Effects of Charge Exchange in Direct Reactions

P Ii Y S I GAL RE VIE W VOLUME &82, NUMBER 4 Effects of Charge Exchange in Direct Reactions* RoRv CoKER AND Tmo TAMURA The Uwzeers~ty of Texes, ANstin, Texus Fb'71Z {Received 13 February 1969) The inclusion of charge-exchange effects in direct-reaction theory is discussed and a nuniber of experimental results are then treated: the threshold effect in ' Zr(fg, p) and 92Mo(d, p) excitation curves in the region of the (d, n) threshold to the. analog of the parent state, and the behavior of the Zr(d, np) and 9 Mo (d ep) total cross sections in the same region; Hamburger's data for 207Pb (d' p) 208Pb and, 6nally, the discrepancy between (fg, n) and (3He, d) spectroscopic factors for analog states in several light nuclei. The theory is successful in describing the threshold effect and Hamburger's data, but does not explain the discrepancy in spectroscopic factors. A number of methods of calculating the analog-residual-state wave functions are critically discussed. I. INTRODUCTION Sec. .. ~ ~HERE now exists a considerable body of experimental evidence'-" indicating that the (d, p) reaction channel leading to a given residual nuclear state is coupled to the (d, Il) reaction channel leading to the isobaric analog of the residual state excited in (d, p') . The simplest theory of such reactions, originally sketched. by Zaidi and von Brentano, and quantitatively extended by Tamura and Watson, requires replacement of the outgoing nucleon-nucleus state in the usual distorted-wave Born-approximation (DWBA) amplitude by the appropriate linear combination of solutions to coupled Lane equations. '4 The correctness of such an approach can now be given a fairly rigorous test, since a variety of experimental results is available. In Sec. II, we discuss the introduction of chargeexchange eGects into the usual DWBA theory. In Sec. III we discuss the general features of the (d, Il) threshold effect observed 111 (d, P) cxcltatloll clll'ves, 'tllc complementary (d, ep) data of Cue and Richard, and the results of calculations. using the theory of " " " * Research supported in part by the U. S. Atomic Energy Commission. ' C. F. Moore, C. E. Watson, S. A. A. Zaidi, J. J. Kent;, and J. 'G; Kulleck, Phys. Rev. Letters 17, 926 (1966). R. Coker and C. F. Moore, Phys. Letters 25, 271 (1967). ' W. R. Heffner, C. Ling, N. Cue, and P. Richard, Phys. Letters 26, 150 (1968). ' C. F. Phys. Letters 25, 408 (1967). ' E. F. Moore, Alexander, C. E. Watson, and N. Shelton, Nucl. Phys. (to be published) . ' J. L. WC11, M. Cosackp M. T. MCElhstrem, J. C. Norman, and M. M. Stautberg, Bull. Am. Phys. Soc. 12, 1196 (1967) . 7 D. G. Martin , and S. A. A. Zaidi, Bull. Am. Phys. Soc. 12, 1196 (1967). g L. S. Michelman and C. F. Moore, Phys. Letters 25, 446 (1968). ' R. G. Clarkson and %. R. Coker, Bull. Am. Phys. Soc. 13, 631 (1958); and unpublished. '0 W. R. Coker and C. F. Moore, Bull. Am. Phys. Soc. 13, 631 " (1968). for Nuclear Studies Annual Report, 1967 (unClarkson, Ph, D. dissertation, University of Texas, 1968 (unpublished); R. G. Clarkson, W. R. Coker, T. A. Griffy, and C. F. Moore, Center for Nuclear Studies Technical Report No. 4, 1968 (unpublished). '2 S. A. A. Zaidi and P. von Brentano, Phys'. Letters 23, 466 Center published); (1966) . R. G. ""T.Tamura and C. E. Watson, Phys. Letters 25, 186 (1967). A. M. Lane, Nucl. Phys. 35, 676 (1962) . "N. Cue and P. Richard, Phys. Rev. 173, 1108 (1968) . ' 182 II. In Sec. IV, the theory is applied to the and finally, in Sec. V, the discrepancy in spectroscopic factors obtRIncd wltll ( Hc, d) and (d, N) reactions to Isobaric analog states'~ is studied with three examples, the "Ni(d, e) "Cu data of Marusak and Percy" the "Mg (d, n) "Al data of Fuchs et al. , and the 'Be(d, n) "B data of Fuchs and Santo. In Sec. VI, we draw some general condusions concerning the importance of charge exchange in direct reactions, and discuss some problems in the present theoretical description of such reactions. ""Pb(d p)1"Pb data of Hamburger" " II. " CCBA AMPLITUDE The scheme for including effects of charge exchange in direct reactions is simply to include the Lane potentiaP' U(r) t T in entrance and/or exit channels. Here t is the isospin of the lighter nucleus, or nucleon, T is the isospin of the heavier nucleus. There is no efkct in a channel where the projectile has t=0, as is the case for deuterons and n particles, but channels with t, T/0 are coupled, through the Lane equations, with their analogous channels. Thus in (d, I) and (d, p) reactions the p+X and II+X (E) final states are coupled. In ('He, d) and (t, d) reactions, the 'He+ and 5+ 7 (Ã) initial states are coupled. In reactions such as ('He, p) and (1, n) both entrance and exit channels are coupled to their analogous channels. In what follows, we will consider mainly the (d, e) and (d, p) reactions. The customary DWBA amplitude is given, for (d, p), by E Here x„&— and x~&+& are the usual distorted waves describing the motion of proton and deuteron, respectively, in the nuclear potential. P„ is usually a simple shell-model state, with quantum numbers n= fej l}, of the stripped neutron, and pz is the internal & "E.W. Hamburger, Phys. Rev. Letters 19, 36 (1967). "R.H. Siemssen, G. C. Morrison, B. Zeidman, and H. Fuchs, Phys. Rev. Letters 10, 1050 (1966). » A. Marusak and F. G. Percy (private communication). '9H. Fuchs, K. Grabisch, P. Kraaz, and G. Roschert, Nucl. Phys. A110, 65 (1968). 