< > SA I OFT ORNL P 2074 i : : TEEEEEEE EEEE 4:25 144 1.6 MICROCOPY RESOLUTION TEST CHART NATIONAL BUREAU OF STANDARDS - 1963 ORN-p-2014 CONF-660306-6., CESTI PRICES H.C: 51. MN.5.0 DIRECT SOLUTION OF LANE EQUATION AND THE ANALOGUE STATE 566 RESONANCE IN (p,p) PROCESS RELEASED POR ANNOUNCEMENT MAY 5 Turo Tamura IN NUCLEAR SCIENCE ABSTRACTS Oak Ridge National Laboratory, Oak Ridge, Tennessee MASTER In order to explain the peaks in the spectrum of neutrons from (p,n) reactions, Lane? intoduced a new term of the forma: U (t•T), (1) into the usual optical model potential. Robson formulated an R-matrix theory by taking into account (1), not explicitly, but somewhat implicitly, and achieved a beautiful success in explaining the experimental excitation fur:ctions around the analogue state resonances in the (p,p),(p,n) and other cross sections." In particular the observed asymmetry of the peaks in the excitation functions was very well accounted for. If (1) is considered explicitly in an optical model calculation, it zi.ves rise to a set of coupled equations between the proton (p) and neutron (n) channels. If the proton energy is sufficiently low, the corresponding "Research sponsored by the U. S. Atomic Energy Commission under contract with the Union Carbide Corporation. J. D. Anderson and C. Wong, Phys. Rev. Letters 7, 250 (1961); 8 442 (1962); J. D. Anderson, c. Wong and J. W. McClure, Phys. Rev. 126, 2170 (1962); 129, 2718 (1963). 2A. M. Lane, Phys. Rev. Letters 8, 171 (1962); Nucl. Phys. 35, 676 (1962). 3D. Robson, Phys. Rev. 137, B505 (1965). *J. D. Fox, C. F. Moore and D. Robson, Phys. Rev. Letters 12, 198 (1964); P. Richards, C. F. Moore, D. Robson and J. D. Fox, ibid 14, 343 (1964). G. R. Satchler, R. M. Drisko and R. EBassel, Phys. Rev. 136, B637 (1964). See also P. E. Hodgson and J. R. Rook, Nucl. Phys. 37, 632 (1962). : 2 neutron energy becomes negative, and at a certain energy the neutron is · . strongly coupled into a bound eigenstate, which has a much larger amplitude - - than at energies off this resonance. Because of the coupling between the (p) and (n) channels, this resonance is reflected into the proton channel and a narrow resonance is expected to occur in the (p,p),(p,n) and other reactions. The purpose of the present paper is to show that, with a reasonable set of the optical model parameters, such resonances cre indeed predicted at the right energies and that the excitation functions thus derived are in good agreement with experiment. The coupled equation to be solved may be written, in the upper(u)- and lower(e)-spin representation, as +1 T - E - U' [1-8+4. -U + U] %=(4. - V.1 mu + 4 talle), (2.1) 1-3+V6 - v: - 1] x = (. - Veletrhy - ef Xa), (2.2) where ty = (27 +1)*, tz = (27./(2T +1)) and the other notations are the same as used by Robson, except that the sim in front of U, has been reversed so that in (2) U and U have the same sign, and that in (2.2) a potential U' wher the other notations are the same has been introduced which is defined as U = + iWe. (3) . Equation (3) means "..at an imaginary potential is assumed only in the bo channel, an assumption that seems reasonable, because the u-channel corresponds to the ground or a very low-lying state in the system consisting of the neutron and the target. LEGAL NOTICE EF . The report mo prepared as an nocount of Government sponsored work. Neither the United Kate, nor the Cornington, nor my person acting on behall of the countantai: A. Make my warrant or repromtatou, emprened or impued, moropact to the ACCU moy, completorno, or watelewe of the information contained aan die reports of that the wee of my information, apparatua, method, or procom deelondta dalo report may not infringe privately owned the or B. A w ay Webudes with repeat to the wool, or for tumagwa runuttag Am the wo of my formation, appunto, method, or proces drolowed to do reporte Ao od to the choro, “porna notte au bobl of the Commission" moludne me . Mogna or controle of the Commission, or employee of met contractor, to the tent that wuch anplegue of contractor of the Comminaton, or employw a woh contractor preparaa, denomination, or provide accoto, any information permet to Wo employment or contract with the Completou, or Me employment will make contractor. 1 . .. * * 16, Since the (ul)-channel representation is unphysical,' we shall now transform into the (pn)-channel representation by using the relation Xu = t Xp + tahu anh Xe = tz Xp - tyXin (4) The resulting equation is [1-8+ V – V – F M - X -- (0 + 25eu, Xp (5.1) [1-2+A - v, - tv, * ]Xn--/ (04 + 2163w, y Xp (5.2) T - E to The technique and the computer program used previously in computing the (elastic and inelastic) scattering cross sections in terms of the coupled- charnel calculations can be used here, with a modification required by the lect that the neutron is in a negative energy state since in our case 274 A detailed numerical calculation has been made to analyze the ex- citation functions of the differential cross section of the YM0(2,p) process, for E at around 5.3 MeV, where an analogue state resonance of * character was found.* The explicit form of the optical model potential assumed is the following: 5o = v./(1+e(r)), U = V •4e(r)/(1+e(r))?, We = W. (r)/(1+ē(r)) ?, (6) e(r) = expl(-r47/3)/a) and a(r) = exp[(r-F_A+/3)/a), OT. Tamura, Rev. Mod. Phys. 37, 679 (1955). 