A N N A L E N D E R PHYSIK 7. F O L G E * B A N D 21, H E F T 3-4 * 1968 Diffraction Scateering and Regge Pole Technique By S. A. EL-WAKILand A. A. KRESNIN~) With 9 Figures Abstract The diffraction scattering of spin zero nuclei was considered in Regge pole representation. The second pair of pole in the complex 2-plane was found to affect the small angle, scattering. This can be shown in the analysis of a-particle scattering on the nuclei Mgz4, Zns4 and S P . The scattering of two identical particle was also considered a t an energy above the Coulomb barrier. Satisfactory result was obtained. 1. Introduction The oscillatory structure of the elastic and inelastic cross-section was assumed t o be associated with the diffraction effects. BLAIR introduced the sharp cut off model [ l ] t o describe the &-particle elastic scattering but this model fails t o reproduce the cross-section behaviour a t large angle of scattering and near diffraction minima. A number of improvements were suggested by MCINTYRE et al.  and FRAHN et al.  which lead t o quite satisfactory results. The inelastic scattering has been investigated by the adiabatic approximation method ,in which the relation between the elastic and inelastic cross-section is easily understood . AUSTERNand BLAIR [S] introduced a new mathematical technique for the adiabatic distorted waves inelastic scattering of arbitrary multipolarity. Some suitable approximations make it possible t o reexpress the inelastic amplitudes in terms of derivatives of the partial wave elastic scattering function. I n the present work it is attempted t o investigate the elastic and inelastic scattering of spin zero nuclei with the aid of the complex angular momentum method . The contribution of the other two pair of poles nearest t o real axis was consider t o analysis the scattering of a-particles on the nuclei Mg24, Zns4 and S F , from which one can see that it affects the values of the cross-sections at small angle of scattering, also the method was used t o study the scattering of two identical nuclei 01s-016a t E = 15, and 8.8 MeV quite good agreement with experimental results was obtained. 2. Elastic Scattering The elastic scattering amplitude of a spinless projectile incident on a spin zero is: l) 8 On leave from Physics-Technical Institute, Kharkov, USSR. Ann. Phyaik. 7. Folge, Bd. 21 114 hna.len der Physik * 7. Folge * Band 21, Heft 3/4 * 1968 The scattering function S (1) can be written as : #(I) = ~ ( 1e2is(l) ) where 6 (1) is the phase shift which represents the effect of the nuclear potential on the scattering. The nuclear phase-shift vanishes for large 1-partial waves and increases as the 1-value decreases. With certain conditions on the analytical properties of q ( I ) in the complex 1-plane the amplitude (2.1) can be expressed by means of the WATSON-SOMMERFELD transform  as the spm of a background integral on the line A B of the contour shown in fig. 1and an expansion in terms of poles in the complex 1-plane. Using this transformation the amplitude f ( 6 ) becomes : i (2v 1) f ( e ) = ,kJmnG(i - x ( Y ) ) p V ( - COB e) av. (2.3) + I Fig. 1. Paths of integration in the v-plane The contribution of the background integral is small, it is of the order e-lola if one takes q (1) in the form of WOODS-SAXON shape i.e. (2.4) INOPIN et al.  calculate f ( 0 ) in the case of two pole approximation of ~ ( 1 ) a n d they had obtained the differential cross-section in the form : 8nR o(e) = k sin 0 l a @ )12e-zae[b2 + Cos2(kRe + y ) ] , where a(1) is the residue of ~ ( 1 at, ) 1 = lo b = sh(2 I m S ( 1 ) ) = s h ~ and , n a c 21, y = - +4- (2.5) + ia,R is the interaction radius, + arg a @ ) . (2.6) In the above calculation the LEGENDRE function P, (- cos 0 ) is replaced by its asymptotic form; for I Y I 1 that is: P+ 1015 and >> cos e) = f& cos((y + +) (Jc - e) - e 5 - 1;' a(1) = ja(1)I e i a r g @ ) . (2.7) Now, we proceed t o consider the effect of the other two pair of poles. I n this case the position of the poles are assumed t o be a t 1, = lo ial and 1, = 1; ia, + + S. A. EL-WAHIL and A. A. KRESNIN: Diffraction Scattering and Regge Pole Technique 115 + + such t h h lo 16 and a1 a2. Following the same way described in  the differential cross-section in the four poles representation is written as : The elastic scattering cross-section