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Two-nucleon transfer reaction mechanisms
TWO-NUCLEON TRANSFER REACTION MECHANISMS
Masamichi IGARASHI Department of Physics, Tokyo Medical College, Shinjuku, Sinjuku-ku, Tokyo 160, Japan Ken-ichi KUBO Department of Physics, Tokyo Metropolitan University, Fukazawa, Setagaya-ku, Tokyo 158, Japan and Kohsuke YAGI Institute of Physics and Tandem Accelerator Center, University of Tsukuba, Tsukuba, Ibaraki 305, Japan
NORTH-HOLLAND
PHYSICS REPORTS (Review Section of Physics Letters) 199, No. 1(1991)1—72. North-Holland
TWO-NUCLEON TRANSFER REACTION MECHANISMS Masamichi IGARASHI* Department of Physics, Tokyo Medical College, Shinjuku, Sinjuku-ku, Tokyo 160, Japan
Ken-ichi KUBO Department of Physics, Tokyo Metropolitan University. Fukazawa. Setagaya-ku, Tokyo 158, Japan
and Kohsuke YAGI Institute of Physics and Tandem Accelerator Center, University of Tsukuba, Tsukuba, Ibaraki 305, Japan Editor: G.E. Brown
Received April 1990
Contents: 1. Introduction .1. Review of previous studies 1.2. Aim of this work 2. Formulation and method of calculation 2.1. The first- and second-order distorted wave Born approximations 2.2. The triton and deuteron wave functions 2.3. The one-step process 2.4. The sequential transfer (p—d—t) process 2.5, The spectroscopic amplitude for the particle transfer reactions 2.6. The differential cross section and analyzing power 2.7. Why analyzing power? 3. Experimental study of (p 1) cross sections and analyzing powers
*
3 3 6 6 6 9 14 25 32 35 37
3.1. The measurement of (p, t) analyzing powers and cross sections 3.2. Experimental results for (p t) analyzing powers and cross sections 3.3. Experimental studies of the (p, d) and (d, t) reactions 4. Results and discussions 4.1. The one-step (p—d) process 4.2. The sequential transfer (p—d—t) process 4.3. Some properties of one- and two-step processes 4.4. Summary and conclusions Appendix A. Hard-core correction to the one-nucleon transfer form factor References
38
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M. Igarashi et al., Two-nucleon transfer reaction mechanisms
3
Abstract: The roles of the one-step (p—t) process in which two particles are transferred simultaneously and the two-step (p—d—t) process in which two particles are transferred sequentially, are quantitatively investigated for two-nucleon transfer (p. t) reactions with realistic triton and deuteron wave functions and realistic finite-range nuclear interactions. A detailed exact finite-range formulation is given, and analyses of both the one-step and the 3D two-step process are carried out. For the intermediate channels in the sequential transfer process, ~~and ~ + 1 unbound deuteron states as well as the ground state are taken into account. A summary of experimental studies of cross sections and analyzing powers for (p, t) reactions is given. It is found that the simultaneous transfer process alone can reproduce neither the experimental absolute cross sections 208Pb(p, nort)°6Pbreactions the analyzing powers, at 22 and thatMeV. 35 the These sequential results transfer are definitely process confirmed is dominantbyfor comparing both thethe natural calculated and unnatural analyzingparity powerstransitions with experimental in the data. The nuclear shell structure dependence of the analyzing power for the natural and unnatural parity state transitions is investigated.
1. Introduction 1.1. Review of previous studies Direct two-nucleon transfer (p, t) and (t, p) reactions are regarded as probes of the two-particle pair correlations in nuclei [1, 21, and a great deal of experimental and theoretical work has been done in the past two decades. However, the mechanism of these reactions has not yet been well confirmed because of the complexity of the reaction processes, in which four bodies a proton, two correlated transferred neutrons and a residual (or target) nucleus participate. The one-step (simultaneous two-neutron transfer) process, the two-step (sequential neutron transfer) process and in some cases inelastic channel coupling are of comparable significance. It is not easy to distinguish among those processes as far as the cross sections of the reactions are concerned since the main features of the angular distribution of the cross sections are determined by the selection rules the total transferred orbital angular momentum which are usually the same in these processes. Therefore different conclusions about the reaction mechanism came from different calculations, which were made by employing various approximations to avoid complexity of computation. In the following two subsections, we review this point for previous work, and then clarify the aim of the present work. —
—
—
—
1.1.1. The one-step DWBA process The (p, t) and (t, p) reactions have first been described by the simultaneous (one-step) two-neutron transfer mechanism. The pioneering studies were made in the plane wave Born approximation (PWBA) [3—5]and reproduced the forward-peaked angular distribution. Later, most analyses were made in the distorted wave Born approximation (DWBA) with zero-range approximation. The latter approximation is based on the assumptions that the interaction which causes the transfer reaction is a short-range central force and that the wave function of the triton is also short range and in a purely space-symmetric S-state, so that the two neutrons are in a singlet spin state. The method of Lin and Yoshida [6] and Glendenning [7] assumes a Gaussian wave function for the triton and a single-particle harmonic oscillator wave function for the neutrons bound in the nucleus. The harmonic oscillator wave functions were transformed to those of the center of mass and of the relative coordinates of the two nucleons using the Moshinsky transformation [8]. The strong absorption of the distorted waves forces the reactions to occur near the surface of the nucleus. Since the harmonic oscillator wave functions decay too rapidly at large distances, the radial functions were connected to Hankel functions of purely imaginary argument with decay constants corresponding to the separation energies of the nucleon pairs from the nucleus. Later, single-particle wave functions in a finite potential well were used. Two methods have been proposed for this purpose. One was the expansion of
4
M. Igarashi et al.. Two-nucleon transfer reaction mechanisms
Woods—Saxon wave functions in terms of harmonic oscillator functions [91,and the other, derived by Bayman and Kallio [101,was a direct numerical method of projection to the relative and the center of mass coordinates. The zero-range DWBA method reproduces the angular distributions and the relative magnitudes of cross sections in most cases. This seemed to support the claim that the reaction mechanism is dominated by the simple direct simultaneous transfer process. The angular distributions show diffraction patterns resulting from strong absorption, with shapes depending on the transferred angular momentum and parity. The angular distributions of the natural parity state transition are little affected by the details of the reaction mechanism, such as inclusion of the two-step mechanism. In many cases, however, the method fails to reproduce the absolute magnitude of the cross sections. The empirical value of the normalization constant of the zero-range DWBA, D~,is about 2—3 x i05 MeV2 fm3 [11, 121. This is an order of magnitude larger than simple theoretical estimates, although the estimates are not entirely reliable. A number of exact finite-range DWBA calculations employing more sophisticated triton wave functions and finite-range interactions have been reported since the work of Bayman [13, 141. He employed the Tang—Herndon triton wave function [15], which was obtained from a variational calculation with a simple state-dependent central two-body interaction potential with a hard core. For the 40’48Ca(t, p)4250Ca(g.s.) reactions, the magnitude of the calculated cross sections was a third of the observed one. In these calculations the pairing enhancement effect was taken into account. They were a great improvement over the zero-range DWBA calculations. However, in most of the calculations including Bayman’s work, triton wave functions generated from two-body interaction potentials with hard cores were used. As has been pointed out by Dobes [161,the use of such singular potentials requires caution and a correction term must be taken into account. This decreases the calculated cross section by a factor of 2.1 to 2.6 [17]. This alters the conclusion that the exact finite-range calculations improve the calculated absolute cross section of the simultaneous transfer mechanism. The wave function of the two transferred nucleons in the target nucleus was usually calculated by the half separation energy method neglecting the correlation between the two nucleons. But this method has no strict justification; the nucleon pair being transferred does not have a finite internal energy near and inside the nucleus, and hence the kinetic energy of the center of mass motion does not have a definite value. This problem was investigated by many authors [18—20],and calculations including the two-body correlation have been carried out. These caculations indicate that the correlation increases the magnitude of the wave function at the nuclear surface, enhancing the cross section especially for transitions to the lowest 0~state. However, the enhancement compared with the wave function with the usual restricted shell model space is several tens of percent and this is not enough to cover the difference between theory and experiment. Transfer reactions to the unnatural parity state of even—even nuclei are completely forbidden in the zero-range one-step DWBA framework. In finite-range DWBA this process is not completely forbidden, but Bayman and Feng [14] got a much smaller cross section than the observed one for the 22Ne(p, t)20Ne 2 reaction. They employed a spatially symmetric S-state triton wave function given by Tang and Herndon. This is questionable since for these reactions relatively large contributions are expected from the minor components such as the mixed symmetry S’- and D-states [21]. According to Nagarajan et al. [21, 22], the finite-range DWBA calculation with the Strayer—Sauer triton wave function can reproduce very well the cross section of 208Pb(p, t)206Pb(3 1.34 MeV) and also (0 g.s.) at 35 MeV, both in absolute magnitude and angular distribution (see also ref. [23]). These works throw doubt on the assumption that the reaction proceeds primarily as a two-step process. This, however, could not be confirmed by the present authors, as described in this paper. ~,
,
M. Igarashi et al., Two-nucleon transfer reaction mechanisms
5
1.1.2. Multi-step processes Many authors have reported that the second- and higher-order multi-step processes play important roles in (p, t) and (t, p) reactions. The effect of inelastic excitation processes on (p, t) reactions has been investigated by Ascuitto and Glendenning [24] and Tamura et al. [25]. If the nuclei involved are deformed (rotational) or have strong collectivity (vibrational), the nuclei are easily excited by inelastic scattering in the incident and/or exit channels. When the parentage of the final state is based more on such an excited state than on the ground state of the target, the transfer reaction to that state is strongly influenced by inelastic excitation. In such cases, the distorted waves are the solutions of the inelastic coupled-channel equations, but the transfer process is treated by the first-order Born term. This is the so-called coupled-channel Born approximation (CCBA). The sign of the interference between the multi-step and the direct one-step amplitudes depends on the details of the structure of the levels [26—28]. The other multi-step process is the sequential nucleon transfer mechanism. The first formulation of this type of transfer has been proposed by Bang and Wollesen [29] and was applied to the t0B(t,p)12B(1~,g.s.)reaction, for which the DWBA calculations had failed to reproduce the cross section. The nucleon transfer multi-step calculation is sometimes referred to as the coupled-reaction-channel (CRC) method. An apparent difference between the shapes of the CCBA and CRC angular distributions was pointed out for the 22Ne(p, t)20Ne(2) transition: the cross section goes to zero or remains finite at zero degrees [30—32](see also ref. [33]). De Takacsy [34, 35] reported that the 208Pb(p, t)206Pb(3~,1.34 MeV) reaction can be well understood by the sequential transfer mechanism. The strict selection rule (S = 0) on the transferred spin for zero-range DWBA is relaxed by spin transfer (S = 1) in two steps. Then, for such unnatural parity state transitions this mechanism gives a larger cross section than the one-step mechanism. The importance of the sequential process in strong natural parity state transitions has also been pointed out by Hashimoto and Kawai [36, 37]. They found in the analysis of the ltSSn(p, t)”4Sn(0, g.s.) transition that the two-step cross section becomes larger than the one-step cross section. Furthermore the phase coherence, existing in the one-step process due to two-nucleon pairing correlation [2], also exists in the two-step amplitudes [36]. The importance of both the CCBA- and CRC-type multi-step processes was reported for the tSO(p t)160(2) transition at 22.4 MeV [32]. Its smooth angular differential cross section can be well understood by the (p—p’—t) inelastic mechanism for the forward angular region and by the (p—d—t) sequential transfer mechanism for the backward region. The general trends of the observed energy dependence of the cross section (E~= 24.4—43.6 MeV) are interpreted as a more rapid decrease of the (p—d—t) mechanism compared with the (p—p’—t) mechanism [38]. The analyzing power data are also consistent with this interpretation [39]. The second-order (p—d—t) amplitude in the prior—prior form representation accompanies the so-called non-orthogonality term, which is due to the non-orthogonality of the d- and t-channels [40, 41]. Hashimoto and Kawai [36, 37] showed by the finite-range calculation that the contribution from this term is relatively small. Kunz and Charlton pointed out the importance of finite-range effects in the multi-step particle transfer amplitude [42]. Charlton [43] and Hashimoto [37]carried out a finite-range two-step calculation using simple S-state wave functions for the triton and deuteron and a simple central interaction, and found that the finite-range effects are small. It has been shown [37, 44] for natural parity transitions that there are important contributions to sequential transfer from the breakup or continuum deuteron intermediate states. However, these calculations used a simple Gaussian triton wave function and simple Gaussian interactions. So the conclusion is to be reinvestigated with amore realistic wave function and interaction.
