PHYSICAL REVIEW C VOLUME 50, NUMBER 2 AUGUST 1994 Model for asymptotic D-state parameters of light nuclei: Application to 4He Sadhan K. Adhikari Instituto de I'zsica Teorica, Universidade Estadual Paulista, 01)05-000 Sao Paulo, Sao Paulo, Brasil T. Frederico Instituto de Estudos Avangados, Centro Tecnico Aeroespacial, 12281—970 Sao Jose dos Campos, Sao Paulo, Brasil I. D. Goldman Departamento de Ez'sica Experimental, Universidade de Sao Paulo, 20516 Sao Paulo, Sao Paulo, Braszl S. Shelly Sharma Departamento de Fz'sica, Universidade Estadual de Londrina, 86020 Londrina, Parana, Brasil (Received 26 October 1993) A simple method for calculating the asymptotic D-state observables for light nuclei is suggested. The method exploits the dominant clusters of the light nuclei. The method is applied to calculate the He asymptotic D to S normalization ratio p and the closely related D-state parameter D2. The study predicts a correlation between Dz and B, and between p and B, where B is the binding energy of He. The present study yields p —0.14 and D —0.12 fm consistent with the correct experimental g and the binding energies of the deuteron, triton, and the n particle, where g is the deuteron D-state to S-state normalization ratio. PACS number(s): 21.45.+v, 03.65.Nk, 21.10.Dr I. INTRODUCTION The role of the deuteron asymptotic D to S normaliza- tion ratio q" has been emphasized recently in making a theoretical estimate of the triton asymptotic D to S nor- malization ratio g [1, 2]. There has been considerable interest in theoretical and experimental determination of the asymptotic D to S normalization ratio of light nu- clei ever since Amado suggested that this ratio should be given the "experimental" status of a single quantity to measure the D state of light nuclei [3]. In this paper we generalize certain ideas used successfully in the two- and three-nucleon systems in order to formulate a model for the asymptotic D to S normalization ratio of light nuclei. We apply these ideas to the study of the asymptotic D to S normalization ratio, p, and the D-state parameter, D, , of 4He. Though a realistic numerical study of the asymptotic D to S normalization ratios of H, H, and He is com- pletely under control [4—9], the same cannot be affirmed in the case of other light nuclei. Even in the case of He, such a task, employing the Faddeev-Yakubovskii dynamical equations, is a formidable, but feasible, one. This is why approximate methods are called for. As the nucleon-nucleon tensor force plays a crucial and funda- mental role in the formation of the D state of light nuclei [1, 2, 7], it is interesting to ask what are the dominant many-body mechanisms that originate the D state. The present study is aimed at shedding light on the above questions. In the case of the D state of the deuteron, exploiting the weak (perturbative) nature of the D state, Ericson and Rosa Clot [7] have demonstrated that the essential ingredients of the asymptotic D to S normalization ra- tio g" are the long-range one-pion-exchange tail of the nucleon-nucleon interaction, the binding energy Bd, and the S-state asymptotic normalization parameter (ANP), C&, of the deuteron. In the case of H we have seen that the long-range one-nucleon-exchange tail of the nucleon- deuteron interaction plays a crucial role in the formation of the trinucleon D state [1, 2]. We have demonstrated that all realistic nucleon-nucleon potentials will virtu- ally yield the same value of g provided that they also yield the same values for the S-state ANP and g" of the deuteron and binding energies of 2H and sH [1,2]. The purpose of the present study is to identify the dominant mechanisms for the formation of the D state in more complex situations. We do not consider the full dynamical problem for our purpose, but, rather, a clus- ter model exploiting the relevant long-range part of the cluster-cluster interaction supposed to be responsible for the formation of the relevant D state. The o. particle or He has a very important role in nuclear physics and a study of its structure deserves special attention. One important aspect of its bound state is its D-state admix- ture for the He~ 2 H channel. There have been many theoretical and experimental activities for measuring the asymptotic D-state to S-state normalization ratio p for this channel [10—20]. In this paper we study the D state of He and make a model independent estimate of p and the closely related parameter D2 . All the observables directly sensitive to the tensor force of the nucleon-nucleon interaction, such as the deuteron quadrupole moment, Q~, g', etc. , have been found to be correlated in numerical calculations with C& through the relation [1,2, 7, 21] 0556-2813/94/50{2)/822{9)/$06. 