2' H. Fuchs and R. Santo, Phys. Letters 25, 186 (1967). 1277' R. COKRR AND IXj(kr)l ov sauvaos )0LU PW2 l- TAMURA normalize the analog state in the same way as the parent neutron state. The physical signi6cance of the two amplitudes is clear, since there can now be two indistinguishable contributions to {d, p): the direct contribution, represented by the Grst term, and a quasielastic contribution from (d, e) neutrons converted into protons via the oE-diagonal part of the Lane potential, given by the second term. Similarly, for (d, e) reactions, '4Zr(ct, p) Cl T. CL 2f"=(x ' X '4 Il' I x '+'0 ) + S 1/2 5/2 0.( ~ma ' 7 8 DEUTERON 9 MeV ENERGY For purposes of the CCBA calculations reported here, the state function Q„~ was calculated exactly within the Lane model, by solving the bound-state Lane equations, ~' in the two examples considered in which the RIIslog sta'tc Is bound: Be(d 's) B slid Mg(d s) Al' For the other reactions, P„" was approximated as explRIncd 111 Rcf. 13 usmg R IIIR'tchIIlg I'Rdlus whIch Is treated as an adjustable parameter. (A discussion of the use of 8, is given in Sec. VI.} Mc~ must be evaluated for a wide range of incident deuteron energies in order to obtain an excitation curve. For Eq less than the (d, e) threshold, , x„&—I &p„ is a negative energy state. For Eg greater than the threshoM, we have a propagating neutron. Exactly at threshold, we are dealing with a zero-energy neutron, and it might be foreseen that details of the neutron strength function will be of importance in understanding the observed phenomena. For completeness, we may mention that the CCBA calculation described here is a special case of the general CCBA formulation originally given by Penny and Satchle. incidentally, in solving the Lane equations for Eq. (2), X„&+I is matched asymptotically to an outgolllg plaIlc wave; fol' Eq. (3), 1t, ls x ~+I wlllcll ls matched. The reader should keep this in mind when comparing Figs. 1(a) and 1(b) . +, I l/2 0.2 — hJ ~ O.l 0 X g~5/2 ~S/a 9 8 DEUTE RON l0 lI " MeV ENERGY Fxo. j, . (a) Amplitudes of s, p, and fE neutron partial graves at 6.i fm for O'Zr(d p)9'Zr, in the energy region above the 90Zr(d, e) "Nb" threshold, according to the Lane equations, with I. the potentials of Table (b) The amplitudes of s, p, and d neutron partial waves at 6.1 fm, for ~Zr(d, n) "Nb~, according to the Lane equations, with the potentials of Table I. See Sec. D. wave function of the deuteron, normally eliminated along with the e-p interaction F ~ by the sero-range approximation V„I,Q&~ep8 (r„I,) The coupled-channel Born approximation (CCBA) for the same process would be Ilfoo=(xn' '4» I , l". I m'+'4~& —(22'+1) "(x ' '4y I l'.y I xI'+'4~& (2) and x„&-~ are the two solutions to the two x~& coupled Lane equations. p„ is again the usual shellmodel bound state, unless othermisc noted, with biading energy equal to the experimental separation energy. @~~ is the state function of the isobaric analog of the state represented by @„.The minus sign arises from the assumption that the deuteron is in an isospin state with 0. The factor (22'+1)-I" is required explicitly, to f~ —— Here & HL THRESHOLD EFFECTS A. Review of (If, P) Results The initial suggestion that charge-exchange eGects might couple analogous (d, p) and (d, n) channels was given in the original paper of Moore e$ u/. , which reported the experimental observation of an abrupt rise in the "Zr (d, p) "Zr (d~~m ground state) excitation. curve at 170', beginning at the ~Zr{d, m) threshold to the rema1n among F12 analog 1n O'Nb. Such experunents' tlM most ilnpresslvc cvldcncc foI such coupl1ng. A pre1iminary CCBA calculation has previously been reported" resulting in a qualitative 6t to the d~~~ excitation curve of Ref. I. A large number of CCBA calculations have since been performed, with more " ~' ~g R. Stock and T. Tamura, Phys. Letters 22, 304 {1966). S. K. Penny and G. R. Satchler, Nucl. Phys. AS3, 145 (1964). complete "Zr data for comparison than mere available previously. Excitation curves, for deuteron energies 5.0—10.5 MCV, at laboratory angles of 80' and 155' for the "Zr dye ground state) sI)e (1.21MeV), da)I (2.06 MeV), and g, &e (2.19 MeV) excited states were particData at 165' are also available for ularly useful. '4Zr(d, P) and e' "Mo(d, P) to ground de~I and first " " spy