of identical particles of spin-zero can be expressed in terms of the scattering amplitude : where f ( 0 ) is defined by (2.1). I n the two pole representations the differential cross-section can be written as: y = kR8 = + y, ~ c ~-( e)n + y = y. n + arga(Z) + 5 2 10 (2.13) 3. Inelastic Scattering The scattering amplitude f o r single excitation of arbitrary multipole order (I)as was derived by AUSTERN and BLAIR  in the absence of the COULOMB 8* 116 Annalen der Physik field has the form: jlMr,OO(e) = + * * 7. Polge Band 21, Heft. 3/4 * 1968 + i p c,(I)5iz-q2r + (3.1) m(u, OO~EO)(I'I-M~~M~~EO)Y~~~(~,O) (21 x i)1/2 ail i n which the differential inelastic scattering cross-section is : o(@= 3 (3.2) /fIM,00(w/2> where (jlml, j2m, 1 j m ) is the CLEBSCH-GORDANcoefficient. I n the limit that I' I the sum over E in (3.1) can be done, and the result is: > jlHI.OO N (i) (21+ 1)1/2 c,m [I:JGIq (21' +1 ~ as (1') aF 2 yF'(e, 01, where , S(1')= q (2') e'WI') and laI .I [ I : M * ]= + + [ ( I - HI)! ( I M')!]1'2 (I ( I - HI)!! ( I HI)!! ' + M I ) even (3.3) 0 , (I MI) odd. C, ( I )is the reduced matrix element which depends on the details of the nuclear function as well as on the order of excitation (n) and on the final angular momentum I. For single excitation n = 1 + + C , ( I ) = ( 2 1 l)-l/ZR/31, (3.4) where is the deformation parameter. C, (I)can be obtained by comparing experimental and theoretical yalues of cross-section a t particular angle. This introduce an independent method for determining the nuclear deformation parameters. Now, the scattering amplitude in (3.1) is transformed into contour integration using SOMMERFELD-WATSON'S transformation as in the elastic scattering. The result of calculation in the case of two pole representation is : d e ) = 5 ( 2 1 + 1) [c,(0i2 - 2 0 [sin yjl sin v1 + (1+ b,)2 COS (yl- ~ , ) 1 ) , where a (E) is the residue of q (1) a t I , b, = sh (2 I m (I,)) = sh, y1 = la(4I2 = ,2 + ia,, (lo + 31) e - $(& + 1) + y11 y1= arga(E,) n + 2zoa + 4 and 1 v,=(zo+&++Y;j y,'=y,-I,. a (3.6) I n the four pole reprFsentation, we consider that we have two poles a t 1, = I, ia,, 1, = 1; ia2 and their complex conjugate, the cross-section in + + S. A. EL-WAKIL and A. A. KRESNIN: Diffraction Scattering end Regge Pole Technique 117 this representation is : + + 0 = 01 02 (3.7) where ( T ~ has , ~ the same form as in ( 3 . 5 ) with y,, 9, and b, as in (3.6). y2,yz are defined by : b, = sh (2 I m S(Z,)) = shx,. The interference term o3 has the form : (3.8) x cos 2(Re 6, - Re 6,) - [sin (y, - y 2 )sh x1 ch xz (3.9) - cos y1 sin y z sh (x, - x2)]sin 2 (Re 6, - Re 8 , ) ) . a&) and a (Z2) are the residues of 7 (I) a t I , and I,. If the nuclear potential phase shift S (1) is a smoothed function, we can write Re 6 (I,) = Re 6 (I2) and hence the interference term reduced t o : (3.10) 4. Discussion 4.1. The elastic scattering The comparison between experimentally observed elastic scattering angular distribution and equation (2.5), (2.12) gives us the possibility t o obtain all parameter of the model for the case of two poles. The effective interaction radius R and the phase shift y are obtained by matching the location of the maxima of eq. (2.5) and (2.12) t o the experimental angular distribution. Similarly, values of N, and b! can be obtained from the observed values of the - 83CR cross-section in the maxima and minima. The factor N = ( a( I ) can be k sin 0 obtained from the absolute value of the experimental cross-section a t the same angle. Comparison was made for elastic scattering of a-particle on the nuclei MgZ4, ZnB4and S@*,from which one can see that, the computed cross-sections are generally in quite good agreement with the experimental angular distributions for larger angles, Moreover, the relative depth of minima in the experimental angular distribution is not constant but rather strongly depends on the angle of scattering. It seems that t.aking into account the second pair of poles we introduce five additional parameters (R', a@,), arg a ( / & ,b,, a z ) . Actually we have only one independent parameter a2 as it is resonable t o suppose t h a t the values of R', a(Z) and arg a(Z) are determined by the proper- l2 118 Annalen der Physik * 7. Folge * Band 21, Heft 3/4 * 1968 ties of a given nucleus and are just the same for all poles. Also one can see that b2 is a dependent parameter, since b, = sh (2 Im S(Z,)) and we have approximately : + I m 6 (Z, iol) = Im [6 (2,) + iol - On the other hand, in the quasiclassical approximation: where 81, is the angle of scattering for particles with a n angular momentum lo. So we have : -xz_-- a2 XI 011 . Moreover, we have that the best agreement with the experimental angular distribution is obtained when the ratio is the same for all investigated nuclei. 