00 50 822 1994 The American Physical Society 50 MODEL FOR ASYMPTOTIC D-STATE PARAMETERS OF LIGHT. . . 823 where 0 stands for Q", g~, or the usual D-state param- eter, D2, for the triton. The function f depends on the relevant binding energies, e.g. , the binding energy of the deuteron in the case of Q~, and the binding en- ergies of the deuteron and triton in the case of g' and D2, while other low-energy on-shell nucleon-nucleon ob- servables are held fixed. If correlation (1) were exact, no new information about the nucleon-nucleon interaction could be obtained &om the study of Q, rP, or D~z, which is not implicit in the values of Bd, , Bt., Cs, and g [1]. However, this correlation is approximate and informa- tion about the nucleon-nucleon tensor interaction might be obtained &om a study of these parameters &om a breakdown of these correlations. In order that such infor- mation could be extracted, however, one should require precise experimental measurements of these observables [1,2]. In this paper we shall be interested to see if correlation (1) extrapolates to the case of other light nuclei, specif- ically, to the case of 4He. We provide a perturbative solution of the problem, which presents a good descrip- tion of the D state. We find that in order to reproduce the correct D-state paraxneters of 4He, the minixnum in- gredients required of a model are the correct low-energy deuteron properties, including C&~ and g" and the triton and He binding energies, Bq and B . The model also provides the essential behavior of D2 and p as function of the binding energy B of the n particle for fixed B~, Bt,, and g". Consistent with the experimental Bg, Bt,, B, and q" we find p = —0.14, and D2 = —0.12 fin2. The model also predicts an ap- proximatelinearcorrelationbetween Dz (p ) and B for fixed Bg, B~, and g to be verified in realistic dynamical four-nucleon calculations. The model for the formation of the D state is given in Sec. II. In Sec. III we present relevant notations for our future developxnent of the D state. In Sec. IV the ana- lytic model for the D state of 4He is presented. Section V deals with the numerical investigation of our model. Finally, in Sec. VI a brief summary and discussion are presented. II. THE MODEL As the exact dynamical studies of the D state for the light nuclear systems employing the connected kernel Faddeev-Yakubovskii equations are usually perforxned in the momentum space, we present our model in the xno- mentum space in terms of the Green functions or propa- gators. Figures 1 and 2 represent a coupled set of dynamical equations between clusters valid for H and H, respec- tively. In the case of the deuteron the dashed line denotes the exchanged xneson. In the case of the triton the ex- changed particle is a nucleon and the double line denotes a deuteron. In both cases a single line denotes a nucleon. In the case of the deuteron these equations are essen- tially the homogeneous version of the momentum space Lippmann-Schwinger equations for the nucleon-nucleon (b) FIG. 1. The coupled Schrodinger equation for the forma- tion of the D state in H. system, which couples the S and D states of the deuteron. Explicitly, these equations are written as go —VQOGogo + V02Gog2) g2 = V2oGogo+ V22Gog2) (2) g2 ——V2P Gogo. (4) Given a reasonable gp and the tensor interaction V2p, Eq. (4) could be utilized for studying various properties of the D state. This equation should determine the asymptotic D to S ratio of deuteron g provided that the model has the correct deuteron binding Bg and the one-pion- exchange tail of the tensor nucleon-nucleon interaction. FIG. 2. The coupled Schrodinger equation for the forma- tion of the D state in H. where g~ = V[/~) (l = 0, 2) represent the relevant form