states. Several facts about the threshold CGect are striking, and worth summarizing. First, thc CGect becomes progressively weaker as the angle decreases, becoming almost unnoticeable experimentally at 130' in all cases so far considered. Second, the effect (in the A 80-100 excitation region) is strongest in the ground-state curve, and rapidly becomes unnoticeable as one goes to higher and higher excited states. In the 155' and 170' excitation curves for "Zr(d, p)9IZr, there is a strong anomaly and thus a large departure from DWBA predictions for the d5~2 g ound state, but the CGect is unimpressive in the sl~e (1.21 MeV) state and almost unnoticeable in the de~I (2.06MeV) state. Compare Figs. 3 and Q, Both features of the threshold eGeet are understandRblc on the bRsls of the mechanism or1glIlally suggested'" as borne out by CCBA calculations. The sllRpc Rlld magnitude of tllc (d, p) RIlglilar distrlbutlon are controBcd predominantly by the kinematics of specific momentum transfer in the forward angles, but Coulomb and nuclear interactions become extremely important away from the stripping peaks. At back, angles, where the stripping contribution becomes very small thc closs scctlon ls scnsltivc to detai18 of the interaction, Indeed, a di6iculty in 6tting back-angle excitation curves is that the DKBA does not, in general, give a. good 6t past 130', particularly if average or extrapolated. optical-potential parameters are used. The second feature is more subtle. It turns out to be important for the incident deuteron near threshold to be wcB below the nuclear Coulomb barrier. Even when a CCBA calculation is done, the change in the direct (d, p) contribution to the cross section near threshold is small, and the burden of the CBcct is borne by the charge-exchange (d, I) contribution; usually, only a single neutron partial wave has a signi6cant amplitude In' the ~Zr and threshold variation with energy. region this is the neutron p wave. See Fig. 1. If the overlap integrals involving the p-wave neutron make up an important fraction of the total number of intcgrals contributing, one expects a noticeable CB'cct; otherwise, not. The former condition holds if the incident deuteron energy E~ is below the Coulomb barrier, since few deuteron partial ~aves are then important. As a rule of thumb, which has had considerable predictive value, no threshold cGect is seen for a given residual state if the (d, p} Q value for excitation of that state satis6es Q&x4&, where & is the Coulomb energy of the residual nucleus. Since the (d, p) threshold. is at &ash h=&, — Q and the Coulomb barrier height is " roughly obvious. 8, gh„ thc physical meaning of the rule is Finally, we discuss brieBy the A dependence of the phenomenon. While only the I'Zr(d, p) 9IZr and ~MO(d, p)"Mo cases are discussed here, data are available from A 80-100 and from 2~40-50. The p-wave neutron strength function peaks at mass 90. Thus if our understanding of the cfkct is correct, the magnitude of the departure from the expected shape wN. diminish as A either decreases or increases from mass 90. This important prediction" has now been vcri6ed with ~SC(d, p)"Se' and "Zr(p, d)"Zr" excitation function data for the ground states (pele and d&~I, respectively), at 160'. In both cases, an effect remains associated with the threshoM, though as predicted it is very much weaker than in the ~Zr example. A similar behavior is possibly seen in the region of the s-wave neutron strength fuIlctlon peRk at A~SOq when IeAr(d, p}"Ar e and "Cr(d, p) "Cr data are compared, although the anoma1ous behavior is much less clear cut in this region because of large-scale Buctuations. %hen A & 120, say, we expect no effect to be observed in any case, since the condition Q& x4& cannot be satisfied even for the ground-state transition. " &. (If, IIp) &ross Section, +~90 Proton decay of analog states formed in the (d, Ii) reaction on a number of targets in the A 90 region Excltatlon hRs bccn observed by Cuc Rnd Richard. functions were obtained at el,b =1'70', from the thresholds up to ~17 MCV. Angular distributions were also taken. Of more interest here, however, are data for the total (d, p} cross sections, which can be related directly cross sections, calculated by CCBA. to the total (d, One might expect to see some CGect due to charge exchange at back angles, in the (d, e) excitation function near threshold, because of coupling with (d, p). However, (d, I) data are not available, and in any case the charge-exchange CBect might not manifest itself as other than an imperceptible variation in the already steeply upward. -sloping excitation curve just above threshold. It is nonetheless interesting to see the relative success of D%BA and CCBA calcuulations in 6tting the Cue-Richard total cross-section data. The total cross section o(p} for proton decay to a given residual state is related to the total cross section for the formation of the analog state via (d, II), o (d, II), by o(P) =R(P)o(d, I), where R(p} is the branching ratio for decay through the particular p channel under consideration. @ It is to be expected that the angle IIltegl'Rtioll I'cqull'cd to obtain o(d, II} from tile cRlculated di6'erential cross section will wash out all eGects of the charge-exchange process. However, the 6tting of the (d, NP} total cross-section data is stin a useful test of the theory, as mill be seen. I ~ L. S. I} MiclIclInan, T, I. HonneI„and J. G. Letters 28, 659 (1969). Kulleck, Phys. R. COKKR TAsr, E I. Optical V 'Beld URA (lgeV} (fm) (fm) 118.0 0. 