were calculated for all pairs o f  excitation functions. The values o f  these coefficients are shown in Table III.TABLE IIIValues o f cross correlation coefficientsAs can be seen from this table, the values o f  cross correlation coefficients are rather small, except for the ground and the sum of the second and third excited states in 28Si. These values are independent o f  the scattering angle.It should be remembered that results o f statistical calculations contain various kinds o f error. According to Ericson and Mayer-Kuckuk [6], the most important errors in a fluctuation analysis are produced by “finite-range deviations” (FRD). They are due to the finite energy range o f the sample from which our cross correlation coefficients were133calculated. The number o f statistically independent points n in the excitation function is given by the measured energy range I divided by F. The FR D  error is equal roughly toAs can be seen from Tables I and II direct interaction processes dominate in the back­ward elastic and inelastic scattering o f a-particles on 24Mg and 28Si nuclei in our energy range.Fig. 2 for 28Si demonstrates the existence o f some gross structure in the averaged elastic cross section.Most conspicuous is the maximum with a width o f about 2 MeV at the energy 26.5MeV. It may indicate the existence o f some intermediate structure. Indeed, it was shown in the analysis o f angular distributions measured in the energy range o f the maximum that the inclusion o f Regge Pole [10] or Breit-Wigner term [11] near or to the I =  10 partial wave, respectively, can improve the fits significantly.On the other hand, it was shown recently [12] that the maxima o f similar width in excitation curves at backward angles can be explained as interference effects between surface waves in scattering from a strongly absorbing potential.The obtained values o f cross correlation'coefficients are small in most cases, with the exception o f the C^aJt3(0) for 28Si. This indicates that most o f the observed fine structure has a statistical character.[1] R. C z a b a ń sk i, A . K a p u śc ik , S. M icek , J. P ło s k o n k a , J. S z m id e r , Z. W r ó b e l, L. Z a s ta w n ia k ,  R. Z y b e r t, IN P report N o  805/PL, Cracow 1972.[2] K. A. E b erh a rd , C. M a y e r -B ó r ic k e , Nuci. Phys., A142, 113 (1970); J. D . A. R o e d e r s , Thesis, Groningen 1971; J. L eg a , C. M a cq , J. Phys. (France), 32, 369 (1971); M. T a k e d a , S. K a to , T. Y a -  m a z a k i, J. Phys. Soc. Jap., 31, 625 (1971); A. W. O b st, K. W. K em p er , Phys. Rev., C6, 1705 (1972).[3] A. S. R in a t , Phys. L ett., 38B, 281 (1972); R. S to c k , G .G a u l,  R. S a n to , M. B erm a s, B. H a rw ey , D. H e n d r ie , J. M a h o n e y , J. S h erm a n , J. S te y a e r t , M. Z ism a n , Phys. Rev., C6, 1226 (1972).[4] A. R u b b in o , Nuovo Cimento, 54B, 37 (1968).[5] R. O. S te p h e n , Clarendon Laboratory Report, Oxford University 1963.[6] T. E r ic so n , T. M a y e r -K u c k u k , Ann. Rev. Nucl. Sci., 16, 183 (1966).[7] P. F e s se n d e n , W. R. G ib b s , R. B. L e a e h m a n , Phys. Rev. L ett., 15, 796 (1965).f8] J. D . A. R o e d e r s , Thesis, Groningen 1971; R. B. L e a eh m a n , P. F e s s e n d e n , W. R. G ib b s , Phys. Rev., C6, 1240 (1972); K. A. E b erh a rd , C. M a y e r -B ó r ic k e , Nuci. Phys., A142, 113 (1970).[9] T. E r ic so n , Ann. Phys., 23, 390 (1963).[10] B. K a m y s, private information.[11] A. B u d z a n o w s k i,  K. C h y ła , R. C z a b a ń sk i,  K. G r o to w s k i ,  L. J a r c z y k , B. K a m y s, A. K a ­p u śc ik , S. M ic ek , J. P ło s k o n k a , A. S tr z a łk o w s k i,  J. S zm id e r , Z. W r ó b e l, L. Z a s ta w n ia k ,  R. Z y b e r t, INP report N o  824/PL, Cracow 1972.[12] A. B u d z a n o w s k i, K. C h y ła , R. C z a b a ń sk i,  K. G r o to w sk i,  L. J a rczy k , B. K a m y s, A. K a­p u śc ik , S. M ic ek , J. P ło s k o n k a , A. S tr z a łk o w s k i,  J. S z m id e r , Z W r ó b e l, L. Z a s ta w n ia k ,  R. Z y b er t, Nuci. Phys., A211, 463 (1973).4. ConclusionsREFERENCES

Statistical analysis of excitation functions for elastic and inelastic scattering of $\alpha$-particles on Mg and Si nuclei

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Statistical analysis of excitation functions for elastic and inelastic scattering of $\alpha$ -particles on Mg and Si nuclei

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