factors for the two states denoted by the two-body bound state wave function P~, Go is the free Green function for propagation, and V's are the relevant potential elements between the S and D states. Figure 1(b) gives the two ways of forming the D state at infinity: (a) in the first term on the right-hand side (rhs), the deuteron breaks up first into two nucleons in the S state which gets changed to two nucleons in the D state via the one-pion-exchange nucleon-nucleon tensor force, (b) in the second term on the rhs, the deuteron breaks up first into two nucleons in the D state which continues the same under the action of the central one-pion-exchange nucleon-nucleon interac- tion. As the D state of the deuteron could be considered to be a perturbative correction on the S state, in Fig. 1(b) the first term on the rhs is supposed to dominate, with the second term providing small correction. Hence, the essential mechanism for the formation of the D state in this case is given by the following equation: 824 ADHIKARI, FREDERICO, GOLDMAN, AND SHARMA 50 In the momentum space representation of Eq. (4), at the bound state energy, g's have the following structure: (i~lgi) - ci', with p = /2m~B~, m~ being the reduced mass, and Ct" the deuteron ANP's for the state of angular momentum l. The ofF-diagonal tensor potential V20 is proportional to g N, where g N is the pion-nucleon coupling constant. From Eqs. (4) and (5), at the bound state energy one has g NxInt,d 2 where Int represents a definite integral determined by the deuteron binding Bd. Hence g" is mainly determined by the deuteron binding energy and the pion-nucleon cou- pling constant [7]. This idea could be readily generalized to more complex situations. In the case of the triton D state, Fig. 2 and Eqs. (2) and (3) are valid. The form factors g~ are to be interpreted as the triton-nucleon-deuteron form factors, the Green function Go represents the &ee propagation of the nucleon-deuteron system, and the potentials V02 and V20 are the Born approximation to the rearrange- ment nucleon-deuteron elastic scattering amplitudes rep- resenting the transition between the relative S and D angular momentum states of the nucleon-deuteron sys- tem. For example, for nucleon-nucleon separable tensor potential, V02 corresponds to the inhomogeneous term of the Amado model [21] for nucleon-deuteron scattering for the transition between S and D states of the nucleon- deuteron system. The essential mechanism for the forma- tion of the D state is again given by Eq. (4). Now in the momentum space representation of Eq. (4), at the bound state energy, g's have essentially the structure given by ('elm) (7) where C&' is the ANP of the triton for the angular mo- mentum state l. In Eq. (4), Vp2 connects a relative nucleon-deuteron S state to a nucleon-deuteron D state in diferent subclusters via a nucleon exchange. Hence the amplitude V02 involves two form factors, one for the deuteron S state and the other for the deuteron D state. Consequently, at the triton pole the momentum space version of Eq. (4) has the following form: where Int~ and Int2 are two definite integrals. The He asymptotic D-state to S-state ratio p is defined by Ccr ~dd a D Cawdd S (1O) It is clear that, unlike in the case of triton, p is deter- mined by two independent terms. Physically, it means that there are two mechanisms that construct the D- state ANP of He. Now recalling the empirical relation iIi—:C&~/C&~ (C&)2iI, we obtain from Eq. (9) (s x Int,p gd If we include these two possibilities of breakup of He, then the principal mechanisms for the formation of the asymptotic deuteron-deuteron states are given in Fig. 3. We have two equations of the type shown in Fig. 3, one for the S state and the other for the D state. In Fig. 3 the contribution of the last term on the rhs is expected to be small. The virtual breakup of He first to two deuterons and their eventual breakup to four nucle- ons to form the four-nucleon-exchange deuteron-deuteron amplitude as in this term is much less probable at neg- ative energies than the virtual breakup of He to a nu- cleon and a trinucleon and its eventual transformation to the deuteron-deuteron cluster as in the first term on the right-hand side of this