6
M. Igarashi ci al.. iwo-nucleon transfer reaction mechanisms
Recently, it has been found [45] that the angular distributions of analyzing powers A ~(O)[whichwe abbreviate as A(O)] of (p, t) reactions are sensitive to the interference between simultaneous and sequential transfer amplitudes. Experimental results [46] provide direct evidence of this interference. The variations of the interference as a function of both target mass and bombarding energy, and the dependence on the final states [47, 48] have been observed. 1.2. Aim of this work The aim of this work is to settle this confusing situation between two completely different mechanisms, one-step (summarized in subsection 1.1.1) and two-step (in subsection 1.1.2), by making finite-range calculations of each process to an equally accurate degree and comparing the calculations with the experimental results of (p, t) cross sections and analyzing powers. For this purpose we need to use realistic wave functions for the light particle systems. The triton wave function employed here is the one obtained by solving the three-body Faddeev equation [49, 50]. For the intermediate channels in the sequential transfer process, the 1S0 and 3S1 + 3D, unbound deuteron states as well as the ground state are taken into account. The deuteron unbound state channels are evaluated by the DiscretizedContinuum-Channel Method, where the couplings between the ‘S 3S~+ 3D 0 and 1 channels are neglected [51].The same microscopic two-body interaction used in the calculations of the light particle wave functions is adopted for the transfer interactions. The analyzing power A(O) is also calculated and compared with data, which provides a useful test of the reaction mechanism. The A(O) for the unnatural parity state (UNPS) transition is still large even when the unbound deuteron components are included in the calculation with realistic wave functions (see subsections 2.4.3 and 4.2.2.2). The nuclear structure dependence of the observed analyzing powers for the natural and unnatural parity state transitions is also discussed.
2. Formulation and method of calculation 2.1. The first- and second-order distorted wave Born approximations The transition amplitude for a two-step processes is obtained from the second-order perturbative approximation in the coupled-channels formalism. The Schrödinger equation for the total system is written H~P=E~P,
(2.1.1)
where H=H~=Ha+T~*+Va~ 1I~=~JtIJaXa,
a=i,f,m.
(2.1.2) (2.1.3)
Here Ha is the intrinsic Hamiltonian of channel a, where the subscripts a = i, f and m denote the initial, final and intermediate channels, respectively. Ta is the kinetic energy operator of relative motion and 1’a is the intrinsic V~is the effective interaction. Xa represents the wave function of relative motion and c
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
7
wave function which satisfies (Ha Ea)tIIa = 0. The intrinsic wave function describes both states of the target and projectile nuclei in the channel a. From eq. (2.1.1), the coupled equations for Xa are —
(2.1.4)
(Ta+U~Ea)Xa=~~ (~H—E~X~), /3 +c*
U
= (t/J
V
~/i)
(2.1.5)
,
Ea=E~Ea~
(2.1.6)
If the coupling between the channels is strong, these coupled integro—differential equations must be solved directly under proper boundary conditions for the scattering states. Here we assume that the couplings are not strong so that the perturbative approach is valid. By solving the above equations by iteration up to second order one obtains the transition amplitude T= (t/i~x~~HEjtIIIX~~~)
(one-step)
—
(2.1.7) +~
(two-step).
~
Here the zeroth-order distorted wave y~ and the Green’s function ~ (Tf
+
U~ E~)x~ = 0,
(Tm + Urn
(2.1.8)
—
—
satisfy the equations
Em)G~~~ = —1,
(2.1.9)
with the boundary conditions that ~ is an incident wave plus an incoming wave at infinity and ~ has the asymptotic form of an outgoing wave. The distorted potentials Uf and Urn are the bare potentials defined by not including the coupling effects in the starting equations (2.1.4) [52,53] and are different from the elastic optical potential in principle. However, for simplicity, we usually replace them by the phenomenological elastic optical potential and we know that in the light-ion reactions in which the momentum matching condition is well satisfied this approximation works quite well. The wave function ~ in eq. (2.1.7) is the elastic scattering solution of the coupled-channel equations (2.1.4). Therefore it may be a much better approximation to substitute for it the elastic solution using a phenomenological optical potential [53, 54]. Hereafter we adopt the conventional DWBA method where we ignore the difference between X~and ~ [solution of the homogeneous equation with potential U1 in eq. (2.1.4)]. If we want to use X~ in the following, U1 is the potential reproducing elastic scattering. From eqs. (2.1.2) and (2.1.8) we get the post—prior symmetry of the transition amplitude for the one-step process, 1~ = (i/JfX~~~VI U,Iifi,x~Y~) (prior form) (2.1.10) T~ or —
T~1~ = (/JIX~~~V~—U~Icb,x~~~) (post form).
,
(2.1.11)
8
M. Igarashi
ci
al., Two-nucleon transfer reaction mechanisms
In the particle transfer process with light-ion projectiles, the calculation method has been well established using the prior form for the pick-up process and the post form for the stripping process. It is customary to approximate V~ U by V~,the interaction between the transferred particle and the projectile for the pick-up process, and V~ U1 by V~,the interaction between the transferred particle —
—
and the ejectile for the stripping process. 12~can be expressed in four different forms depending on The channel second-order transition which is referred to for amplitude the two H’sT in eq. (2.1.7); either H~’~ (prior) or H(m) (post) for the first H, and H(m) (prior) or H~1~ (post) for the second H. In the p—d—t process, the choice of H~’1and H(m) (prior and prior) representation is used since the interactions for the smaller particle systems are better known. Hence the p—d—t transition amplitude can be written T~~dI= 1Xt’d’d
(prior-prior)
~
(2.1.12) (prior-non),
K~iX~dd~VPkfrPX~
using eqs. (2.1.2), (2.1.8) and (2.1.9). The first term is the “genuine” two-step term and the second one is the non-orthogonality (NO) correction term. In a rearrangement collision the intrinsic states of different channels are not orthogonal to each other so this second term does not vanish. We should note that the overall minus sign appears in the NO term. If we sum over a complete set of states of both nuclei in the intermediate channel, then the NO term becomes equal and opposite in sign to the one-step amplitude [40, 41]. Therefore the “genuine” two-step term (which should also be summed over the complete set of intermediate states) remains as the lowest-order contribution to the transition. In practice, however, it is not allowed to make a complete sum over all the intermediate states t/Id because the i-I’ in eq. (2.1.3), ~I’= ~I’~+ II~+ ~ in the present case, then encounters the overcompleteness problem. Even if we regard the “genuine” two-step term as only the lowest term, it is not feasible to evaluate all the contributions arising from a complete sum over the intermediate channels, although some of these strengths may be significant in several channels, and others are spread over many channels. If a limited model space is chosen, the tendency remains for the one-step and non-orthogonality terms to interfere destructively. In the independentparticle model limit of the two-nucleon transfer form factor calculation, in which no correlation between the two transferred nucleons is considered, the one-step and NO amplitudes cancel each other in each intermediate state [55, 56]. The concept of a “step” is rather ambiguous, when we employ the presentation that brings about the non-orthogonality term. The practical criterion to choose the intermediate reaction channels m is as follows. We first choose the states which have large parentage coefficients. A plausible approach is thus to find such states that give large cross sections for i m and m f. The theoretical arguments about the intermediate states, however, are not so simple. If states which are not orthogonal are included in the intermediate channels, the problem of “overcounting” possibly occurs. For example, the m = d state in the (p—d—t) process and the m’ = t’ state in the (p—t’—t) process are not orthogonal to each other. Therefore in principle we must not take a simple sum of the two two-step amplitudes, otherwise we overcount. This problem cannot easily be overcome in the framework of the second-order DWBA. One takes care of the overlap between the channels. The problem, however, is automatically overcome by the method of coupled channels for multi-step rearrangement processes [57]. —~
—*
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
9
2.2. The triton and deuteron wave functions
To calculate the absolute magnitude of the cross sections for the transfer reactions, it is essential to employ realistic wave functions for the light ions and interaction potentials which cause the transfer. The consistency between the interaction potential and that used in deriving the light-ion wave function is important, especially in the case of a realistic interaction potential with hard or soft core [16, 17]. The existence of minor components in the triton wave function such as the spatially mixed symmetric S’-state and D-state components has a significant role in the simultaneous two-nucleon transfer process leading to the unnatural-parity states of even—even nuclei, where the contribution from the main component (symmetric S-state) is hindered by the selection rule [22]. 2.2.1. The triton wave function The triton wave function employed here is the one obtained by Sasakawa, Sawada and Ishikawa [49, 50] by solving the three-body Faddeev equation with a realistic nucleon—nucleon interaction. The totally antisymmetric wave function obtained from the three-body Faddeev equation in coordinate space is given by a sum of three functions which are represented by the Jacobi coordinate system (see fig. 2.1) =
t~I(12,3)
+
~(23, 1) + 1(31, 2).
(2.2.1)
The three functions are solutions of the simultaneous equations (E
—
H0) 1(12,3)
=
V,2 ~
(E
—
H0)’P(23, 1)
= V23 ~P,
(E
—
H0) b(31, 2)
= V.~,~P
(2.2.2) where E is the total binding energy, H0 is the free Hamiltonian of the three-body system and V1.1 is the interaction between particles I and j. We can see from eq. (2.2.2) that ~ satisfies the three-body Schrödinger equation, (2.2.3)
(E—H0—V~2—V23—V31)~P~=0.
We write ~(12, 3) as a sum of possible three-body states, ~(12, 3)
=
~ (l~s~)j~t~, (l~1 /2)j~1/2; stttmtni~a(x3,y3),
where the sum runs over several internal states a. The
~
2~
(2.2.4)
~ are spin—angular—isospin functions. An
3
2~3
Fig. 2.1. Jacobi coordinate system of the triton.
10
M. Igarashi et al.. Two-nucleon transfer reaction mechanisms
interacting pair 12 is coupled to form a state [(l~s1)j5t~],where s1 + s2 = s~and t, + t2 = t~,and the spectator particle 3 is in state [(lv 1 !2)j~1/2]. s~and t~are the total spin and isospin of the triton and m1 and n1 are the z-components of the total spin and isospin, respectively. The cba(xi, y3) are radial wave functions of the Jacobi coordinate x3 and y3. The functions P(23, 1) and ‘1(31, 2) are given by analogous forms. The classification of the components of the triton wave function by the symmetry group is carried out as follows. We separate the wave function into spatial and spin—isospin parts, Since the triton has isospin t~= 1/2, we have altogether six mutually orthogonal spin—isospin functions, Z~(12,3)
=
(ST) ~ (12, 3); S0t1M~n1~
(2.2.5)
,
where the interacting pair 12 has the intrinsic spin s~= S and isospin t~= T. S0 and t~(= 1/2) are, respectively, the total intrinsic spin and isospin of the triton. The superscript n specifies the various sets of S, T and S0 as
Z~’~(12,3)for 12~(12,3) for Z 31(12,3) for Z~ 141(12,3) for Z 51(12,3) for Z~ Z~61(12,3) for
T=1, S=0, S0=1/2, T=0, S=1, S, 1=1/2, T=0, S=l, S 0=3!2, T=1, S=l, S 03/2, T=0, S=0, S~ 1/2,
(2.2.6)
T=1, S=1, S 01/2.
We can rewrite eq. (2.2.4) in the form 1’~1(12,3),
(2.2.7)
~(12, 3) = ~ g(tl)(xs, y3)Z where the g1’~(x 3,y3) are the spatial parts corresponding to Z~(12, 3). We take appropriate linear combinations of these six functions to form the following six orthogonalized spin—isospin functions of proper symmetries: 5~(12, 3) + Z~61(12,3)1,
Ws(12, 3) = (1/~)[Z~
WA(12, 3) = —(1 /~)[Z”~(12,3)
—
5~(12, 3)
W~
Z~2~(12, 3)],
Z16~(12,3)],
—
12(12,3) = —(1 /~)[Z~
1(12,3)
W~
2(12,3)
W
=
(1 I~)[Z”
141(12, 3),
172(12, 3) = —Z
+
Z~2~(12, 3)],
W~
31(12,3).