886 0.907 5. 80 1.57 0. 77 1.32 0.57 11.0 1.32 0. 345 5. 0 0. 84 15.0 1.34 0. 68 0. 0 1.258 0. 48 0. 0 26AI+n 1.305 '6Mg+ p 0. 66 4. 36 0. 86 ssNi+4 "Cu+n 1.28 90Zr+d 9'Zr+p 53. 0 "Nb+e 52. 0 1.28 1.22 1.27 92Mo+8 54. 2 O'Mo+P 93Tc+n and Ot 17.5 1.25 45. 0 "Nil p 0. 72 0. 65 0. 66 0. 72 0. 607 0. 66 15.95 13.5 8. 10b 14.0 8. 10b =1.2. E„ed E. Eichler, 0.47 7. 50 1.41 1.25 1.27 0. 694 0. 47 0. 47 0. 00 6. 75 1.41 1.293 1.27 0. 694 0. 737 0. 47 b Below neutron CUBA calculations have been performed for the reactions "Mo(d, p) "Mo (dsp ground state) ' and "Mo(d ep)" proceeding through the d~t, analog in "Tc, using the potentials given in Table I. The deuteron and proton optical potentials are extrapolated from Refs. 24 and 11.The neutron potential is discussed later. The 6ts to the available data are shown in Fig. 2. A radius of E, =9.5 fm was used. It was pleasing that the choice of any other value for this radius gave a very much poorer 6t to both (d, p) and (d, ep) data. Indeed, the calculated shapes are very sensitive to this parameter, and one would have little con6dence in them were such agreement not seen. In the "Mo(d, ep) example, Fig. 2 (b), the same theoretical result is obtained with or without coupling, so that with the optical potentials known only E, may be varied to 6t the data. Since the 8, obtained in this way is unique, we believe it is meanusin the (d, p) CCBA, with ingful that the same the same potentials, gives a good description of the (d, p) data at165 . Although a vast amount of experimental data is available on the "Zr(d, p) reaction, and numerous 0%HA analyses are found in the literature, it proved impossible to find there a single set of optical potentials for deuteron and proton channels giving an adequate description of the available data from 5 to I2 MeV Nucl Phys. A101, 408 (1967}. (MeV) 1.32 11.0 C. CCBA Calculations for (d, P) and (d, nP) and V„ {fm) 80. 0 '5Mg+4 W (fm} 10Be+p J. K. Dickens M (MeV) "8+m '4 TA potentials. For all coupled-channel calculations, a Lane potential of the surface type was chosen such that, in the notation of Ref. 32, VI = 108 MeV, and VI'= VIEjnu. Thus, V1'= VI. Channel P(F) =P'-OtP T. AND +so (fm) (fm) (fm) 0. 907 0. 886 1.30 0. 345 1.32 i. 305 19 36 0. 75 0. 4 1.25 1.25 0. 65 1.25 0. 42 0. 66 1.25 1.27 0. 66 1.3 1.22 1.27 ~ ~ ~ 1.20 1.27 7. 20 0 0 1.12 1.27 7. 9 7. 20 threshold, Ref. TV'~0. 0. incident deuteron energy. Therefore, the results of a recent and detailed analysis by Clarkson" were utilized. Clarkson performed optical-model analyses of deuteron "Zr at 5—12 MeV, and elastic scattering from proton elastic scattering from the same targets at 10-13 MeV. A neutron potential was obtained by 6tting the pwave neutron strength function in the mass-90 region; the resulting potential is similar to one suggested by Buck and Percy. Xo single potential could be founc which described both proton elastic scattering and thf neutron strength function, whether a symmetry tern was included or not. The potentials are summarized i Table I. The ~ozr data are surprisingly dif6cult to descril with the present theory. The moderate success of tl calculation presented in Ref. 13 is not preserved wht minor numerical errors committed therein are correcte However, just as in the "Mo case, a particular choice R, is found to provide a reasonable good description both (d, p) and (d, ep) data. Figure 3(a) shows ' 155' data for "Zr(d, p)"Zr (d5/2 ground state), DWBA and CCBA predictions using the potential. Table I. Figure 3(b) illustrates the CCBA 6t for "Zr(d, eP) data of Ref. 15. Finally, in Fig. 4 are sh~ the 6ts to the "Zr(d, p) d5~2 ground-state excita curve at j. /{} and I10', "" " 26 B. Buck and F. G. Percy, Phys. Rev. Letters 8, 444 (i CHARGE KX CHANGE 182 IN DI RECT RKA CTION S I In order to produce the agreement vnth experiment Figs. 3 and 4 iIl thc 1cglon 6-7 McV, 1t was necessary to vary the surface imaginary depth of the sho%'n optical potential. vrith energy smoothly from at 6.25 MeV, to its threshold value of 8.1 MeV. The only other part of any optical potential which vms varied with energy in these cakulations was the real-vill depth of the deuteron optical potential. In the vrork of Clarion, and herein, an energy dependence of the form V= Vo — cxE was used in fItting the (d, d) elastic data from 5 to 12 MeV. neutron O.OMeV, 4, - th E bICI s Rc " 6S fm 8.0 7.0 6.0 5.0 9.0 IO,O tMeV» (a) 5— I Re ~ I ~ ~ ~ 9. 