equation. For this reason we shall neglect the last term of Fig. 3 in the present treat- ment. As in the three-nucleon case the amplitudes in Fig. 3 are the Born approximations to rearrangement amplitudes between different subclusters, which connect different angular momentum states, e.g. , S and D. We notice that in the first term of Fig. 3 either of the vertices has to be a D state so that the passage from S to D state is allowed in this diagram. Consequently, at the pole of the He bound state the momentum space version of Fig. 3 has two contributions corresponding to the deuteron (triton) vertex on the right-hand side being the 8 state and the triton (deuteron) vertex being the D state so that we may write Cn~dd Cn~Nt Cd Ct x Int + Ccx~Nt Cd Ctn1 S D S n2 (9) CD Cs CD Cs x Int, (8) where Int represents the remaining definite integral now expected to be determined essentially by the deuteron and triton binding energies and other low- energy nucleon-nucleon observables. Recalling that g C&/C&, with q" defined similarly, Eq. (8) reduces to Eq. (1). Hence, this simple consideration shows that the ra- tio i1~/q" is a universal one satisfying Eq. (1) determined essentially by the deuteron and triton binding energies and the deuteron S wave ANP Cs. Next let us consider the example of He, where the two deuterons could appear asymptotically either in a relative S or a D state. However, asymptotically the nu- cleon and the trinucleon could exist only in the relative S state. In this case the lowest scattering thresholds are the nucleon-trinucleon and the deuteron-deuteron ones. S, D FIG. 3. The present model for the formation of the S and D states in He. In numerical calculation only the first term on the right-hand side of this diagram is retained. 50 MODEL FOR ASYMPTOTIC D-STATE PARAMETERS OF LIGHT. . . 825 where (g is determined by the S-state asymptotic nor- malizations Cg, Cs. ', C&, C&, and Int represents integrals which are essentially determined by the binding energies Bp, Bz, and B . Hence, in the case of He Eq. (1) gets modified to the form of Eq. (11). study we set g(q2) = 1 so that we have /t ~q\ vrNm~z ~i p, ) (18) III. DEFINITIONS AND NOTATIONS In this section we present notations and de6nitions which we shall use for future development. The asymp- totic wave function for a two-body bound state, «(bind- ing energy B), in a potential V is given by lim (r/j l«) = — lim (q/jlV l«), /27r mR e (12) where / is the relative orbital angular momentum, j is the total final spin of the system (the intrinsic spin of the system is not shown), and lq/j) is the momentum space wave function. The asymptotic normalization parameter C~~ for this state is de6ned by (13) Here N represents the number of ways a particular asymptotic con6guration can be constructed &om its constituents in the same channel. For example, in the channel, sH -+ n+ H, as we can combine the proton with either of the two neutrons to form H, N = 2. Sim- ilarly in the 4He ~ 22H channel, the deuteron can be formed in two different ways and N = 2. But in the channels He ~ H + H and He ~ n+ 3He, neglecting Coulomb interaction, there are four different possibili- ties for constructing the outgoing channel components, so that N = 4. From Eqs. (12) and (13), we obtain In Eq. (18) apart &om a kinematical factor that takes into account the centrifugal barrier, the form factor is assumed to be independent of the relative momentum of the two components forming the bound state, consistent with the minimal three-body model [1].In particular, the form factors for the formation of 4He &om nucleon(N)- triton(t) channel and &om two deuterons(dd) are, respec- tively, and with Nt~ x V / &t ~a +Nt- 3 dd( ) Q/ dd Camdd ( ~2~ ( tPdd ) (20) 3(B.