2(12,3) = Z~
(2.2.8)
M. Igarashi et al., Two-nucleon transfer reaction mechanisms
11
Ws is totally symmetric, WA is totally antisymmetric, W~12is of a mixed symmetry with (12) symmetric, W~ is of a mixed symmetry with (12) antisymmetric, W~ is symmetric in spin space but of mixed symmetry in isospin space with (12) isospin symmetric, and finally W~,is symmetric in spin space but of a mixed symmetry in isospin space with (12) isospin antisymmetric. If we decompose the wave function ‘1 into these six spin—isospin components, we obtain the Derrick—Blatt classification [58], ~(12, 3) =
~AWS
+ ~SWA
+
+
+
+ ~12W~12.
(2.2.9)
Here, the cP are the spatial parts in the Derrick—Blatt classification and can be rewritten in terms of the g~(x,y) as follows: 5~(x, y) + g16~(x,~)] = _(1/~)[gW(x, y) g~2~(x, ~)] (1!~)[g~ —
1/2
~3/2
= —(1 I~)[g~5~(x, ~) =
g~4~(x, ~)‘
—
~3/2
g~(x~ ~)]~
~
1/2
= (1 /~)[g( ‘~(x,y)
+
g~2~(x, y)],
(2.2.10)
= g13~(x,y).
We truncate the function space at either 3 or 5 channels. In the case of a three-channel wave function, the spectator is in S-states and the interacting pair is in ‘S 3S, + 3D, states. In the case of 0 and and the interacting pair is in a five-channel wave function, the spectator can also be in a D-state, 3~, + 3D, states. The channels are further classified by the total spatial angular momentum L 0 and the total intrinsic spin S,~.These are represented by the index4~(x, k y), in table 2.1. g~5~(x, y) and g~6~(x, y) vanish, and so do Since we restrict l~and 1~to even values, the terms g~ 113A ~ and ~~2• Bui the total wave function 1I~of eq. (2.2.1) has non-zero values of and ~‘3I2’ because the other partitions ‘13(23, 1) and P(31, 2) contribute. However, there is no contribution to ~pA, since the contributions from ‘1(23, 1) and ‘13(31,2) cancel each other. We define a probability ~‘t/2
P(L0, r) =
ff
~r~2
dx dy
(2.2.11)
for each Derrick—Blatt classification symmetry r [which specifies six different components in eq.
Table 2.1 Classification of states cP~(i2,3) with five channels a 32 4 5 6 7 8
channel k
2~*
32
3D,
4 5 5 5 5
3S
‘S 3S,0
3D, 3D, 3D, 3D,
pair(12) (1,, j,)
particle 3 (L0, S~)
Z”
(0, 1/2) (0,1/2) (0,1/2) (2, 3/2) (2,3/2) (2,3/2) (2,3/2) (2, 3/2)
(0,1/2) (2,3/2) (2, 3/2) (0,1/2) (1,1/2) (1,3/2) (2, 3/2)
Z’3’ Z~’
t(12, 3)
12
M. Igarashi et al.. Two-nucleon transfer reaction mechanisms Table 2.2 Binding energies (B) and state probabilities (P) of the triton (on the left-hand side) and the deuteron (on the right-hand side) wave functions for different nucleon—nucleon forces triton B,
RSC3 RSC5 URG3 URGS UG3 UG5 TRS3 TRS5 TH
P
(MeV)
0 (%)
P0 (%)
6.389 7.031 6.945 7.480 6.844 7.395 6.829 7.464 7.42
90.08 88.90 92.30 91.25 91.59 90.48 91.09 89.87 1001)0
1.91 1.67 1.52 1.38 1.36 1.23 1.67 1.49
deuteron P~ (%) 0.09 0.05 0.07 0.06
P,, (%)
Bd (MeV)
(%)
8.01 9.34 6.19 7.32 7.05 8.22 723 8.53
2.225 2.225 2.150 2.150 2.082 2.082 2.226 2.226 2.207
6.47 6.47 4.95 4.95 5.51 5.51 5.92 5.92 0.00
(2.2.10)], and the spatial total angular momentum L0. The probabilities of admixture of the S, S’, P and D-states in table 2.2 are defined by 0J~)~
Ps.P(0,r—1/2~)+P(0,r-lI2),
~s~(
P~=P(1,r=S)+P(1,r=1/2~)+P(1,r=1/2)+P(1,r=3/2~)+P(1,r=3I2), (2.2.12) =
P(2, r3/2*)
+
P(2, r3!2).
Here we can show that P(L 0, r = 1/2 ~) = P(L0, r = 1/2 ) and P(L0, r = 3 /2k) = P(L0, r = 3/2) for all L,,. The P-state admixture appears in the case of the five-channel wave function but this is very small. To investigate the dependence of the absolute cross section of the one-step (p, t) process on the triton wave function, we tested various wave functions obtained with different nucleon—nucleon interaction potentials and different truncation of the spin—angular momentum space. The corresponding binding energies and the probabilities of admixture of S, 5’, P and D-states in the triton bound state wave function used in this work are listed in table 2.2. RSC, URG, UG and TRS represent the wave function for the Reid soft-core [59], Ueda—Riewe—Green model II [60], Ueda—Green model I [61]and Tourreil—Rouben—Sprung [62] nucleon—nucleon interaction, respectively. The numbers 3 and 5 mean the three-channel and five-channel wave functions, respectively. TH stands for the variational wave function with Tang—Herndon [15] interaction, which is also listed in the table for comparison. All calculated binding energies of the triton, B~,are smaller than the experimental value 8.482 MeV. Various additional sources of binding energy have been suggested such as three-body forces [50],but we do not discuss this further. This insufficiency of the binding energy decreases the magnitude of the transfer reaction cross section, since it indicates that the overlap between interaction and wave function is insufficient. The xy~~(x,y)’s of the RSC5 triton wave function are shown in figs. 2.2 and 2.3. The functions are contracted in the x-direction and surprisinglywith long-ranged in athe Thesoft-core ‘S~andradius the 3~, 3S, component 1,, = 2 have nodey-direction. at around the as components with l~ = 0 and the a function of x, which is caused by the strong repulsive core of the RSC potential coupled with a particle exchange effect of the Faddeev equation.
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
13
0.15
‘N
y=1.98fm
/ /
0.10
\\ \\
~
/~
~0.05
,1/
3S 1
_su2N..
—
“~
~
f: : ~ —0.10
0
1
2
3
4
5
6
x ( fm ) Fig. 2.2. The RSC5 triton wave function. The xy~(x,y)’s of eq. (2.2.4) as a function of x at y
1.98fm.
2.2.2. The deuteron wave function The sequential transfer (p—d—t) process passes through the intermediate deuteron unbound state as well as the ground state. The wave functions used are evaluated with the same realistic nucleon— nucleon interactions with non-central components that we used for the triton wave function calculation. The binding energies and the D-state admixture probabilities in the ground state are listed in table 2.2 for various interaction potentials. 0.15
x=1.5fm /
\
/
0.10 /
/ >,0.05
~:
\
I,---., I, //3/
~i
~1 N
-‘
D, —S,,2 D1-d312
/,
3c,
S~1~
\
N .
~:::~ —0.10 0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15
y ( Im ) Fig. 2.3. The RSC5 triton wave function. The xyç&jx, y)’s as a function of y at x = 1.5 fm.
14
M. Igarashi et al. - Two-nucleon transfer reaction mechanisms
The unbound ‘S0 and 3~,+ 3D1 states are calculated with the same realistic nucleon—nucleon interactions at appropriate incident energies. The coupled-channel equations (with a tensor force) are solved exactly for the bound and the scattering states. For the case of a momentum dependent force the method of numerical calculation given by Sawada and Sasakawa [63] is used. These deuteron wave functions are used for the calculation of the projectile form factors connected with the one-nucleon transfer process and the method will be described in sections 2.4.1 and 2.4.4. 2.3. The one-step process The transition amplitude for the simultaneous two-nucleon transfer process [one-step (p, t) process] is given by
~(~
T10 = KxH(k,, r,)~~( r~,rB(~B)~V~ ~,
~)
r~,r2)~~(k0, ~
(2.3.1)
Here x ~ (x is the proton (triton) distorted wave, and r~(ri) is the relative coordinate of the proton (triton) with respect to the target (residual) nucleus. The ~(i’sare the internal wave functions of the proton, the triton, and the target and residual nuclei. ~ r, r) and tI~A(~B,r,, r2) describe the internal motion of the transferred two-nucleon pair in the triton and the target nucleus. V is the residual interaction causing the two-nucleon transfer, V= + V2,, + VBP UAP in the prior form representation, where V,,~and V2,, are the interactions between the proton and transferred nucleons, VBP is the one between the proton and the residual nucleus, and Ui,,., is the optical potential used to generate the distorted wave x In the DWBA method, the approximation (xit/Jlt/JB~VBP UAP~tIJPt/JAXP)=0 is commonly used. The adequacy of this approximation has been argued on both theoretical and practical grounds [64] (other references are cited in ref. [52]). With this approximation the form factor of the projectile—ejectile system is given by —
—
(~t(~p’
r~,~
+
~
(2.3.2)
2.3.1. The one-step (p, t) form factor When the same interactions V,~and V2~,are used in both the evaluation of the triton wave function by the Faddeev equation and the calculation of the reaction form factor, the product (V1~+ ~ can be evaluated easily with the aid of the Faddeev equation (2.2.2), (V,3
+
V,3)~I-’~ = (E
—
H0)[13(13, 2)
+
‘13(23, 1)] .
(2.3.3)
On the left-hand side of this equation, ~ consists of three components, and both the wave function and the interaction potential have angular momenta, if the interaction potential has the tensor term. Many terms must therefore be evaluated, and the treatment of the angular momentum coupling is tediously complicated. On the other hand, the right-hand side can be evaluated simply, because E Hf) is a scalar operator. The method, however, requires a better accuracy of the wave function for the evaluation of H013 than is the case using the left-hand side of eq. (2.3.3). The coordinates used to describe the reaction are shown in fig. 2.4. There are three independent coordinate vectors, which describe the relative positions of the four particles participating in the reaction. In order to eliminate one coordinate vector and express the form factors as a function of the —
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
15
r
Fig. 2.4. Radial coordinates describing the one-step two-nucleon transfer process A(p, t)B.
channel vectors r~and r,, we use the coordinate transformation twice; once here and again just after eq. (2.3.28). First we transform them to the relative and center of mass coordinates of the two transferred neutrons. For the triton system we take the transformations (x,, y,)
~s
(R’, r)
,
R’ = ~y,
+
~x,
,
r=y,
(x2,
~‘
(R’,r)
,
R’ = ~y2
—
~x2
,
r= —y2—
y2)
~x,
—
(2.3.4a)
,
,
(2.3.4b)
and for the target system (r,, r2)
~
(R, r)
R = ~(r, + r2)
,
,
r = r,
—
r2
(2.3.5)
.
Corresponding transformations of the wave function are performed in the generalized Bayman—Kallio method [10, 65]. First we consider the transformation of the form factor of the triton system, eq. (2.3.2). According to eq. (2.3.3), the products of the interaction and the triton wave function are replaced by the sum of two functions, [V31(x2)+ V23(x,)]W = (E— H0)[t13(x2, y2)+ cP(x,, y,)].
(2.3.6)
We transform the second term on the right-hand side, (E H0)~(x,,y1), to the (R’, r) coordinate system, where the pair 1 and 2 are coupled first and then particle 3 is coupled to them in the angular—spin—isospin space. We separate the orbital and intrinsic spin parts of the wave function by transforming it to the L—S coupling scheme, 2)J~1/2; s~t im mi) a4~a(xi, y,) 13(23,1) = ~ (l~s~) j~t1,(l~lI ~x 5x ~x = ~ l~ 1/2 j~ (s 1/2)5’M~.}(23,1)~I(t~1/2)t,n~}(23,1)~ aL’S’Ms. L’ S’ st —
x (L’ m~ M —
5. S’MsIsim~)4 ~
y1),
(2.3.7)
16
M. Igarashi et a!.. Two-nucleon transfer reaction mechanisms
where the LS—jj transformation bracket is related to the 9j symbol by 11
~! 11
12
S
L
S
2
li
=
~,
________________________________________ 11~ Si + 1)(2j, + 1)(2L + 1)(25 + 1)1 12 ~2 12
.
(2.3.8)
J
~L S J 3, 1)) is the three-body spin function in which pair 23 is coupled to the intrinsic spin state Ks~1 s~ /2)S’Mç.}(2 and totally to the state S’. (t~1/2)t~n~}(23, 1)) is the analogous isospin function. The intrinsic spin and isospin functions are transformed from the (23, 1) coupling scheme to the (12, 3) coupling scheme using Racah algebra, (s~1/2)S’M~.}(23,1))~(ç1/2)t 1n,}(23, 1)) =
~ (s,s2(s)s3S’~s,s3(s~)s1S’)~(s~ 1 I2)S’M~.}(12,3)) x t,t2(t~)t3t,It2t3(t~)t,t,) (t 1 /2)t5n,}(12, 3)).