1 VNT. 1 witi. the parameters V. = 54.67 MeV, W = 2 MeV, V = 541/A MeV, r. = 1.25 Fm, (7) 4. = 1.332/a1/3ten -= 13.37 Mev, a = 0.65 Fom, ā = 0.47 Fin. The theoretical excitation fwictions thus obtained are compared with ex- periment s' in Fig. 1, and the agreement is seen to be very good. The good agreement obtained in Fig. 1 indicates that our theory predicted not only the correct position of the resonance, but also the correct shape of the resonance, namely, an asymetry with a longer tail at the lower energy side. It should be noted that we would get an asymmetric resonance with a longer tail at the higher energy side, if a very small w. had been used, or if w; were sufficiently large but it was assumed only to appear in the left hand side of (5) and not in the coupling term in the right hand side. . It should also be remarked here that, in obtaining the theoretical curves in Fig. 2., Eq. (5) was not solved as it stands, but was solved after the operators on the right hand side were multiplied by a coustant q, say, and q = 0.8 was used. The reason of this artifice is explained as follows. It is known that the spectroscopic factor S of the st orbit in the ground state of M0, the analogue of which is the resonant state of interest here, is known to be 0.70, and we expect that our resonant state has a « The author is indebted to Dr. J. D. Fox for sesiding the detail of the experimental data. os. A. Hjorth and B. L. Cohen, Phys. Rev. 135, B920 (1964). similar value for s. On the other hand it is clear that Eq. (5) as it stands is an equation in which s = 1 is assumed, since no mechanism is considered there which prohibits the filling of the sį state with its fully possible amplitude. If Eq. (5) were solved as it stands, we indeed have a resonance which is too wide, and we cannot fit the experimental excitation function shown in Fig. 1. Since the use of a with a value less than unity gives rise to an rescnance which is narrower than that obtained with q = 1, it can be con- sidered a very practical way of taking account of the fact that S is less than unity into our calculation. If then it becomes possible to relate the value of a that gives agreement between our excitation curves and experiment i a (a = 0.8 in the present example), with a value of S, this artifice can be considered a useful way of obtaining the spectroscopic factor from the analysis of the analogue state resonance in the (pp) cross sections. If W' were put equal to zero in (5) and if we pl function of E, s the amplitude of the scattered st wave as defined by the relation Wip = F. * 6+ (6 + 1F,) in the asymptotic region), then the width that can be read off this plot is just to the partial width for the emission of the sų particle from the analogue state resonance, since with W = 0 r. equals the total width of that resonance. We thus computed r. for q = 0.8 and ç = 1.0, and obtained the ratio 1 (0.8)/5 (1.0) which is to be interpreted as the value of S that we are seeking. The result is S = 0.53, in close agreement with the (a,p) data.? The role that a non-vanishing Wi plays in data fitting is to reduce the peak value of from unity (a value obtained with W' = 0) and broaden the resonance (in addition to giving a correct asymmetry in the resonance curve as mentioned above). It shoulà be noted that the increase in the width thus obtained is much smaller than the value of W; itself. In spite of the success of the above artifice, it is desirable to develop a method in which the derivation of S can be made in a less artificial : way. For that purpose we have extended our basic equation (5) so that in the p- (and n)-channel the state, in which an sų particle coexists with the target (and its analogue) in its ground state, is coupled with another state in which the particle in a dalo orbit coexists with the target in its first excited 2* state. In other words the two-channel coupled equation (5) is extended to a four channel coupied equation. We should now keep q = 1, but we can get a sufficiently narrow resonance if a surficiently large value is taken for B, the coupling constant between the st-ground and 05/2-2* channels. It was found that an equally good fit to the excitation curves as shown in Fig. I can be obtained in this way, when the optical model parameters as given in (7) are used, and B = 0.12 is taken. (This value of B is in agreement with that derived from a Coulomb excitation experiment; B = 0.12 + 0.2). After this good agreement was obtained we performed cal- culations of To with W = 0, for ß = 0 and B = 0.12 and obtained s in the same way as was done for the above artificial treatment. The obtained result is S = 0.55, very close to the previous value showing that the above artificial method is indeed a useful, practical way, even though it seems to lack a Pormal basis. P. H. Stelson and L. Grodzins, Nuclear Data (to be published). When the above four-channel calculation is made, and the value of 3 is fixea, another way exists that allows us to compute s far more definitely than the above method that evaluated S via the evaluation of In Namely, - we can solve a bound state problem of a neutron in Mo in which the above coupling of the si-ground and d5 19-2 states are taken into account. The probability of the former state is nothing but the value of S, we are seeking. A computer program to solve this problem has almost been completed and the result will soon be reported. . In addition to the possibility of this more reasonable way of cal- culation of S, another and very important merit of the four (and more) channel calculation is that it allo's one to calculate the effects of the resonance on the (p,p') process. The above four channel calculation has already given an indication of a resonant behavior in the inelastic scattering cross section. The author is indebted to Drs. P. von Brentano, D. Robson, J. P. Schiffer and G. R. Satchler for stimulating and helpful discussions. ORNL-DWG 66-2432 325 ..... . . .. . .... ... . . .. 92Mo(p, Pol 300 - - రాణం 273 -30 - - - - dicin (til it. vict?) 1250 | 'హం 00 23 Goon :00 / - 75 Ho008, 1650/6097 1 50 -- 5. 5.2 5.4 5.3 ELAB (Mev) Fig. 1. Comparison between the theoretical and experimental excitation functions. A END .. .':.: DATE FILMED 6 / 16 /66