5 fm —. 8 Zr(d, np) Zr E X 0 I- 6 N O Ky Ch 4I K I I I I LLI I 2 Mo(d, np» 'Mo 8 t 2 E t I I l 6 ENERGY ABOVE THRESHOLD IO (MOV» l- 6 III Fn. 3. (a) "Zr (d, p}»Zr {dg~ ground state) excitation function at 155', from Ref. 11. The solid line is a CCBA calculation with O K R, =6.5 fm and the potentials given in Table I. The dashed line is an ordinary DWBA calculation using the same potentials. Normalization is absolute. (b) ~Zr(d, np™)~Zr integrated cross section, from Ref. 15.The solid line is the integrated ~Zr (d, n) O'Nb~ cross section calculated by CCBA with R, =6.5 fm and the potentials of Table I, arbitrarily normalized. o& O I- ~2 lZ 6 ENERGY ABOVE THRESHOLD B IO {MeV) FIG. 2. (a)»Mo (d', p}»Mo(d'51, ground state') excitat»n«nction at 165', from Ref. 3. The solid line is a CCBA calculation with R, =9.5 fm and the potentials given in Table I. The dashed line is an ordinary DWBA calculation using the same potentials. Normalization is absolute. {b)»Mo(d, Np) 92Mo integrated cross section, from Ref. 15. The solid line is the integrated 92Mo(d, e) 93Tc~ cross section calculated by CCBA with R =9.5 fm and the potentials of Table I, arbitrarily normalized. R, *6.5 fm 0604b) CI G2- Sozr(d, p) g. s„d5~.„, l70 15 Clarkson et al "obtained'. Ozr(d, p) excitation curves at 80', 95', 110', 140', 155', and 170', from 5.0 to 10,9 MCV incident deuteron energy, for nine states in "Zr. In Fig. 5 are sholem, the 155' excitation curves for 1.21-MeV s~/2 and 2.06-MeV d3/2 states, illustrating the Q-value rule stated in Sec. III A. We have 416, =3.0 MeV, while Q(sII2) = 3.76 MeV and Q(d3I2) = 2.9 MeV. As expected, no threshold eGect is noticeable 111 tile d8)2 excl'tatlo11-flIIlctloI1 data. Since llo (d, Np) data are available, the CCBA results in Fig. 5 are shoran for various choices of E,. A radius somewhat less than 6.5 fm is seen to be satisfactory in both cases. The IO E bl cI CC . 05 Zi|,'d, p) 6.0 INCIDEN 7 g. s., d5&2, IlO . DEUTERON 8.0 IO.O MeV ENERGY Fxo. 4. O'Zr(d, p)9'Zr(dg~ ground state) excitation function at 110' and 170', illustrating the angle dependence of the threshold eEect as discussed in Sec. III A. The solid line is the CCBA calculation with R, =6.5 fm, and the dashed line a corresponding DWBA calculation, in both cases. Potentials are those of Table I. R. COKER 0.5-60 6.5 G4-7.0 E 5 j oa02 90Zr~d p), Qi 2.06 MeV, o: OA-R,a ~&"= (x. &' '4- ~-. x.'"e.) —(2T+I) '"(x ' '4 6.0 03-6,5 7.0 0.1- I , a+ 02cs & x, d5/2 05- &t YAM URA resonance in the sssPb+ p system has T =45/2, and we should not expect a resonance in (d, p). %e want to show that the expression for M c allows such an anomaly. First note that Moo I Eq. (2) can be written as M&co+. 24 &cc where ~&co is obtained from Eq. (2) byreplacingx„~ and x„& byx„, &&-& and && &, respectively. is similarly de6ned. g~,~t ~ M& and x„,&& & are the solutions of the Lane equations with T=Tg+I, x~,&~ ~ and x„,&~ ~ the solutions with T=T(. Thus we get 06- g~ T. AND Zr(d, p), I.2I MeV, sizz 5.0 7.0 iNCiDENT DEUTERON I 9.0 I 1.0 ENERGY (MeV) Fxo. 5. 9'Zr(cd, p)ii'Zr(j. .21 MeV sII2 and 2.06 MeV dgI2) excitation functions at 155', illustrating the Q dependence of the threshold eGect as discussed in Sec. III A. The solid lines are CCBA calculations for various choices of E, The dashed line is the usual D%BA result. The potentials are those of Table I. Normalization is absolute. . vertical arrow appearing in Figs. 3-5 marks the position of the appropriate (d, n) threshold. In Fig. 6 is shown the result of using the proton potential of Ref. 13 in both nucleon channels. Such a potential predicts the p-wave neutron strength function peak at about mass 110. Note that an an effect remains associated with the (d, e) threshold, and that a second minimum occurs near 9.5 MeV. For all coupled-channel calculations discussed in this and the next two sections, a surface-type Lane potential was chosen, with a depth in each case such that in the proton channel it has a strength of 27(E — Z)/A MeV. See Table I. I x~'+'A) as (4) (analog '"x ' ', (5) so that M& =0. The isospin selection rule is thus already built into Eq. (2). However, the Lane equations do not conserve isospin. This means, in particular, that the second equality in (5) does not hold over all space. Indeed, its violation becomes signiicant in the region just outside the nuclear surface, which is the region where the largest contributions to the (d, p) stripping process arise. It is to be expected, then, that the CCBA as embodied in Eq. (2) will explain the phenomenon observed by Hamburger. In Fig. 7(a) the CCBA excitation curves are compared with Hamburger's data, while in Fig. /(b) the 'ssPb(P, P) data of Zaidi ei al.s~ are compared with the et =155' o I CP hJ Hamburger" P CO CO Cl Q' " Soc. Japan Suppl. 