The spatial part is transformed to the form (E
—
)
~
13(~,
H0)~’~’~’~’(x1, y~)[y’oI(~ x y
=
(2.3.9)
A~.
g~~’(r, R~)[YA(~)x YA’(~l)]~’,, (2.3.10)
where lX 1~ 1),, A and A’ are the orbital angular momenta associated with coordinates x,, y,, rand R’, respectively, and L’ is the total one. The summation over A and A’ runs in principle up to infinity, but the contributions from the terms decrease rapidly as A and A’ increase. To obtain (r, R’) we set the z-axis along the ~ direction as shown in fig. 2.5 and multiply both sides of eq. (2.3.10) by (—)‘ +M (L’ _MfAlM~~A0)Y~,(R~)* and integrate over R’( 0, and then sum over M’. This choice of z-axis is different from those in other works [10, 65]. The merit of this choice is discussed later. Since
4)
g~~,f31)
E~I e
e
r
X2 R 2
Fig. 2.5. The integration of eq. (2.3.11) is performed with I=z and
~1
andx, in the x—z plane.
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
17
Y~,(~~)V(2A+1)/4ir8(m,0), we obtain R’)
g~Yt)L(r
xf (E
—
=
E (—)~M’(L’_M’A’M’~A0)\/2L 1~’(I,) x
H
1
y’Yl(fl]L’yL’(~l)*
y,)[Y
0)’’~’~(x,,
di’.
(2.3.11)
The explicit expression for the spherical harmonics is
~)etm4d~o(0)V(21
Y~(0,
=
+
1)/4~,
(2.3.12)
where d~~(0) is the reduced rotation matrix. The vectors x,, y 2 and R’ are now in a plane containing the z-axis. We can choose ~ = = = 0 without loss of generality. Finally, then, expression eq. (2.3.11) becomes (l~,1~)L’
g~~ (r, R ~=
1)(2l~+ 1)(2A + 1)(2A + 1) 2(2L’ + 1)
\/(2l~
+
x ~
(AOA’M’IL’M’)(lxmxly M’
x
f
—
mx~L’M’)
t~~’(x,,y,)d~
(E
—
0(~jd~m0(00)dZ0(0) d cos 0,
H0)
(2.3.13)
in which 2 + R’2 R’rcos 0, y, = ~\/~r2 + R’2 + 3R’rcos 0 x, ~~r cos0~=(—~r+R’cos0)/x,, cos0~=~(~r+R’cos0)Iy,. —
(2.3.14)
This implies the selection rule lx+ly+A+A’=even,
(2.3.15)
which ensures parity conservation. We can show that the second term (E
—
H
0)c13(x2, y2) of eq. (2.3.6) has exactly the same contribution as (E H0)11(x1, y1). We can express the Jacobi coordinate system (x2, y2) in terms of r, R’ and 0 as follows: 2 + R’2 + R’r cos 0, y 2 3R’r cos 0, x2 = ~~r 2= ~ + R’ (2.3.16) cos = —(~r+R’ cos 0)/x 2 cos = —~(~rR’ cos 0)/y2 —
—
,
—
18
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
According to eqs. (2.3.14) and (2.3.16), replacement of 0 by
ir
—
0 interchanges (x2, 0.,~,,y2, 0~.)and
(x,, 0~,y,,ir—00). Because d~1(7r 0) —
= (....)t_md/
-
~(0)
-
(2.3.17a)
,
we have and
4~,=0.
(2.3.17b)
We get the same type of integrand for the second term as on the right-hand side of eq. (2.3.13), differing only by a phase factor (_)bx~, where we put l~= l~= l~.Furthermore, the transformation coefficients for the intrinsic spin and isospin spaces have the following relations: (s,s2(s~)s3S’~s3s,(s5)s2S’) =
(2.3.18a)
s,s2(s~)s3S’~S2s3(s~)s1S’) ,
(_)50
(t1t1(t~)t3t~lt3t,(t~)t2t,) = (—)‘~‘~( t,t2(t~)t3t,~t2t3(t~)t1t,) .
(2.3.18b)
In the full space, the two terms of eq. (2.3.6) differ only by a factor (_)1x+A+so+so+tx+to~ This factor equals + I because of the antisymmetric nature of the pair of fermion particles. As a result the two terms give an identical contribution. As the final result the projectile form factor becomes lx
(~i(~p’ r, ~
+
~
=
~x
Ix
l~ s~ j~, I.~J0~~’S’M5S~2T12L’ 5’ s~
x (s,s2(S,2)s3S’Is2s3(s~)s,S’)~t,t2(T,2)t3t~~t2t3(t5)t,t,) 5~~5’V(25’ + 1)(2s + 1)W(s~S, x ~ (—) 2s~L’; S’s) x (L’ m,
—
M5.
M5. — m~Ism, — m~)(s~m0s m~— m~js~m~)(T,2 n~ n,, t~n~It~n,)
~12
—
5’(”,)X X~2(h1
x [x
2)]~,_mp[Xu1(Tt) X t~’(r,
x ~
R’)[YA(~) x Y 4’(~’)]~-Mç..
g~
(2.3.19)
For the target system we expand the wave function and the two-nucleon wave functions ~2N12(r,,
I/llaMa T8N~
~jAMA,TANAB’i’2)
~
in terms of the wave function r2) in the j—j coupling scheme,
t/JJAMATANA
SBA(ala2 jT,2)
a1a2jT12
x (JBMBI
MA
—
M81 JAMA)(TBNBT,2
~JBMB,TBNB~~i~2
NA — NB~TANA)
Ti2NA_NB(r,, r2)
,
(2.3.20)
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
where the
19
a2 jT,2)’s are the parentage coefficients of the two-nucleon configurations in the target nucleus IA MA.TANA• The normalized two-nucleon wave function I/r1mT12~’~2(r,, r2) is transformed ai a2 into the L—S coupling scheme, SBA(a,
I/’•
r2)
,j,JmT12NI2(
a1a2
1’
x
1/2(1 +1 ôaa)
=
(r,)I/i~ (r
[~.
2)
j1m1
—
1/2(1 + ôai.a2) x (L m
[
(_)i_T12~fjimi(T2)t~ri2m2(Tl)1[Xii(Ti)X
—
~
iI2II
12 1/2 L ~i2
LS12M51,
Ms S,2Msi2~jm)[~S1(u,)X
x {[~fri) X
12(T2)JN1~i2 1 1.
x
1/2 1,1
1,
1 =
~ (j,m1j2m2~jm) m1(m2)
I/’I2(r2)]~M5
()si2+
—
X52(~T2)]~[Xu1(Ti) X Xt2(T2)].~~
T12[I/~ (r2) X
I/~2(r,)]~M5}.
(2.3.21)
We transform the radial part to the form [I/i,(r1)X I/i~(r2)]rn —Ms ~
—
J l.A
= An f~1”2~(r,
x I/t2(r~)]rn — M5
()5i2 + Ti2[ t/i()
2)
X ~‘A(1~)]rn_M~
(2.3.22)
.
R)[Y5(i
As before, we set the z-axis along the ~ direction, and get f(t~.l2)L(r
R)
=
1
—
(_Y~2+Ti2+l
2
V(2l~+ 1)(2l2 + 1)(2A + 1)(2A + 1) 2L+1
x ~ (AOAMjLM)(1
1m,12M—mJLM)
Mm1
x
f
ut(rt)ut(r2)d~1o(0,)di2 M—m1,0’,‘02)d~0(0)d cos 0,
(2.3.23)
where 2+R2+rRcos0, r, =\I~r cos0 1=(~r+Rcos0)/r1,
r
2+R2 rRcos0, —
2=~~r
(2.3.24)
cos02=(—~r+Rcos0)/r2.
This also brings the selection rule 11+12+A+A=even.
(2.3.25)
We integrate over the spin and isospin functions of the transferred two-nucleon pair 12 and also over the vector i~to eliminate the dependence of the form factor on it, and we obtain
20
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
J
1t”t~~(r,R)[YA(~) x (t~,t )L’ {g~ m,—M 4,~(r, R’)[YA(~)x y(~~)]L* ~ 5.J AA =~
(
)..i+AAy(2L
)A+mt~Ms.~
+
AMB+mPMSJ
1)(2L’ + 1)W(LL’AA’; IA)
A
x
(L MA
x
g(~l~)L~ (r,
—
MB +
mp
—
M~.L’ —m,
+
Ms.~lMA
—
MB + m0 —
m~)
R’)f~,,”2~~(r, R)[Y4(1~)x YA(k’)]~M÷
(2.3.26)
We can sum easily over A, because the spatial parts of the functions g and f do not depend on A. Then, ~
W(LL ‘AA’; lA)g~5’.~Y)L (r, Rl)f
(_)l.
t2)L(r
R)
A
=
—L
1
x
21
x
+
+
1
—
1)(212
(
2
)S17+Tis+ti+t2+L
—
2
(.,~Si2+Ti7+tr+tv+L’
1)(2A + 1)(2A’ 1)(2L’ + 1)
+ 1)(21~+ 1)(2l~+
1
~
1
(2L
+
+
1)
(LML’M’IlM+M’)(AMA’M’llM+M’)
MM’ m1 m,
x (l,m,12 M
J J
X
—
mjLM)(l1m~l~ M’
u~(r,)u~(r 2 )dtim1.0 (01)d
—
m~lL’M’)
m10(02)d~0(0)dcos 0
—1
x
(E
—
H0)
txty~’(X,, Yi)~,o(0xi)~i~mx.0(0yi)~,o(0’)
dcos 0’.
(2.3.27)
—1
We get the final expression of the transfer form factor, which is a function of the coordinates R and R’, as follows:
~ =
r~,r~)t/l8(~8)IV,~ + V2P p(~p)~A(~B, T~, t(yP~mP(JMi MA — MBIJAMA)(l MA ~
—
MB +
mp — m~s m~ m —
211 MB
i
tsjAfl’
x (s~m~s m~ mPIs~m~)(TBNB T,2 NA —
—
NBI
TANA)(T,2 n1
—
n~t0n~It~n1)
jtl +t2~tx~ty~t
~
X
sBA(aia2
jT,2) 1/2(1
aia2j~]~ LL’S,2S
X
[1, 112 [L
+ 6a1,a2)
1/2 j,][ l~ Sx 1x1 1/2 12 [l~ 1/2 j~W(s~S,2s,L’;S’s)W(LL’is; IS,2) ~i2 ~ L’ 5’ s,j
I
x (SiS2(Si2)515’1S253(Sx)StS’) (t,t2(T,2)t3t~It2t3(t~)t,t~)
—
MA)
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
x (_)~P~5~t(_)S5i2(25+ 1)1/(21 + 1)(2L —
x
1
()1~~+t~L’
—
+
1)(2L’
+
21
1)(25’ + 1)
S
()
12+Ti2+1,+12+L
—
1
S,2+T,2+t~+t~+L’
() —
2 2 1 /(21, +1)(2l2+1)(2/~+1)(2l~—1)(2A+1)(2A+1) x21+1 ~ (2L+1)(2L’+1) x
~
(LML’M’IlM
+
M’)(AMA’M’IlM
+
M’)
MM’mim.1
x (l,m,l2 M — m1~LM)(l1m~l~ M’ m~IL’M’) —
x
x
f J
r2(J ui(ri)ut(r2)d~ 0(0)d~m 0(02)d~0(0)d cos 0
(E
—
H0)
x [Y~(’~)x
x.ty)L’(x,,
yi)d’~o(0i)4~rn o(0yi)~o(0’)d cos o’) dr
}‘A~(R)]MA_MB+mp_mt~
(2.3.28)
This transfer form factor, which is represented in the coordinate system (R, R’), is transformed once more to the channel vector (re, r~)coordinate system by the ordinary method of calculation of finite-range DWBA form factors [66]. 2.3.2. The selection rules for the one-step process From eqs. (2.3.27) and (2.3.28) we can get the relations of the angular momentum coupling as shown in fig. 2.6. An additional restriction is that A + ~12 is even as a consequence of the antisymmetry of the two transferred neutrons. The relative orbital angular momentum A is even for ~i2 = 0 or odd for ~12 = 1. The parity conservation imposes the restrictions that 1~~~1y = (_)A+A’ = +1 (2.3.29) = () ,
Fig. 2.6. The angular momentum coupling illustrating the selection rules.