—B,) PNt = 2 Pdd = /2(B —2Bd), (21) where B and Bd are the binding energies of He and. H, respectively and we assume h = m~„Q]eQ„1. For the case where angular momentum states S and D states are mixed, the probability amplitude for a given l value is proportional to the corresponding spherical harmonic Yt, (q). Defining the spin-angular momentum functions P, t~ (q) as &t'-(q) =(Yt(q)& ), = ). C",' . Yt-, (q)4-. As the partial wave t matrix may be expressed as (q/jltlq/j) = lim l(q/jlvl«) I' quis E+ B (14) (22) where Q, is the spin state of the system, denotes an- gular momentum coupling, the vertex functions in the minimal model for t -+ Nd and d -+ NN vertices take the form the parameter C~~ is related to the residue at the t-matrix pole by (q/jltlq/j) R- =—»m 1«/jlVI«) I' = quips srNm~2 / 7]. (&i) (23) With this definition, in the limit of y, ~ 0, C~t ~ 1 [1]. For the two-particle bound state, the vertex function g(q) for a definite angular momentum / (and j) can be written as p (plg-t(q') = — N, C, i —. g(q'),s.Nm2~ (i p, ) where the kinematical factor which takes into account the centrifugal barrier has been explicitly shown. The function g(q ) essentially provides the momenti~~ depen- dence of the vertex function. In the present qualitative g,=, , (~.)=- „C.~-. (p.)NN ~ 4PNN d qdp2—,'X»i (J.) . ~NN (24) Here C& and C& are the asymptotic normalization pa- rameters for the 8 state of the deuteron and triton, re- spectively, whereas g" and g are the ratios of correspond- ing D-state ANP's with the S-state ANP's. The relative moment»m of the nucleon with respect to the deuteron 826 ADHIKARI, FREDERICO, GOLDMAN, AND SHARMA 50 is pq, whereas the relative momentum of the two nucleon system is pq. IV. D-STATE PARAMETERS OF He Having defined the relevant vertex functions, we can write the equation for constructing the S and D states I of He in the He + 2 H channel. The first line of Fig. 3 represents diagrammatically the present model for the formation of the asymptotic S and D states of He. Using the notation of previous section, we see that the explicit partial wave form of the present model could be written as gD~i (ti3) = f ~a'ii ff ~oql~i)q3 ) ( SIL, (P3) '@Xl 4 'I +l, (ti3)) Li,L3 1 Nd ~ N g L„(pi) X" o(qi) B B , , goo'(qi)Bd B q$ 4q3 qi q3 2 0 (25) -AN( ) 4PNN d 3 L 11 (p3) (26) where pi ———(sqi + qs) is the relative momentum be- tween the exchanged nucleon 2 and the structureless deuteron 3, g7s ——(qi + qs/2) is the relative momentum between the spectator nucleon 1 and nucleon 2, qq is the momentum carried by nucleon 1 and q3 is the momen- tum carried by the structureless deuteron of momentum q3. The indices 1 and 2 refer to the spectator nucleon and the exchanged nucleon; and 3 refers to the structureless deuteron. Here l3 is the angular momentum state of He; ls ——0 (2) corresponds to the 8 (D) states of He. L3 is the relative angular momentum of the two nucleons forming the deuteron of momentum —qq, Lq is the rela- tive angular momentum of the structureless deuteron of momentum qs and the nucleon 2 forming the triton of mo- mentum —q~, y, is the spin state of nucleon 1 with mo- 2 mentum qz, and yz is the spin state of the structureless deuteron with momentum q3. We dropped the index m of the form factors at NN, Nd, and Nt vertices, because of the angular momentum coupling notation employed. Here g~L, is the L component of the vertex defined in Eqs. (23) and (24). For example, wh~~~ CL,.= Cs ( Csrl ps/p~w) for Ls ——0 (2). The rhs of Eq. (25) is the first term on the rhs of Fig. 3. The term goo is the ¹ form factor, (Bi —B —sqi) represents the propagation of the two-particle triton-nucleon state at a four-particle en- ergy E = —B, the energy for propagation of the two- particle triton-nucleon state being B~ —B . The term (Bd —B —qi —3qs/4 —qi qs) represents the prop- agation of the three-particle nucleon-nucleon-deuteron state at a four-particle energy E = —B, the energy for propagation of the three-particle nucleon-nucleon- deuteron state being Bd —B . There are two angular momentum-spin coupling coeKcients. The one involving gz&&L gives the angular momentum coupling to form the1/2Lg triton and its coupling to nucleon 1 to give the final zero total angular momentum of He. The one involving g I gives the spin-angular momentum coupling of nucleons 1 and 2 to form the deuteron of momentum —q3 and its coupling to the structureless deuteron 3 to give the final zero total angular momentum of He. Finally, there is summation over the internal angular momenta Ii and L3, and integrations over the internal loop momentum qq and angles of q3. Substituting the values of vertex functions and rear- ranging Eq. (25), we obtain the properly normalized function A&d(qs) given by Pdd 2CSCSCS ljmiV PiVdpiVi ~ ~&. (qs, qi) 7r pdd o Bg —B — 3 qy such that at the He pole it gives the asymptotic normalization parameters of He: Pdd ( ) Cn +dd- In Eq. (27) +4 (Vs tii) =f ~ii..'iii., ((ill. (is) @x(lt. »i.'