24, 288 11M'/). 1' x. ' '=(2T+I) CO "T.Tamura, ~J. Phys. I and the analogous equation for 3f&cc. The familiar expression symbolized state) =T (parent state) requires that IV. EFFECT OF ANALOG RESONANCES IN (d, P) observed. a marked energy dependence in '+Pd(d p)' 'Pb excitation curves, at deuteron energies such that an isobaric analog resonance is excited in the outgoing proton channel. %e shall show briefly that this phenomenon can be fairly well described within the framework of the CCBA as developed in Sec. II. The analysis given here has been summarily and is included for compIeteness. reported previously, It may seem obvious that if a strong resonance occurs in the proton channel, its'eGect will be observed'. readily in the (d, p) excitation curve. However, the isobaric analog resonance is a very special type of resonance, and Hamburger's observation of the anomaly in (d, p) is in fact quite surprising when conservation of isobaric spin is considered. In the incident channel, the total isospin 0. In the is just the isospin T» of the target, since td — '"Pb(d, p)~Pb case, T~=43/2. However, the analog I 6p l55 o eoZr(d, p)9IZr(g. L) I CO OA CO CO OP. P, IE CP I l 6 7 INCIDENT I 8 I 9 DEUTERON IP II M6y ENERGY »0 ~ ~zr(d p)"Zr (4&2 ground state) and 90Zr(d n)»Nb~ analog) cross sections at 155', using the potential of Ref. 13 in both nucleon channels. (d'g2 ~7 S. A. A. Zaidi, J. L. Parish, J. G. Kulleck, C. F. Moore, and P. von Srentano, Phys, Rev. 155,'„'1312 (1968). CHARGE EXCHANGE IN DIRECT REACTION 8 predicted elastic scattering in the proton chamuel, calculated by straightforward solution of the Lane equations. ' The functions x~& ' and g„& ', obtained by fitting the region of the analog resonance in this way, were used in Eq. (2) to obtain the CCBA results shown in Fig. 7(a) The agreement is by no means perfect, but shows that our approach is basically correct. A signi6cant aspect of the calculation is its demonstration that Hamburger's data may be described completely within the framework of direct reactions, so that it is not necessary to introduce phenomenological resonance terms as originally suggested by Hamburger. & 28T. Tamura, in Isobaric Spic irl, 1Vuclear Physics, edited by J. Fox and D. Robson, (Academic Press Inc. , New York, 1966), p. 447; J. P. Bondor8, H. Lutken, and S. Jagare, ibid. , p. 576. '9 T. Phys. Rev. Letters 19, 321 (1967). Phys. Rev. 165, 1123 (1968). S. Cohen and D. Kurath, Nucl. Phys. AIOI, 1 (1967) . '2 G. R. Satchler, R. M. Drisko, and R. H. Bassel, Phys. Rev. Tamura, "T. Tamura, " 136, B637 (1964). 90 o ) ( ~t ~o ~o oiI 50 ~ ~ I i 208pb ~o ~ ~ og 0. I ~ I~ ~oooo ~ ooo O f4 o 80 70 Cg 60 60 " 60— p 90 attention to Some time ago, Siemssen evidence for a systematic discrepancy between the factors extracted from conventional spectroscopic DWBA analysis of ('He, d) and (d, n) reactions on the same target nucleus when the residual nuclear state is an isobaric analog state. Although good agreement is usually found between the spectroscopic factors obtained for, say, the ground state, one may Gnd the ratio of the spectroscopic factors to be S(d, e)/S('He, d) 0.5 for the first analog state. It seems natural to investigate what effect a CCBA calculation of the (d, e) cross section has on this discrepancy. ' 30%e investigate three cases for which detailed DWBA analyses have previously been carried out. For the two reactions 'Be(d, e) "Be(1.74 MeV, p@,) and 'Be('He, d) ' B (1.74 MeV pgq), to the mirror of the "Be ground state, one 6nds S(d, e) 1.0, while S('He, d) ~2.5 in somewhat better agreement with the prediction of Cohen and Kurath, 2.36. A calculation has been reported by one of us" which indeed seemed to explain the discrepancy by reducing the CCBA (d, e) cross section by a factor of 0.44. However, a numerical error was made in the calculation, and the actual state of affairs is rather different. Taking the optical potentials of Ref. 29, and the isospin coupling strength suggested by Satchler et el. ,32 the result is that the CCSA cross section is irlcreused by a factor of 1.35. Thus the discrepancy becomes somewhat worse. In an effort to understand the result, we changed the sign of the t T term, which reduced the CCSA cross section by only 6% relative to DWBA. Even with the coupling term four times the value normally quoted, and of opposite sign, the CCBA/DWBA ratio is only 0.79. Changing the optical potentials, in particular the I I 40 70 "8 Reaction 70 &20 V. SPECTROSCOPIC FACTORS A. 'Be(d, n) pb —. f40 —, 'b et al."directed 207pb(d +) ti& . " 80 1283 o ~o f2 ~ f25 ~ &250 p 'o«a4 f0 ~o ~ f, ot ~ ' ~ ooo II,iiooooi o" ~$ ~ 4 50 40 f f, 3 f f.5 ' 7 f f. f f.9 f2.f 0 o f6.2 f64 E&' '(MeV) f6.6 f6.8 '~ (MeV) E& (a) ~ f70o jo st 30 i f70 f7. 