M. Igarashi ci a!., Two-nucleon transfer reaction mechanisms
22
.
(2.3.30) 7rB designate the intrinsic parities of the proton, triton and the states of the target Here ~ ~“A and and residual nuclei A and B, respectively. The natural parity transitions satisfy the condition ~A
~B
= ~l1+12
= ()A+A
= ~IT
~,
= ~A~B
=
(~)‘~
(2.3.31)
where j is the transferred total angular momentum, while unnatural parity transitions satisfy the condition (2.3.32)
~A~B().
The component in which A equals zero contributes only to the natural parity transition. If A and ~12 are zero, then A = L and j = L, so that the parity change is given by the relation = ~A~B
= ()A+,.t
= ~
=
(—)‘.
(2.3.33)
However, if A $0, z~ir= (—)‘~ is allowed and unnatural parity transitions can occur. The A = 0 component is the main component of the triton wave function [thedefinition of A is accord with eq. (2.3.10)]. Therefore unnatural parity transitions are much weaker than natural parity transitions. The selection rules for each component of the triton wave function with a particular symmetry type is given as follows. Since we use eq. (2.3.3) in this analysis, we need to classify the symmetry type of (E H 0)[13(13, 2) + 13(23, 1)]. This classification is therefore not purely about the triton wave function, but rather about the product of the transfer interaction and the triton wave function. The interaction potentials have a dependence on the spin and angular momentum. We consider only the term (E H0)13(23, 1), since the two terms of eq. (2.3.3) obey completely the same selection rules. The recoupling coefficients (eq. 2.3.9) from the spin—isospin function Wr(23, 1) of eq. (2.2.8) to the z~~~(12, 3) are given in table 2.3. In the (p, t) reactions the isospin of the two transferred particles is 12~(12, 3), and Z~3~(12, 3) and Z15~(12, 3) are allowed to be[see onlyeq. in triplet states 1). Therefore not allowed (2.2.6)]. For(T,2 this=reason only S Z 12 = 0 transitions are allowed for the spatially symmetric state and only S,2 = 1 transitions are allowed for the mixed symmetry state with total spin —
—
S0=3/2.
The selection rules for each particular symmetry type are summarized in table 2.4. The spatial parts of each triton state are also given in the same table, where the ~k’~ represent the spatial part of the function indexed by k, which appears in table 2.1. In the case of the triton wave function with the three-channel configuration the transferred spin s in the reaction is always equal to the total spin ~i2 of the two transferred neutrons. Hence, A = even for s = 0 and A = odd for s = 1. The P-states are allowed in the case of the five-channel wave function, but the probabilities of the P-state are very small. For the P-states the s = 1 transition is allowed in the case of ~i2 = 0. Namely, both even and odd A are allowed
WA(23,1) W~ 2(23,1) W~,,,(23,1)
1(12,3) Z~’ —1/V~ —~/~/4 0
Table 2.3 The recoupling coefficients of spin—isospin functions Zt2~(12,3) Z’31(12. 3) Z’41(12, 3)
Z’”(12, 3)
Z16(12, 3)
1/V~
0
0
0
0
—‘~ñ/4 0
0 —t/2
0
\ro/4 0
0
v’~/2
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
23
Table 2.4 Selection rules for particular symmetry types of the triton wave function state
(L
S sym. S mix.
(0,1/2) (0,1/2)
Psym. P mix. (S0=1/2)
(1,1/2) (1,1/2)
P mix. (S0=3/2) D mix.
(1,3/2) (2, 3/2)
0, S~)
S,2
spatial part
—(1/’v’~)(~,—~2—~5) 0 (1/V’~)(~, + + f~) 0 1 ~is/~ 0 ~6/v’~ 0 1 ~ I ~ +~+ 1
A
s
even even odd even, 0 even, 0 odd odd odd
0 0 1 1 1 0 and/or 1 0 and/or 1 1
Table 2.5 Allowed values of the relative angular momentum of the transferred two nucleons 0~—s0~
0*3*
(!sj)
A
(!sj)
A
(000)
0,2,4,... (1,3,...)
(303)
2,4,6,... (1,3,...)
(110)
1,3,5,... (2,4,...)
(213) (313) (413)
1,3,5,... (2,4,...)
for the transferred spin s = 1 transition, but A = 0 is not allowed. For the P-states both s = 0 and s = 1 transitions are allowed in the case of ~12 = 1. The allowed values of the relative angular momentum A are listed in table 2.5, where the set (lsj) stands for the transferred orbital, spin and total angular momenta of the transfer form factor. The odd A’s for the s = 0 transition, and the even A’s for the s = 1 transition are allowed in the case of the five-channel triton wave function, but not allowed in the case of the three-channel wave function. 2.3.3. The hard-core correction to the form factor for particle transfer processes Several exact finite-range DWBA calculations [13, 14, 67, 68] with various sophisticated triton wave functions and finite-range interactions have been reported. The resulting cross sections of all these calculations are smaller than the observed ones, but the differences among them are not very large. Most of these calculations, however, employ two-body interactions with a hard core like the Tang— Herndon interaction [15]. As has been pointed out by Dobes [16], the use of such a singular potential for rearrangement collisions requires a significant correction. The form factor for rearrangement collisions contains an overlap integral of the type (I/tuncorrL1”k//corr), in which the interaction V is put between the wave function çt’corr, which correlates with the interaction in question, and the uncorrelated wave function I/’uncorr~An overlap integral of this type has a finite contribution from the inner region of the hard core, which is shown by the hatched area in fig. 2.7. If we consider a square potential with finite V~,and thickness r, and increase V~to infinity, the potential times the correlated S-state scattering wave function inside the potential becomes
M. Igarashi et a!.. Two-nucleon transfer reaction mechanisms
24
r~
r
Fig. 2.7. Hard-core effect.
lirn V(r)I/i(r) = lirn
k’ sec(kr~)e’~,
(2.3.34)
where E = h2k2/2~and V0 E = h2k’2/2/L, where E and ~sare the scattering energy and the reduced mass of the scattering particle, respectively. This may be represented as a discontinuity of the slope of the wave function at r = r~,and is described by the following additional term, the so-called pseudopotential term: —
V’(r)=
)
-~-- 8(r—r —~-— r. 2~sr~ c ar..
-~—
(2.3.35)
This hard-core correction term has opposite sign compared with the usual overlap integral from the region of attractive potential and reduces the total magnitude of the overlap integral. This term is familiar in Brueckner theory [69], but it has not been used in transfer reactions. In the case of (p, t) reactions the overlap integral with hard-core corrections is given by +
V~~(r~~) + V~~(r~~) + ~
(2.3.36)
These correction terms are calculated exactly for the case of the triton wave function obtained by Tang and Herndon (TH). Its spatial part is assumed to be totally symmetric, and expressed as ~~=N~f(Ir,_r 2I)xf(Ir2_T3I)xf(H_r,~),
(2.3.37)
where N, is the normalization constant. The two-body interaction used by TH is V(r)
=
P0V0(r)
+
P,V,(r)
(2.3.38)
,
where P0 (P,) is the spin singlet (triplet) projection operator, and V1(r) (i = 0, 1) is given by ‘71(r)
=
(r ~ d)
,
V,(r)
=
—V~exp[— K~(r— re)]
(r
r~).
(2.3.39)
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
The parameters used are r~= 0.45 fm, V~= 277.07 MeV, V? 2.735
=
549.26 MeV,
25 K
0 = 2.211 fm’ and K1 = fm’. The radial function f(r) in eq. (2.3.37) has the form 2[exp(—ar)+ B~exp(—f3r)] (r d), (2.3.40) f(r) = u(r)/r (r~d), f(r) = A~r~’
and u(r) is obtained as a solution of the differential equation Fl 22du ——
—i
+[~(V (2.3.41)
0+1/~)—e]u(r)=0.
The constants A and B, are determined so as to match smoothly the two expressions at r = d. The other parameters are determined to minimize the binding energy of the triton in a variational sense [15]. TH gave a few sets of parameters and we here choose the following set: a = 0.265 fm’, /3 = 3.60 fm’, d = 1.20 fm and e = —10.0 MeV. In this case the pseudopotential is given by V’(r) = 150.435 MeV fm x ~5(r re).
(2.3.42)
—
The importance of the hard-core correction terms in the (d, t) form factors is illustrated in appendix A. 2.4. The sequential transfer (p—d—t) process The second-order DWBA amplitude for the sequential transfer (p—d—t) process is given by TIdP
=
~
~
“~~~)~frB(~B)IV,2 + V,P~d(~P, ~
X G~(r~, T~)(~d(~P, T4~C(~B, TI)I V~p -
Kx~(k,,~
x
(I/~(~,
+
exchange terms
r~,r
V2~
T~hc(~B, T,)~
(1.E-32),
T))
p(~p)~A(~B’ TI,
B(~B)k~d(~P, ~c(~B~ P(~P)~A(~B,Ti,
T2)X~~~(k~, re))
r))
r2)x~(k~, Tn)) (2.4.1)
where G~(rd,r~)is the distorted wave Green function of the intermediate deuteron channel and the second term is the channel non-orthogonality term. For the calculation of these amplitudes we need three kinds of form factors of the projectile system: (a) (~PjV23JI/Jd(r)),
(b) (çll~(T)~V~2 + V,3~I/t,(r,P))r’ (c) (I/id(r)kit(T, p))r’ where the subscript T stands for an integration over the coordinate T. We evaluate these form factors employing the same realistic triton and deuteron wave functions as well as the realistic transfer interaction as for the one-step process. Our purpose is to calculate the one and two-step processes with the same degree of accuracy. 2.4.1. The one-particle transfer form factors The RSC5 triton wave function, and the deuteron ground state and unbound state wave functions
26
M. Igarashi ci al.. Two-nucleon transfer reaction mechanisms
obtained with the Reid soft-core potential are used in the present calculation. Since these wave functions include the D-state and the interaction includes L—S and tensor components, the transfer form factors of the projectile—ejectile system consist of I = 0 and I = 2 angular momentum components. The projectile form factor of the (p, d) process can be calculated directly as a product of the interaction and the deuteron wave function [70]. The form factors between the deuteron and triton are evaluated using the relations (I/i0~V,2+ V3jI/,)
(I/’~~ I/’1)
=
(I/i0(12)~E— H0~’1(12,3) +
(I/.~~(12)~ 13(12,3)
=
13(23, 1)
+
+
cP(31, 2))
(2.4.2)
,
13(31, 2)) .
(2.4.3)
2.4.1.1. The projectile form factor for the (p, d) process. We define the deuteron wave function, which includes the D-state component, by Isdtdmdnd)
=
(lms~m~
~
—
(~~)
mlsdmd)[X,,
x
X
~(~n)]~m[~,(Tp)
X,(T~)1~i’Y~(r)u1(r),
ims~~
(2.4.4) where u1(r) is the radial wave function of each component. The nuclear interaction has central, L~S and tensor force components, V~~(r) =
1’~(r)+ VLS(r)L S
+
‘
VT(r)S,. .
(2.4.5)
The projectile form factor is given by (s0t~m~n0 ~
(Im~md — mp
~
= I!
Isdtdmdnd)
x (t0n~t5n0 5P
x
—
m~Jm0
—
m~)(s~m0Jm~— mP~Sdmd)
‘.,
-I
—
flp tdnd )i
m a, —,n Y~ (r)~2,’
n’Pn(_)5P~J’dV(2s
+
—m
(o~~ )x,~ P(r~) 1)(2j + 1) W(ISflSdsP; ~
-
(2.4.6)
(_)
We sum over I’ and s~and get
E
(IsPflsdmd
~
0m0)
V~(r)u0(r)— =
I ‘S~~s
\/~
—V’s VT(r)uO(r) + V~(r)u0(r)
(I = 0, ~
VT(r)u2(r) 3VLS(r) [V~(r)
—
—
2VT(r)]u 2(r)
=
1),
(I = 2, s,,,, = 1), (1 = 0, s~= 0).