(t)~))00 ([I I, (pi) X"lo Yo(qi))oo- (28) Bd —B —q, ' —-', q,' — q~ q3 2 The values of l3 ——0, 2 yield the S and D state of He, respectively. For evaluating the integral X, we expand the energy propagator in terms of spherical harmonics as below, = 2n ) (—I) Kl, (qi, qs)/2L+ I [Yl, (qi) (I Yl. (q&)]oo,&d —&- — q& —4q3 —m . q3 (29) 50 MODEL FOR ASYMPTOTIC D-STATE PARAMETERS OF LIGHT. . . 827 where KL(qg, q3) = PL(Z) dx (3o)— q~ —-q3 —R q3 & By using the angular momentum algebra techniques [22], the intrinsic spin dependence of the integrand and the part containing the spherical harmonics in Eq. (27) are easily separated out and evaluated independent of each other. Next the same procedure is adopted to separate the qz- and q3-dependent parts of the integrand. After integrating over angles we get the following result for A&". 1 p~~ o & —& —-,'q,' 2 OO ( )L /2 ( )L /2 yP I +La P— 2 +P L+—1 ) ) KL(q3 a) I P L 0 kP'NN ) kP't ) x C000 C00'0 C00'0 " U(1L31l3j 1Lq)U — Lql; 1—Sq U(aL3 aLgl3) L3&) U(ap/3L& —p; LzL) (2 2 (2P+ 1)(2ls+ 1) (2L3 + 1)!2L~! (2L + 1)(2p + 1) (2a + 1)!(2L3 —2a)!(2p + 1)!(2I g —2p)! 1 2 (31) Here U(j&j3jsj4, JK) —= (2J+ 1)(2K+ 1)W(j,j&jsj4, JK) are renormalized 6j symbols. As Lq ——0 or 2 and I3 ——0 or 2, the left-hand side of the above equation contains four terms. We retain the three terms linear in D state and neglect the term containing a product of rid and rl~. [In the limit q3 ~ ip, , analytic expressions are easily obtained for KL(qz, q3) (relevant L values in the present context being L = 0, 1,2).] The asymptotic D-state to S-state ratio for He is defined by 2 (I&&) (32) A0d" (i Pdd) After substituting numerical values of various angular momentum coupling coefficients for allowed values of angular momenta in Eq. (31), we evaluate p as Z, t' gd 4 g' ) Z, /' ~d 4 q' l V'NN pNd ) 0 (pNN 3 pNd ) +~ k pNN pNd ) where I+2 p~— KL(n, ~udd). 0 B& —8 ——',q' Similarly, the D2 parameter of He is defined as ~o."(q3) . A".'(q3) D2 ——hm 2 ~~ ——lcm" 'q3g00(q. ) " 'A0 (q.)q. The integrals appearing in Eq. (34) are performed analytically for q3 ~ 0 and the result for D3 is ( g g' ) 121JNg+ v/B —Bd /' rl 4 q' & ( pNN pNd) 6 pN|'+ QB~ —Bd E&NN pNd) 2 p4+ sIN~&B- Bd+3(B----Bd) & n" 4 n' & (PNg + QB~ —Bd) E&NN PNd ) (34) Equations (33) and (35) are the principal results of the present study. V. NUMERICAL RESULTS The numerical results for the D-state parameters of 4He based on Eqs. (33) and (35) are expected to be more I reasonable than that for 3H of Ref. [1] because of three reasons. Firstly, the approximate analytical treatment of Ref. [1] employing the diagrammatic equation of Fig. 2 for H is more approximate than the present treatment employing Fig. 3 for He. This is because in the former case the neglect of the spin singlet two-nucleon state as an intermediate state is too drastic; whereas in the latter 828 ADHIKARI, FREDERICO, GOLDMAN, AND SHARMA 50 case there are no other competing channels if we permit only exchange of one nucleon as shown in Fig. 2. The ex- change of two nucleons is possible but is much less likely and is usually neglected in the treatment of four nucleon dynamics [23]. Secondly, the minimal cluster model we are using is expected to work better when the nucleus is strongly bound and the constituents ( H and H) are loosely bound. As He is strongly bound this approxi- mation is more true in He than in H. Finally and most importantly, in the present model we are taking the dif- ferent vertices to be essentially constants as in Eqs. (24) and (25) which corresponds to taking the vertex form factors unity. This reduces the dynamical equations es- sentially to algebraic relations between the asymptotic normalization parameters. In so doing systematic errors are introduced. The calculation of the triton asymptotic D to S ratio g' in Ref. [1] will have the above error. But the 4He asymptotic D to S ratio p of Eq. (33) and D2 of Eq. (35) are obtained by dividing two equations of type (28), one for ls ——0 and the other for ls ——2, where exactly identical approximations are made. This