2 (b) Fzo. 7. (a) "'Pb(d, p}'"Pb cross section at 90', 125', and lines are CCBA predictions. (b) Isobaric analog resonances in "Pb(P, P) at 90, 125', and 170'. The solid lines are obtained by solution of the Lane equations. 169'. The solid neutron surface absorptive potential, which might be expected to be smaller for CCBA, had no meaningful effect. Nor was the shape Qt to the (d, e) angular distribution improved for any of the variations performed. We conclude that the charge-exchange coupling of analogous (d, p) and (d, e) channels does not explain the discrepancy observed in 'Be(d, e) "B. B. "Mg(d n)"Ai As another application Reaction of the CCBA we conside the "Mg(d, m)"Al ' and "Mg( He, d)2'Al" reactions. The states of interest are the 0+ (0.23 MeV) and the 2+ (3.16 MeV) levels of "Al, both of which have 2=1. "Mg(d, p)"Mg data for the parent analogs are also available for comparison. DWBA analysis is somewhat complicated in this case, since "Mg is deformed. Nilsson wave functions have been used in the reported analyses'9+ but the spectroscopic factors extracted by this means do not show better agreement with theoretical expectations than those given by ordinary spherical shell-model states, which we have therefore used for simplicity in the CCBA calculations. Table II summarizes the results 34 It is seen that, as in the 'Be(d, e) "B case; the CCBA result is not appreciably different from the DWBA result, and again yields a "A. %eidinger, R. H. Siemssen, G. C. Morrison, and B. Zeidman, Nucl. Phys. AI08, 547 (1968}. '4S. Hinds, H. Marchant, and R. Middleton, Nucl. Phys. A62', 257 (1965); H. F. Lutz and S. F. Eccles, ibid. A88, 513 (1966). R. COK ER 10,0, Ni(d, n) Cu(g. s.) 5.0 E, = tO. OMev I.OE 0.5- I TAM URA I I t 50 IO 50 70 ec~&"&~ 90 I IO I50 {a) obtained by Percy and Marusak, at E&=5 and j.o MeV. A great deal of work has been done with "Ni('He, d) "Cu, and spectroscopic factors are readily available in the literature" for bound and continuum states up to 7.0-MeV excitation in "Cu. We con. sider the pa~2 ground state and the just-unbound (by 0.48 MeV) pm~& state at 3.9-MeV excitation, which is presumably the isobaric analog of the "Ni ground state. Since the PereyMarusak data had not been previously analyzed, both regular DWBA and CCBA calculations were performed. Excellent fits to the 10.0-MeV angular distributions were obtained using the ' Ni+d potential of Schwandt and Haberli, and a neutron potential due to Percy. ' See Fig. . 8. The analog state spectroscopic factor was obtained by smooth extrapolation to the continuum energy, since this was the method used in the ('He, d) analyses. The shape fit in Fig. 8 for the analog-state transition was obtained by performing a calculation with very weak binding, 0.5 MeV. Both CCBA and DWBA were normalized by the same factor (S= 0.332), so that the difference in magnitude of' the two angular distributions is apparent. The ratio of the analog state spectroscopic factor 5& to that of the "Cu ground state So is S&/SO=0. 19 from two ('He, d) analyses, in reasonable agreement with the theoretical expectation of 0.17. The result Gf the (d, e) DWBA analysis is (S&/Sp) nws~ 0.33. A coupled-channel calculation, assuming normal coupling, gives essentially no change: (S&/Sp)cong 0.30. Thus the discrepancy is not explained. However it is worth noting that the result of a third ('He, d) analysis88 0.29, so that the reality of a gives (S&/So)nws~ discrepancy is unclea, r in this case. " IO. O 5.0 I, O- D. Summary b)4 The three cases considered here are sufficient to lead to the conclusion that although the charge-exchange coupling might be suspected to have a noticeable eGect on the magnitude of the (d, e) cross section to an analog state, the direct (d, e) term in fact remains predominant and is essentially unchanged by the coupling, at least in the lighter nuclei (A & 50) . 0.5- O, I 20 40 80 80 IOO I20 VI. GENERAL CONCLUSIONS i@0 e~g(de&) (b) Pro. 8. (a) ' Ni(d, n) "Cu(pg/g ground state) cross section at Ed=10.0 MeV, from Ref. 18. The D%BA prediction is shown as a solid line. (b) "Ni(d, n)"Cu(p3/2 analog) cross section at Eq=10.0 MeV, from Ref. 18. DKBA and CCBA calculations, assuming the analog state is bound by 0.5 MeV, are shown as solid and dashed lines, respectively. spectroscopic factor somewhat smaller, rather than larger. The optical potentials used were those of Ref. 19. See Table I. C. "Ni(d, n)"Cu Reaction ~ 182 " b}g 0 T. AND As a final example, we consider the (d, e) reaction on Ni, for which angular distributions have recently been In Secs. III and IV, it has been seen that the scheme of Sec. II explains fairly well the anomalous energy dependence of (d, p'j cross sections in cases where effects can play a signi6cant role. charge-exchange Section V showed, on the other hand, that CCBA is not sufhcient to explain the anomalous 5& spectroscopic factors, particularly for (d, e) reactions. Siemssen" has stressed that the experimental data bearing upon the S& anomaly are few as yet, and that drawing conclusions J. Vervier, Nucl. Data Sect. B 2, 34 (1968). and %. Haberli, Nucl. Phys. A110, 585 (1968). ' G. Morrison and J. P. Schi8er, in Ref. 28, p. 748; A. G. Blair and D. D. Armstrong, Phys. Letters 16, 57 (1965) . '8 D. J. Pullen and B. Rosner, Phys. Rev. 170, 1034 (1968) . 