(2.4.7)
Substituting (2.4.7) into (2.4.6) we get two form factors with (1= 0, j= 1/2) and (1= 2, j=312). 2.4.1.2. The projectile form factor for the (d, t) process. The numerical calculation method is
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
27
analogous to the case of the one-step (p, t) form factor. The final expression is given by
(~~(12)J V23 + V~jI/~) =
~ (Ims~m~jm + ~1
m~)(J0m~ m —
.1
x (T,2 n1
—
m
“
—
m
m~jm
+
m~~s~m~)
n
n~t,,n~t~n~)i Y, ~
x ~ \/~(2Jx+1)(2Jy+1)(2lx+1)(2ly+1)(2l+1)(2j+1)(2J0+1)(2L’+1) tXSxIr LL’S’
x (2S’ + 1)K5i52(Si2)53S’~5253(5x)5iS’)(t1t2(T,2)t3tjt2t3(ç)t,t1) 1x ~ j~ L 1 L’ 1 x 1~ s~ j~ ~12 ~ 2 jtx+Iy~/~L L’ S Si ~d .1 5i —
~‘
x ~ (—)t~M(lml~ M
x
fJ
(E
—
—
m~~L’M)(L’ —MlM~L0)
H 0)
Y)d~O(0x)4~mO(0y)d~M ~(0) d cos
tty)L’(x
This form factor also has two components (1 = 0, j = 1/2) and (I
=
2, j
0 uL(r)r dr.
=
(2.4.8)
3 /2).
2.4.1.3. The non-orthogonality overlap form factor for the (d, t)process. The direct overlap integral appears in the overlap form factor in the non-orthogonality term of the two-step transition amplitude. This direct overlap integral is easily calculated as
j m + mfl)(Jd m~ m — m~j m + m~js1m~)
(c~d(12)~(12, 3)) = ~ (ims~
—
tjL
x
(T,2 n~ n~~
The other terms (~d(12)~(23, 1)
—
+
f
2 dr. (2.4.9) ~LI(
p)uL(r)r
~(31, 2)) are calculated in an analogous way as in subsection
2.4.1.2.
The calculated form factors of the projectile—ejectile system for the ground state (~d= 1, t 0
=
0)
deuteron channel are shown in fig. 2.8. 2.4.2. The finite-range form factor for the one-particle transfer process The finite-range form factor of each one-particle transfer step in the two-step transition amplitudes is calculated by the code TWOFNR [71]. This code uses the method of coordinate transformation given by Austern et a!. [66] for the calculation of the finite-range form factors. Let the orbital and total angular momenta of the transferred nucleon in the target nucleus and the projectile nucleus be (1,, and (12, 12)’ respectively, as shown in fig. 2.9. The allowed transferred
1~)
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
28 10~ ~
1
~2
(b)
~a) 102
,,,
101
.
~p IVnP I~d)
~l
-
\,
~
‘V
~ ~
-
L=2
(~d~Vnn~V~pNI’t)
‘~s\
I0l
-
L=0
/
.
“S
N
~
102
N
-
/‘
-
10~ I
L—2
3
<~dI’4’t)
/
‘~
io10~ ~ 0
1
2
3
4
5
6
7
8
0
1
2
R(FM)
3
4
5
6
7
8
R(FM)
Fig. 2.8. Form factors of the projectile—ejectile system evaluated with the RSC potential, the RSC ground state deuteron and the RSC5 triton wave function. (a) Projectile form factors of the (p. d) process. (b) Ejectile form factors of the (d. t) process. (c) Overlap form factor of the (d, t) process.
angular momenta, usually denoted by 1, s and j, are given by
11
1—I2H
ii
121
<—~I+i2,
~I2,
~
(2.4.10)
The transferred angular momenta (Is]) corresponding to the first and second steps are referred to as (lsj),0, and (Isj)2fl0, respectively. The inclusion of the (12 = 2,12 = 3/2) component of the projectile form factor compels us to calculate many transition amplitudes. Such transition amplitudes arising from minor admixtures are not important for the absolute cross section at low incident energy, but they play important roles in the tensor polarizations and analyzing powers [72].
Fig. 2.9. The coordinates and angular momenta involved in the one-particle pick-up reaction A(a, c)C.
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
29
2.4.3. The spin—isospin closure approximation The deuteron is a loosely bound particle and it is easily broken up. Furthermore, if we use the spin—isospin closure approximation in the intermediate deuteron channels, the selection rule for the transferred spin s is quite different from the case in which only the ground state deuteron channel is taken into account, as shown below. Therefore, for the intermediate channels of sequential transfer processes, we take into account the ‘S0 and ~ + 3D, unbound deuteron states as well as the ground state. As Pinkston and Satchler [73] pointed out, ‘S 0 channels in the spin—isospin closure approximation drastically change the spin transfer selection rule in the sequential transfer process from that when only the ground state channel is taken into account, namely the s = 1 transfer component disappears. The assumptions implicit in this approximation are the following: (a) The triplet (s~= 1, t~= 0) and singlet (s~= 0, t,~= 1) deuteron states are degenerate. (b) The triton and deuteron are purely in the symmetric S-state and the interactions which cause the transfers are purely central forces. (c) The optical potential of the intermediate deuteron channel does not have a spin—orbit coupling term. Under these assumptions, the transition amplitude through the singlet (s~= 0, t, = 1) deuteron intermediate channels has the same radial overlap factor as the one through the triplet (s~= 1, t, = 0) deuteron channels and we can easily sum the spin—isospin kinematical factor of these two deuteron channels. This kinematical factor is given by I, s, j, __________________ 12 ~2 12 V(2s~+ 1)(2t~+ 1)W(sas,s8s2 5c5Y~4’(tat1t8t2 tat) Is] X
\/n~Ksctc{l(sata);s,t,)-\/7c~sbtb{I(sCtC);s2t2)
(2.4.11)
,
where Ii’ 5, and], for i = 1, 2 are the angular momenta transferred in the first (i = 1) and second (i = 2) steps, respectively, and 1, s and j are the transferred angular momenta through the whole sequential process. The spin and angular momentum relations are shown in the fig. 2.10. The values without the 9-] symbols are listed in table 2.6. If we choose only the triplet deuteron ground state channel, both s = 0 and s = 1 transferred spin transition amplitudes are allowed. But if we sum the transition amplitudes from both triplet and singlet intermediate channels, the s = 1 transferred spin transition amplitudes cancel. Therefore the two-step amplitude has no s = 1 transfer amplitude and no distinctive selection tc
~c
S6
intermediate (lsj~-~ ~a
~a
“A
incident
“N~(ls~)2nd lsjt
~
8b
tb
final
Fig. 2.10. The spin and angular momentum scheme of the two-step process.
~B
30
M. Igarashi ci a!.. Two-nucleon transfer reaction mechanisms Table 2.6 spin—isospin kinematical factors Deuteron channel Transferred s, t
(s
(s = 0. (s = 1.
-3v~!4 -V3/4
t
=
t
=
1) 1)
1.
t
0)
(s =0, ç
=
1)
(s
=
1.
t
=0)+ (s =0, ç
=
I)
.\/~i4
\‘~/4
0
rule from the one-step process. This fact is very important for polarization phenomena. In the case of transitions to the unnatural parity states, the cross section of a sequential transfer calculation under these assumptions is strongly reduced and the analyzing power vanishes exactly if no spin—orbit term is included in any of the distorting potentials. So we must check the validity of these three assumptions of the spin—isospin closure approximation. The reality of the closure approximation will be checked by using realistic unbound deuteron wave functions and realistic transfer interactions. In our calculation the three assumptions mentioned above are found to break down. The analyzing power data of the 205Pb(p, t)206Pb 3 transitions are useful for this check, as discussed in subsection 4.2.2.2. 2.4.4. The unbound intermediate deuteron channels The intermediate channels of the unbound deuteron states are treated in the following way. Let an unbound deuteron wave function be $(k, T). It is a positive-energy solution of the Schrodinger equation for the p—n system, [T~
2/2,a~)k2]4.(k, r) = 0 (2.4.12) 2+ V~,,(r) (h where r, T~ 2,V~and k are the relative coordinate vector, the kinetic energy, the interaction potential and the asymptotic wave number of relative motion between p and n, respectively. Here we use a realistic nucleon—nucleon interaction for Vu,,. namely the Reid soft-core potential. We truncate the continuous spectrum of 4(k, r) at a certain value kmax, and divide the range [0, kmax] into a finite number, N, of bins with equal widths ~k = k, k, . We take the average of cb(k, r) in each bin, ,
—
—
=
~
J
~(k,
T)
dk,
.-
(2.4.13)
4
to2P~/2~R, representis equal in the bin,average and assume the c.m. to the energythat in each bin. energy of the p—n pair corresponding to h Thus, 112k2
E
0=
112P2
(2.4.14) 2~R 2 + ~(~k 2. (2.4.15) = ~(k~+ k,1) 1) The “wave packet” ~ 1(r)has a faster decay of amplitude along the relative distance r than the original —i- +
2,a~
—i-
.
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
31
scattering wave ~(k, T). We calculate the projectile form factor of each one-particle transfer step using the wave packet function. Namely (a) (~~IV23l~~(r)), (b) (4~(T)lV,2+ V131I/i1(r, p)),. and (c) (~~(r)l ~f,(T, P))r are used for each discretized unbound intermediate deuteron channel. The value of kmax depends on the spin state of the p—n system and also on whether one calculates the interaction term or the non-orthogonality term of the T-matrix element. We used theinteraction following values: 3D, state for the terms, kmax= 1.Ofm’ for the ‘S0 state and kmax=2.0fm~’ for the 35~ + and kmax = 1.Ofm for the S 0 state and kmax =3.Ofm for the 5, + D, state for the nonorthogonality terms. The criterion for these choices is the convergence of the transition amplitudes. In the present calculation, ~k is chosen to be kmax!8 and the k-space truncations and discretizations are shown in fig. 2.11. Figure 2.12 shows the contributions of individual bins (1—8 corresponding to the range 0.1—1.0fm~) in the singlet unbound deuteron (‘S0) channel to the interaction term. The convergence with respect to a change of kmax is quite rapid in this case, but depends on the spin state of the deuteron channels and on whether the amplitude is the interaction term or the non-orthogonality term. The choice of the distorted potential for the c.m. scattering wave calculation in the discretized unbound channels has not been well established. Since the coupling among the continuum channels is strong, the folding potential of the proton and neutron optical potentials with the “wave packet” functions may not be reliable for the second-order DWBA calculation [51]. We should evaluate the coupling effects between the continuum channels with the distorted potential. For simplicity, however, we use the elastic scattering optical potential at the energy corresponding to the ground state deuteron channel for all unbound state channels. A theoretical approach for the excited intermediate deuteron states has also been proposed by Austern and Kawai [74] based on the closure property (the adiabatic method).
Interaction
Non-orthogoncility
term
term
1 frrc
Li ~ is
0
3D 1 Fig. 2.11. Truncation and discretization of k-space.
32
M. igarashi et a!.. Two-nucleon transfer reaction mechanisms
~
1~
~ -deutert TOtQ~-~....jnterQctbfl
—-.-------
i:
I~:
Si::
iO~
~ax~°fm1
0°
20°
N
40°
60°
80°
~9c.m. 10°Pb(p—d—t)205Pb3’ process at E~= 35MeV. The Fig. 2.12. The contributions to the interaction terms from individual bins for the case of the intermediate deuteron state was spin singlet and k,,.~,= 1.Ofm’. The first neutron is picked up from the 1,,, state and the second one from the p.. state.
2.5.
The spectrscopic amplitude for the particle transfer reactions
2.5.1. The antisymmetrization of the DWBA amplitude The completelyantisymmetrized wave function of the total system I/i~,A can be expanded as a linear combination of products of the separately antisymmetrized projectile I/la and target I/’A wave functions. Let P’A(i) represent the operator that interchanges 2m particles between the two groups a and A, i.e., PaA(l) consists of m transpositions. The number of independent m interchange operators Pa~(1) is N’A = (~)( ~,‘), where i = 1, 2, . . - , ~ Then the total number of possible interchanges is given by (2.5.1) where we assume a ~ A. One obtains for the completely antisymmetric wave function with the proper normalization / I/IA
=
)
+A~~2 1
a
a
m N~’A
m=0
m
-
PaA(1)I/II/1AXaA
()
=1
,
(2.5.2)
where XaA is the wave function of relative motion between a and A. The nuclear matrix element of the particle transfer reaction, which is represented by the prior form, is reduced to
M. Igarashi et al., Two-nuc!eon transfer reaction mechanisms —1/2
(~/ib,BIVaA
(a +A)
UaA~aA~
-
Na”A
~
)
(a a
a
33
~b,BIVaA
~
-
UaAlP~(~)~a~AXaA)
(2.5.3)
K~BUaAXaA).