10' 10 t def 1 10-1 10 20 30 40 50 60 -+eO Fig. 2. Angular distributions for elastic scattering of 41 MeV a-particles by MgZ4 20 30 40 50 60 70 80-ten Fig. 3. Angular distribution for elastic scattering of 21 MeV a-particles by ZnS4 The result of calculations is shown in fig. 2, 3, 4 for the case of two poles and 4 poles from which one can see that the next two poles improved the values of the cross-section a t small angles. S . A. EL-WAKILand A. A. KRESNIN:Diffraction Scattering and Regge Pole Technique 119 The expression (2.12) was also used to calculate the angular distribution for the elastic scattering of 0l6and 0l6at 8.8 MeV. where the MOTTpredictions  are good and a t the energy 15MeV., where the angular distribution show 10' 10 10-1 Fig. 4. Angular distribution for elastic scattering of 41 MeV &-particlesby Srs8 10 lo-* 20 40 30 50 8 O 60 70 80 -+ ------l I 16 16 0 to 7Y e ,$ 5 La \ E .C %I$ * 10' 10 M 50 70 90 ecm. 110 130 Fig. 5 . Elastic scattering angular distribution of 01* Ox6 a t = 8.8 MeV. The solid curve is the calculated curve. The spots is the experimental points + 'h 50 30 70 90 ll0 130 flcm. Fig. 6. Elastic scattering angular distribution of OI6 0 I 6 a t Ec.m. = 15 MeV. The solid curve is the calculated curve. The spots is the experimental points + 120 h n a l e n der Physik 8 7. Fo!ge * Band 21, Heft 3/4 * 1968 characteristic decrease in the cross-section below the MOTTprediction [lo]. The results of calculation given in fig. 5 , 6 a t both energies show satisfactory agreement with experimental results. 4.2. The inelastic scattering The inelastic scattering of &-particles on the nuclei Mg24, Z d 4 and Eras with the excitation of levels I = 2+ are considered. I n those calculation we have used just the same parameters as for the case of the elastic scattering. The parameter C,(I)which appears in eq. ( 3 . 5 ) and (3.7) can be obtained by comparing experimental and theoretical values of cross-section a t a definite angle. The result of calculation for both cases of two and foul poles are shown in figs. 7, 8 and 9. As in the case of the elastic scattering, taking into account the second 10 20 40 30 50 60 +en 20 Fig. 7. Angular distributions for inelastic scattering of 41 MeV a-particles by MgZ4 30 40 50 60 70 so-+eO Fig. 8. Angular distribution for inelastic scattering of 41 MeV a-particles by ZnB* pairs of poles improves the agreement between experimental and theoretical angular distributions especially for small angles. The values of the deformation parameter obtained are listed in table 1 from which one see t h a t they are in a close a,greement with the values obtained in ref. . Table 1 target Mg24 Zn64 Srn8 I 2 poles 1 1 1 0.254 0.18 0.041 4 poles 0.28 0.16 0.042 i ~ 1 ref.  0.24 0.19 0.08 S. A. EL-WARILand A. A. KRESNIN:Diffraction Scattering and Regge Pole Technique 121 1 t O(S) 10-1 10-2 10-8 Fig. 9. Angular distribution for inelastic scattering of 41 MeV &-particles by Sras 30 40 50 60 70 80 9O-f References [l] BLAIR,J. S., Phys. Rev. 95 (1954) 1218.  MCINTYRE, J. A., K. H. WANGand L. C. BEKER,Phys. Rev. 117 (1960) 1337. 131 FRAHN, W. E., and R. H. VENTER,Ann. Phys. (N.Y.) 24 (1963) 243. S. I.,J E T P (USSR) 28 (1955) 734, 736 [translated JETP 1 (1955) 588, 6911.  DROZDOV, INOPIN, E. V., JETP (USSR) 31 (1956) 901 [translated JETP 4 (1957) 7841. BLAIR,J. S., Phys. Rev. 115 (1959) 928. [ 5 ] BLAIR,J. S., et al., Nucl. Phys. 17 (1960) 614, 641.  BLAIR,J. S., and N. AUSTERN, Ann. Phys. (N.Y.) 33 (1965) 15.  INOPIN, E. V., and A. A. KRESNIN, J E T P (USRR) 48 (1965) 1620. A., partial differential equation in physics (Academic Press, New York  SOMMERBELD, (1942) p. 282.  MOTT, N. F., Proc. Roy. SOC.(London) A 126 (1930). [lo] BROMLEY, D. A., J. A. KTTERNER and E. ALMIQVIST, Phys. Rev. 123 (1961) 878. Cairo (U.A.R.),Atomic Energy Establishment. Bei der Redaktion eingegangen am 11.Juli 1967.