39 R. H. Siemssen (private communication). '~ '6 P. Schwandt IN DIRECT REACTIONS CHARGE EXCHANGE TABLE II. Absolute spectroscopic factors. Excitation, J~ (MeV) Spectroscopic factors "Al (& 3. 16, 2+ a Reference p)' 2. 96, 2+ (d', n)' CCSA(d', n) l. 00 2. 46 0. 34 1.26 0.37 0. 25+0. 14 0. 65 0.34+0. 17 0.30 1„=0 ~ -~ 1„=2 Reference 19. 34. is somewhat dangerous. Nonetheless, it seems unlikely that the sole explanation of the phenomenon rests upon charge exchange. Another explanation must be sought. An unsatisfactory aspect of the analysis of threshold eGects discussed in Sec. III is the use of the adjustable parameter R, This arbitrariness was difBcult to avoid, for several reasons. Consider for explicitness the 'DZr (d, p) 9'Zr example. In order to set up Lane equations for the 92nd nucleon to be solved for x„& and x„~ &, we assumed "Nb~=['T ( 'Zr) j/(2K+1)'I'. Hence it is required that at least the major portion of p~ must be equal to P„)see Eq. (2) It is important to notice that p„ is a real function, except for an over-all phase. A way to avoid use of R, is to solve the Lane equations again for the 91st nucleon" but since the proton described by is in the continuum, P„ is complex. A naive application of the Lane equations is thus dangerous, and we thought it preferable to set p„=p„» exactly for r~ E, and to connect ))))~ smoothly with a positive energy Coulomb function for r&R, . All such problems can be avoided once one has a scheme to construct the wave function of a proton in an isobaric analog resonance, since this is precisely the residual state An approach which seems plausible is the introduction of the so-called "ideal analog state" which is obtained by literally operating with T upon the parent neutron state. The equality f~~ =P„ is then exact over all space. In the language of the shellmodel approach to nuclear reactions, 4' is a bound state embedded in the continuum, to which it is coupled. Introducing the continuum wave functions f~&+)(E), the "real analog state" @„ is given as . & ]. P„. f„ 4:+ f 0 ~)z)p,"~'=(E)d&, (6) where the mixing coefficient a(E) can be calculated exactly, in principle, given a shell-model Hamiltonian. as defined by Eq. (6), has the requisite properties to describe the resonant state. p„, J. Zimanyi fg) b 0. 14 Reference 33. 4' ('He, 1.85 0. 0, 0+ 0. 23, 0+ and B. Gyarmati, Phys. Letters 2'V, 120 U. I'"ano, Phys. Rev. 124, 1866 (1961). (1968). Recall that in fitting "Zr(d, p) and "Mo (d, p) data simultaneously, quite diGerent values of R, had to be used: R, 9.5 fm for 2Mo and E, 6.5 fm for ' Zr. The radically differing (d, np) total cross sections for these two nuclei- suggest that we take this difference in R, seriously. The explanation of the observed diGerences thus lies in p~~. In the language of Eq. (6), a decreasing R, represents an increasing contribution from the second term, representing coupling to the continuum. Thus the "Nb" state is required to be coupled more strongly to the continuum than the "Tc" state. This result is consistent with the energetics of the two systems. The Coulomb displacement 6, for the "Zr-"Nb~ system is about I1.9 MeV. For the "Mo-93Tc~ system it is 12.4 MeV, so that the Coulomb barrier is half an MeV higher in the latter case. However, the groundstate analog )t) decay c.m. energy is 4.7 MeV for "Nb", and 4.3 MeV for 'Tc 42 Thus the decaying 3Tc proton is eGectively handicapped by an MeV, relative to the "Nb" proton, in penetrating the Coulomb barrier. It is quite plausible that the "Tc state has a considerably longer lifetime, as compared to O'Nb", i.e., that the second term of Eq. (6) is small. It is to be expected, then, that E, ( 3Tc") E ("Nb"). The development of a theory of isobaric analog resonances based on the shell-model approach4' is now under way by one of us (T.T.) . It is intended that 4~ can be evaluated as indicated by Eq. (6). It will then be possible to redo and extend the calculations reported in Sec. III, using the improved Also, the availability of such a resonance theory will permit reanalysis of such (d, p) data as Hamburger's, discussed in Sec. IV, by improving the description of the state of the 209th nucleon. It is hoped that the role of the isospin selection rule can be made transparent in such a description of . ) p„. the process. 4' D. D. Long, P. Richard, C. I . Moore, Rev. 149, 906 (1966). 4'T. Tamura, and J. D. Fox, Phys. in Proceedings of the International Symposium on Nuclear Structure, Dubne (International Atomic Energy Agency, Vienna, 1968), p. 213; Phys. Rev. (to be published).