The last equation may be obtained as follows. The potential VaA — UaA can be replaced by H E as already mentioned in section 2.1. Then the operator ~~A(~) commutes with the full Hamiltonian H, and we also use the fact that PamA(~)t/IbB= ()mIfrb,B. 4’b,B is expanded similarly, —
N~
b
\t/2
/
,, ,.~ U
=
)
(b + B)
5
v
z1 (—) ~ n0 i=t b
—1/2 ~
,, P8~(t)ç&~ç1J8x~fi
b—a ~ (_)(n-P)(b-a) p=O
(_)fl
0
N~PN~~ ~ i=t 1=t
~ P~(i)P~B(])~BxbB,
(2.5.4)
n=
where b > a and P~B(]) stands for p interchanges~ofparticles between the group ä~=(b — a), i.e., the particles belong to b but not to a, and B [74, 75]. The phase factor ()(1~P)(l~~) results from the antisymmetry of the wave function of nucleus b when the (n — p) particles pass through group e~. Inserting eq. (2.5.4) in eq. (2.5.3), 1/2 ~/ib,BIVaA
-
UaAl~aA) =
(a +A)
1/2
(b
+
B)
b
b—a
~
~
~
~(n-P)(b-a+1)
K
x XbBI/1bh/1Bl1~a; (1)(VaA — UaA)IcllaclJAxaA~, where we have used
P~B(j)4’A = (—YI/’A.
K~/ib,BIVaA- UaA~aA)
=
1/2
~
x
~
(2.5.5)
Writing m = n —p, then
(~)(~) 1/2
(b; a)(B)
1/2
a (_)m(b-a+t)(b
(XbBI/lbt/IBIPaB(l)(VaA
—
~
-
a+ m)
.
(2.5.6)
Thus the nuclear matrix element is seen to have (a + 1) terms; the m = 0 term is the direct term; the second one corresponds to one particle in a being exchanged with one in core B; the third corresponds to two particles being exchanged, and so on. The exchange terms are in general expected to be considerably smaller than the direct term, although this fact has not been carefully examined. Recently a theoretical consideration of exchange effects in DWBA was proposed by Kozak and Levin [64]. In this work only the direct term will be considered. For the case of the direct term the spectroscopic amplitude is given by b 1/2 1/2 St12 = ()(b~a)B() (a(sa, ta), b — a(s, t)I}bs 8, tb)(B) KB(JB, TB), A — B(], t)I}AJ~,TA~. (2.5.7)
34
M. Igarashi ci a!., Two-nucleon transfer reaction mechanisms
The phase factor ()(ha)B is required for expressing S 1/2 in terms of the c.f.p., because the particles in A need be reordered from a + 1, ~, b,. . . , b + B to b + 1,. . - b + B, a + 1, ~, b, which requires (b a)B transpositions. The phase factor is positive for the one-step two-nucleon transfer process, but is negative for the sequential two-nucleon transfer process, because the same kind of spectroscopic amplitudes appear twice. The phase factors of one of them is negative and the other is positive, because one of the nuclei is odd and the other is even. This phase difference is important for the coherent sum of the one- and two-step processes. ,
—
2.5.2. The spectroscopic amplitudes of each process
The spectroscopic amplitudes of the projectile system are listed in table 2.7. The c.f.p. for the decomposition of the projectile system is included in the radial part of the form factor in this work. Here only the Clebsch—Gordan coefficient for the isospin transformation and the statistical factor are tabulated. The spectroscopic amplitude of the target system is given by 51/2(] t)= (~)‘2(B(J8, TB), A
—
B(j, t)l}AJA, TA).
(2.5.8)
For the case of (p, t) pick-up reactions from a doubly closed shell nucleus the spectroscopic amplitudes in the J representation (separate treatment of protons and neutrons) are given as follows. (i) A 0~ J transition of the (p, t) process, in which two neutrons are transferred from the same shell 11’ —+
2(j~, ~B)
S~
=
(2]i+
1)12(
]~J1i(JB);
i~(JB)l}J~’~’ (a))
=
1/~J~ +1.
(2.5.9)
This factor is independent of whether 11 is an outermost or inner shell. (ii) A 0~ J transition of the (p, t) process in which one neutron is transferred from an outer shell 12’ and another one from an inner shell —*
],,
S~!~2(jj2;~B)
= (_)212(211+
1 )12(2i 2+
1)1 /2(
.21i(.)
i~ I)
j~”~’(o))
]~]~ 0
=
.212 - -2j,+i (0)) 12‘ X 12 (12); 121)12
12
0
lB
~B
0
—1/2J~+ 1.
(2.5.10)
Table 2.7 The spectroscopic amplitudes of the projectile system Reaction (p.1) (p,d) (sd = 1, (p,d*) (Sd =0. (dl) (sd = 1, (d*.t) (sd =0,
(çn~tn~t~n~) Id
td td td
=0) = 1) =0) = I)
-V~7~ —iiV~
(1 )‘~
—1
1/V’~ 1
2 S~ -v~
1 \/~ —l
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
35
(iii) A one-particle transfer from an outermost shell 12’ 5T1/2 (12)
=
v~Ki-n,—l 2 (u2); 121}122(a2)) U(a, U2JAJ2 fl
(2.5.11)
JBa2),
where a1 is the core spin of target A; ~A= at x ]~~(a2), and the recoupling coefficient of angular momenta U is given by
U(abcd; ef)=~(2e+ 1)(2f+
1)
W(abcd; ef).
(2.5.12)
(iv) A one-particle transfer from an inner shell j,, 2(j~) = ()JAJ
ai
],~}j~(a,))U(a
1(_)n2~Kifi’(u);
S~/
(2.5.13)
2~,J~j,; jBa,),
where n2 is the neutron number of the outermost shell and this shell is not always fully filled and has spin a2. The spectroscopic amplitude of the (p—d—t) process used for the pure configuration [j,, j,]J is given
by 2’1’\/2(2J + 1). S~/~(j~) x 5~2(],) = (—) The spectroscopic amplitude of the (p—d—t) process used for the pure configuration
(2.5.14)
=
by
1!2
=51/2(J)
x
S~/2(j2) = (_)11~2~V2J +
1~
— V2J
+
1
[],,j
2]J is given
ôj.J7~
(2.5.15)
S
where j’ denotes the total angular momentum of the intermediate channel. In actual calculations we construct the transition amplitudes using these pure configuration spectroscopic amplitudes and the configuration mixing coefficients given in table 4.2. 2.6. The differential cross section and analyzing power The transition amplitudes of the one-step process (eq. 2.3.1) can be written in the form ~
=
~ (2] + 1)”2(JAMA1 MB — MAlJBMB)B’~’°(0),
(2.6.1)
tsj
B°~”°(o)= ~ S2f3;rlia(kb,k),
(2.6.2)
shell
where m = MB — MA + mb — ma is the possible magnetic substate quantum number of the transferred orbital angular momentum 1. The spectroscopic amplitude S1t2 and the reduced amplitude f.3 depend on the nuclear shell configurations. The sequential transition amplitude TIdP (eq. 2.4.1) is written in the same form but the function B(0) is now B~a(0)=
5~2X$~m~(k 8,k). shett,(tsj) is,.(tsi)2fld
(2.6.3)
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
36
The angular differential cross section is defined by du.
=
U~hf
~
k Tr(rtTp’T~), ~a
j~ 2 2
(21Tt1
)
(2.6.4)
where T is the sum of ~ and ~ and the is’s are the reduced masses. In the case of equal probability of observation of all final spin substates for equally prepared initial spin substates, the efficiency and density matrix elements take the following forms: rnbMB,mhMB =
(2.6.5)
mhmbMBMB~
P~nMA.m~M~ = 25a + 1
2~A
+
1
(2.6.6)
3mam~6MAM~’
Hence the cross section is given by do
V
2jB+l
11
kb
P~aPb
=
(2~h2)2 ka (25a + 1)(2JA + 1)
jmmm~
lB 1
where B. is the sum of ~
2
mmm
b
a(O)l
and BtdP for all possible l and
(2.6.7a)
,
s,
m”m”(o)= ~[B B7
1~°(0)+B~1(0)].
(2.6.7b)
tst s
In the case of vector polarized projectiles, the density matrix is given by P
2Sa+1 2JA+1 (i+
+1
(2.6.8)
~~MAM~’
and hence the differential cross section is da~— dUf1 dfl — ~Ifl
P~aPb
+ (21Th2)2
_____
1T’P
k,, 1 T ka (25a+1)(2JA +1) 5a+1 r~ k
a
.
~ T ~1 Sal
2 69 .
~
where ~a is the vector polarization of the projectiles. The analyzing power Ab is defined by P •A
— do~Id~1 —1do~1/dQ1 d~1/dQ1 —
3
—
+
If
1
a
the polarization
b)
~a
—
is prepared
~a
Tr[T(Pa Sa)T~1 +1 Tr(TT~) -
2 6 10 ( . . )
along the y-axis, which is perpendicular to the scattering plane zx,
A~(0)becomes [114] ~1mmbma
A~(0)°°
V(~a
—ma)(sa
+
ma + 1) ~
Bmmbma(OY
5 a
Jmm5m,,
j
\ I
.
(2.6.11)
M. Igarashi et a!., Two-nuc!eon transfer reaction mechanisms
37
2.7. Why analyzing power? For the reaction A(a, b)B, let us denote the transition amplitudes corresponding to each different path of the reaction process by ~ ~ Let m = MB MA + mb — ma, as in eqs. (2.6.2) and (2.6.3), and let the superscripts indicate the paths of the different processes. Figure 2.13 shows the four different paths as an example of two-nucleon pick-up reaction mechanisms. We express the differential cross section (2.6.7) and the analyzing power (2.6.11) symbolically by the forms —
r(0)
=
~
~ T~
A(0)=Im ~ m,~m’
=
~
(~
T~2 + ~ T~T~’)*),
(2.7.1)
(~
(2.7.2)
T~T~*)/cr(0).
pp
As the m = 0 component is usually predominant, we explicitly write o~and A in the forms ff(0) = I T~I2+ I T~2~2 + 2 Re[T~ T~2)*]+ . A(0)
=
. .
(2.7.3)
,
Im(T~’)T~’)* + T~T~*+ T~,l)T~2)* + . . .)/~r(O)
(2.7.4)
Suppose that process (1) is the main contribution and process (2) is one of the correction terms such as higher-order contributions. Then we see that the correction (2) is more sensitively involved in A(0). The main term is the cross term of the m = 0 (spin-nonflip) and m = 1 (spin-flip) components in process (1); the interference terms between processes (1) and (2) therefore give rise to more sensitive contributions to A(0). This is particularly so for unnatural parity state transitions in (p, t) reactions, where the one-step m = 0 transition is almost forbidden, and the sequential transfer m = 0 transition is allowed. Hence the interference term T~!)T~* makes the main contribution to A(0). By utilizing this characteristic of analyzing powers, we have extensively studied nuclear reaction mechanisms as well as microscopic structures of nuclei and effective nuclear forces using the 22 MeV polarized proton beams accelerated by the University of Tsukuba 12 MV Tandem Accelerator.
X Fig. 2.13. Various competing processes in the reaction XN+ 2(p, t)YN. The direct process (1), the sequential transfer (p—d—t) two-step processes (3), (4), and the inelastic two-step process (2) are indicated.