division is expected to reduce the above-mentioned sys- tematic error and Eqs. (33) and (35) are likely to lead to a more reliable estimate of He D state compared to the estimate of sH D state obtained in Ref. [1]. Equation (33) or (35) yields that for fixed Bt, , Bg, g, and g', p and D2 are correlated with B . Specification of B alone is not enough to determine the He D-state parameters. We have established in Refs. [1, 2, ll] that in a dynamical calculation g' is proportional to g" for fixed Bg and Bt, from which the theoretical estimate of g~/g" was made. This was relevant because of the un- certainty in the experimental value of g . If this result is used in Eqs. (33) or (35), it follows that p and D2 are proportional to g" for fixed Bg, B~, and B . Next the results of the present calculation using Eqs. (33) and (35) are presented. In Eqs. (33) and (35), in actual numerical calculation both g" and g' are taken to be positive. The positive sign of rl~ is consistent with the order of angular momentum coupling we use in the present study [ll]. In Fig. 4 we plot p versus B cal- culated using Eq. (33) for different values of Bq and for g" = 0.027 and B" = 2.225 MeV. The value g" = 0.027 is the average experimental value reported in Ref. [7]. There is a recent experimental finding: g" = 0.0256 in Ref. [9]. For the present illustration we shall, however, use g" = 0.027. Though the final estimate of the asymp- totic D-state parameters of He will depend on the value of q employed, the general conclusions of this paper will not depend on the choice of this experimental value of g". The five lines in this figure correspond to B& ——7.0, 7.5, 8.0, 8.5, and 9.0 MeV. The numerical value of g for a particular Bt is taken from the correlation in Ref. [1]. We find that the magnitude of p increases with the in- crease of B for a fixed Bq and with the decrease of Bq for a fixed B . This should be compared with the corre- lation of q' with Bt in Ref. [1]. We also calculated D2 using Eq. (35) for different values of Bt, and B More results of our calculation using Eqs. (33) and (35) are exhibited in Table I. We employed different val- ues of Bq and B . The values B = 28.3 MeV and Bq —0.14 —0. 'I 8 26 28 30 B (MeV) FIG. 4. The p versus B correlation for fixed B~ ——7, 7.5, 8, 8.5, and 9 MeV and with g = 0.027 using Eq. {33). The curves are labeled by the Et, values. The g' values for each line are taken from the g' versus Et, correlation of Ref. p —0.14, D2 —0.12 fm, (36) p /p~~D2 = l. In Ref. [24] it has been estimated that p /(p&&D2) 0.9 in agreement to the present finding. Next we would like to compare the present result with other ("experimental" ) evaluations of these asymptotic parameters. Santos et al. [13] evaluated p Rom an analysis of tensor analyzing powers for (d, n) reactions on S and Ar. They employed a simple one-step transfer mechanism, plane-wave scattering states, zero-range or asymptotic bound states. Keeping only the dominant angular momentum states for the transferred deuteron they predicted p = —0.21. In another study Santos et al [14] considere. d the ten- sor analyzing power of reaction 2H(d, p)4He and con- = 8.48 MeV are the experimental values. The other val- ues of B and Bq are considered as they are identical with results of theoretical calculation of Ref. [19]. As the values of the binding energies are crucial [1, 2] for a correct specification of the D-state parameters we de- cided to consider these binding energies obtained in Ref. [19]. For example, B, = 8.15 MeV is the mean of H and He binding energies obtained in Ref. [19] with the Urbana potential. For the same potential they obtained B = 28.2 MeV and D2 ———0.24 fm, to be compared with the present D2 ———0.15 fm . For the Argonne po- tential they obtained mean B~ ——8.04 MeV, B = 27.8 MeV, and D2 ———0.16 fm, to be compared with the present D2 ———0.12 fm . But the large change of D2 in Ref. [19] from one case to the other is in contradiction with the present study. The first row of Table I is the re- sult of our calculation for p and D2 consistent with the correct experimental Bq and B and using g" = 0.027, rP/rI" = 1.68: 50 MODEL FOR ASYMPTOTIC D-STATE PARAMETERS OF LIGHT. . . 829 TABLE I. Results for D-state parameters of He calculated using Eqs. (33) and (35). B (MeV) Bg (MeV) Da (fm') P P I ddDQ 28.3 28.2 27.8 8.48 8.15 8.04 0.027 0.025 0.0266 0.0454 0.0514 0.0430 —0.12 —0.15 —0.12 —0.14 —0.17 —0.14 1.07 0.98 1.05 eluded that agreement with experiment could be ob- tained for —0.5 ( p ( —0.4. Earp et al. [15] studied tensor analyzing powers for (d, n) reactions on Y. They employed simple shell- model configurations for the nuclei involved, performed a finite-range DWBA calculation, and represented the 4He ~ 22H overlap by an effective two-body model. From an analysis of the experimental data the authors concluded D; = —0.3+ 0.1 fm'. Tostevin et al. [16]studied tensor analyzing powers for (d, a) reactions on Ca. They performed a DWBA cal- culation with local energy approximation and concluded D2 = —0.31 fm2 and p = —0.22. But Tostevin [17] later warned that this value of D2 may have large error. Merz et al. [18] has performed an analysis in order to make a more reliable estimate of these parameters. From a study of the 4oCa(d, n)ssK reaction at 20 MeV bombarding energy employing a full finite-range DWBA calculation they predicted D2~ ———0.19 6 0.04 fm . Recently, Piekarewicz and Koonin [20] performed a phenomenological fit to the experimental data of the H(d, p) He reaction and predicted p = —0.4. From a study of cross section of the same reaction, however, Weller et aL [12] predicted p = —0.2 + 0.05. Considering the qualitative nature of the present study we find that there is reasonably good agreement between the present and other studies. This assures that we have correctly included the essential mechanisms of the forma- tion of the D state. Unlike in the case of H, there are two distinct ingre- dients for the formation of the D state of He: g~ and This is clear from expressions (33) and (35). This possibility did not exist in the case of H where g is determined uniquely by g" apart &om the binding ener- gies. In the usual optical potential study of the D state of 4He as in Ref. [24] the dependence of p on rl~ is always neglected. This dependence will be explicit in a micro- scopic four-particle treatment of 4He. Such microscopic calculations are welcome in the future for establishing the conclusions of the present study. VI. SUMMARY We have calculated in a simple model the asymptotic D state parameters for 4He. The present investigation generalizes the consideration of universality as presented in Refs. [1] and [2] for the trinucleon system. The univer- sal trend of the theoretical calculations on the trinucleon system and the consequent correlations are generalized here to the case of the D-state observables of He. The essential results of our calculation appear in Eq. (36). We have used a minimal cluster model in our calculation where essentially the bound state form factors are ne- glected, thus transforming the dynamical equation into an algebraic relation between the different asymptotic parameters. Dividing two such equations, one for the asymptotic S state of 4He and other for the asymptotic D state of 4He, the estimates of Eq. (36) are arrived. As we have pointed out in Sec. V, such a division should re- duce the systematic error of the approximation scheme. Dynamical calculation using a realistic four-body model should be performed in order to see whether the present estimate (36) is reasonable. At the same time accurate experimental results are called for. We have predicted correlations between p and B, and between D2 and B to be found in actual dynamical calculations. Such correlations, though appearing to be extremely plausible in view of the calculation of rlt of [1],can only be verified after performing actual dynamical calculations. ACKNOWLEDGMENTS This work was supported in part by the Conselho Na- cional de Desenvolvimento Cientifico e Tecnologico and Financiadoras de Estudos e Projetos of Brazil. [1] S. K. Adhikari and T. Frederico, Phys. Rev. C 42, 128 (1990). [2] T. Frederico, S. K. Adhikari, and M. S. Hussein, Phys. Rev. C 37, 364 (1988). [3] R. D. Amado, Phys. Rev. C 19, 1473 (1979); R. D. Amado, M. P. Locher, and M. Simonius, ibid. 17, 403 (1978). [4] B. A. Girard and M. G. Fuda, Phys. Rev. C 19, 583 (1970). [5] J. L. 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