38
M. Igarashi et al.. Two-nucleon
3. Experimental
transfer reaction mechamms
study of (p, t) cross sections and analyzing powers
3.1. The measurement of (p, t) analyzing powers and cross sections We explain the measurement of the analyzing power for the reaction 2”8Pb(p, t)‘OhPb as a typical example of a (p, t) analyzing power experiment. Polarized proton beams were produced by a Lamb-shift-type polarized ion source with nuclear-spin filter [77]. The 22.0 * 0.05 MeV polarized proton beams were obtained using the University of Tsukuba Tandem Accelerator. The beams were analyzed by the 90” analyzer magnet and focused on a *“‘Pb target by a switching magnet and a pair of quadrupole magnets. The beam intensity on the target was 50-100 nA with a spot size diameter of less than 1 mm. The energy spread of the beam was 0.5 keV. We used a 208Pb target, 0.48 -t 0.02 mgicm’ thick, backed by a 3 pm aluminum foil. The enrichment of the ‘08Pb target was 98.69% ‘08Pb (0.26% for *06Pb and 1.03% for *“‘Pb). The thickness of the ““Pb target was measured by weighing and checked by comparing the proton elastic scattering data with the optical model prediction. The experimental setup of the detection systems is shown in fig. 3.1. We measured and monitored the beam polarization P by using the reaction ‘Li(p, a)4He at 0,,,, = 130”, where its analyzing power A(0) is known to be 0.80 t 0.01 [78]. The polarization monitor target LiF was placed upstream of the 20sPb target. The monitor target was 60 kg/cm2 thick LiF evaporated onto a carbon foil. We measured the left-right asymmetry of the alpha particle yield of the reaction ‘Li(p, cy)4He using two solid-state detectors at 0,,,, = 130” (see fig. 3.1). The up and down spin direction of the incident protons was flipped every 100 nC of beam current integration by the fast spin state interchange system (FASSICS) [79]. This system controls the spin-flipping pulses of the polarized ion source and supplies gate signals to the particle counting system for each spin state. The typical beam polarization was P = 0.80, and the error in the beam polarization was kO.02. The degree of beam polarization thus obtained was consistent with that obtained by the quench-ratio method [77]. Tritons from the (p, t) were momentum analyzed by a magnetic spectrograph [80] and detected by a single-wire position-sensitive proportional counter (SWPC) system [81]. The horizontal acceptance of Magnetic
spectrograph
\
Bcm Fig. 3.1. Experimental
setup for the measurement
=130°
of analyzing powers and cross sections for the reaction “‘Pb(p, t)2”hPb at E, = 22 MeV,
M. Igarashi et al., Two-nucleon transfer reaction mechanisms
39
the magnetic spectrograph was A8 = 2.0”, which corresponds to a solid angle of 2.0 msr. The SWPC system has a sensitive length of 30 cm and is composed of a position counter and a AE counter. This system covers an excitation energy range up to about 4 MeV in “‘Pb for a given magnetic field. The coincidence requirement between the position and AE signals lowered the background level in the triton spectrum. The vector analyzing power A(8) for the (p, t) reaction is obtained from the relation
N+(8) - N_(8) F N+(8) + N_(8) ’ 1
A(@) =
(3.1.1)
where N+(8) [N_(8)] is the number of tritons at the reaction angle 8 produced by the spin-up [spin-down] proton beams with degree of polarization P. The differential cross section ~(0) with the unpolarized proton beam is given by g(0) =d@)ldfl=
&[N+(e) + N_(0)],
(3.1.2)
where the factor (Yis obtained from the target thickness, the number of incident protons and the solid angles of the spectrometer. We measured the angular distributions of the analyzing power A(B) and the cross section g(0) from 19~~~ = 5” to 90” in 5” steps. 3.2. Experimental results for (p, t) analyzing powers and cross sections We show in fig. 3.2 a typical momentum spectrum of tritons from the reaction 208Pb(p,t)206Pbwith a = 25”. The full width at half maximum (FWHM) of the triton 22 MeV spin-up proton beam at 19,~~ spectrum is 30 keV, which is mostly determined by the energy losses in the 208Pb target. All the observed triton peaks were identified: the excitation energies of the levels of 206Pbobtained from the peak positions in the triton spectrum coincide with those obtained by previous experimental work [82] with the experimental error of t10 keV In the following sections, the statistical errors in the observed analyzing powers and cross sections are always indicated by error bars in the figures of their angular distributions. Usually the error bars for the cross section data do not exceed the size of the data points (circles etc.). The absolute errors of the analyzing powers (cross sections) are estimated to be about 4% (15%). We list our (p, t) experiments in table 3.1. Most experiments have been done at an incident proton energy E, = 22 MeV, but some at energies of 17.0, 18.5, 20.0, 21.0, 21.5, 22.5,28.0,35.0, and 50.0 MeV in order to investigate the effect of the energy dependence of the (p, t) transitions. 3.3. Experimental studies of the (p, d) and (d, t) reactions In order to evaluate the contribution of sequential transfer (p-d-t) two-step processes to the (p, t) reaction, we made (p, d) and (d, t) experiments. In the study of 208Pb(p,t)206Pb(3:) at E, = 22 MeV, for example, we measured the cross sections ~(0) and the vector analyzing powers A(B) for the transitions 208Pb(p,d)207Pb(p,,,, p 312, f 512, f 712, h 9/2, i 1312 ) at E, = 22 MeV, and for the transition 207Pb(d,t)206Pb(3:) at E, = 17 MeV. In addition, we measured (~(0) and the vector and tensor analyzing powers iT,,(8), T20(0), T2,(0) and T22(0) for the transitions 208Pb(d,t)207Pb(pl/2, p 312, f S/2, f712, i 1312 ) at E, = 17 MeV so as to study single-particle transitions. We measured the tensor analyzing powers
M. Igarashi et al., Two-nucleon transfer reactlon mechanisms
40
208Pb(p,t)206Pb 8 l&,=25’
Ep=22.0
MeV
SPIN UP
800 CHANNEL
1200
1600
2000
NUMBER
Fig. 3.2. The triton momentum spectrum of the 2”8Pb(fi, t)*O’Pb reaction at E, = 22 MeV, O,,, = 25”, with spin-up protons.
T2,(0) to obtain precise nuclear structure information concerning the deuterons and tritons involved in the reactions; the tensor analyzing powers TZq(0), especially T2r(0), are sensitive to the non-S components such as the D component of the deuteron and/or triton. Our (p, d) and (d, t) experiments are listed in table 3.2. The experimental cross section and analyzing-power data have all been analyzed in terms of the first-order DWBA method in the references cited and they provide fundamental information for the analysis of the (p-d-t) two-step processes. In addition to the (p, d) and the (d, t) experiments, we carried out systematic measurements of the elastic and inelastic scattering of polarized protons from various nuclei [83] and also of the elastic scattering of vector- and tensor-polarized deuterons [84]. The aim of the elastic scattering experiments is to obtain the optical potential parameters for protons and deuterons while that of the proton inelastic scattering experiments is to fix the strength of the inelastic scattering part of the (p-p’-t) two-step process. For the triton channels, we use the Los Alamos triton elastic scattering data [117].
4. Results and discussions 4.1. The one-step (p-d)
process
The absolute values of the calculated cross section of one-step (p, t) processes have been investigated by many authors. The results, however, scatter over a wide range, which obscures our understanding of
M. Igarashi et al., Two-nucleon transfer reaction mechanisms
41
Table 3.1 The (p, t) reactions for which analyzing powers and cross sections have been measured. Analyses in terms of one- and two-step processes are given in the cited references, E, is in the lab. system, Qp is the Q-value for the ground state transition and .I: the final state spin and parity of the nth excited state of excitation energy E, Target nucleus
Final nucleus
EP (MeV)
*“Pb
‘“‘Pb
22.0
J:(K) -
$eV) -5.63
Ref.
(MW
45, 47
O;, O’(1.167) 3;(1.341),
3;(3.12)
2;(0.803),
2;(1.467),
4;(1.684),
4;(1.998),
7,(2.200),
7;(2.865)
48, 85-89 2;(1.784), 4;(2.928),
2:(2.149), 5;(2.783),
2;(2.424),
87
5,(3.014),
35.0
3;(1.341)
48
50.0
3l(1.341)
48 47
“‘Pb
‘04Pb
22.0
-6.34
0;
“‘Pb
2”‘Pb
22.0
-6.70
06
47
14’Nd
‘42Nd
22.0
-5.47
0:
45, 90
“‘Ba
““Ba
18.5
-7.035
0;
20.0
0;
21.0
0: 0; , 2; (0.816)
22.0
91 91 91 91, 92 91
28.5 ‘Ye
‘Ye
22.0
-6.02
0; , 2; (0.743)
“‘Te
“‘Te
22.0
-6.59
O;, 2;(0.666)
28, 45, 92-94
rZ2Te
12”Te
22.0
-8.55
0;) 2;(0.560)
92, 93
““Sn
““Sn
22.0
-7.19
O;, 2;(1.294)
92, 93
l16Cd
““Cd “‘Cd
22.0
-6.36
0;) 2; (0.558)
92, 93
22.0
-7.10
O;, 2;(0.617)
92
“‘Cd
““Cd
22.0
-7.89
06, 2; (0.374)
92
““Pd
‘“‘Pd
22.0
-6.48
0;) 2; (0.434)
28, 45, 92-94
0;) 2;(0.434)
92
“kd
55.2
92
‘““Pd
22.0
-7.28
O;, 2;(0.512)
92
‘04Pd
22.0
-8.16
0;) 2;(0.556)
92
rU2Pd
22.0
-9.15
0;) 2;(0.557)
45, 92, 93
O;, 2;(0.557)
92
55.2 lU2Ru
22.0
-6.65
O;, 2;(0.475)
92
“mRu
22.0
-7.54
0;) 2;(0.540)
92
“Ru
22.0
-8.70
O;, 2;(0.653)
45, 92, 93
9hM~
22.0
-6.98
0;) 2; (0.778)
92
y4Mo
22.0
-8.05
O;, 2;(0.871)
92
=Mo
22.0
-9.26
0;. 2;(1.509)
46, 92, 95, 96
“Zr
17.0
-7.35
0;
45, 91, 95, 96
18.5
91
20.1
91
21.5
91
22.0
91
22.5
91 91
28.5 @Ni
“*Ni
“Ni
‘“Ni
22.0
-8.02
18.0
-9.94
97 97 97
20.0
97
22.0 ‘“Ni
‘“Ni
22.0
-11.90
97
58Ni
‘“Ni
22.0
- 13.97
97
42
M. igurashi
et (II.,
Two-nucleon
transfer
reuctum
mechanisms
Table 3.2 (p. d) and (d, I) reactions for which analyzing powers and cross sections have been measured. The definitions of E, or E,, Q,. Jz(E,) are the same as in table 3.1. Analyses in terms of DWBA are given in the cited references E, or E,
Q,
J:(k)
Reaction
(Me”)
(MeV)
(MeV)
“‘HPb(p, d)“,~Pb
22.0
-5.14
-‘“XPb(d,t)““Pb
17.0
-1.11
li2-(0), 3i2 j/2 (0.570), 13/2’(1.633). I K(U), 3:2 512 (0.570). o;, 0;(1.167)
X?Pb(d, t)?“hPh lJ’Nd(p. d)‘“Nd lJ’Nd(p. d)li’Nd “hSn(p. d)“‘Sn “‘Cd@. d)“‘Cd “‘Cd(p, d)“‘Cd “lXPd(p, d)““Pd
17.0 22.0 22.0 22.0 22.0 22.1) 22.0
-0.48 -5.59 -7.59 -7.11 ~6.82 -7.18 m-7.00
““Pd(p, d)‘“‘Pd “‘2R~(p, d)“” Ru “Zr(p, d)“Zr
22.0 22.0 22.0
-7.32 Ph.99 -6.42
Ref. (0.898) 7’2 (2.340) 9K (3.409) ~(0.898) 712 (2.340), 13/2’(1.1633)
2;(0.803), 21’(1.467). 7i2 (0). 3/2 ~(0.742) 3i2’(0), 112’(0.194). Ii?‘(O), 3/2 (0.16), l/?‘(O) Ii2 (0) 5/2’(O). 1/2’(0.12) lli2~(0.21) Si?‘(O) 5/2’(O) 5/2’(O), 5!2’(1.47),
3,‘(1.304) 1112 (0.754) 1112 (0.32)
l/2-(1.21).
7/2’(1.88)
98 98 98 99 99 100 87 98 101 101 IO1 101 101 101 101 101 101
the reaction mechanism of (p, t) reactions. In this section we carefully study the absolute values of calculated cross sections, in particular their dependence on different triton wave functions. Test of the computer code and comparison with the results obtained with the Tang-Herndon triton wave function As a test of our computer code PTFFITWOFNR [102, 711, we repeated the same calculations as the ones undertaken by Bayman and Feng [14] employing direct integration by the Monte Carlo method. We obtained perfect agreement with their results. We compared the results with the code by Takemasa [68]*’ in detail employing an expansion in harmonic oscillator wave functions. After correction of an error in his code, we got good agreement. Previous calculations [13, 14, 67, 681 do not include the hard-core correction terms mentioned in section 2.3.3. For a detailed study of the hard-core correction, we investigate the ground state transition of the “)Ca(t, p)42Ca reaction at E, = 10.1 MeV as an example. In table 4.1 cross sections of simultaneous transfer calculated under different assumptions are compared with the measured one. The optical potential parameter set II of ref. [14] is used. The radius, diffuseness parameter and the spin-orbit strength of the single-particle binding potential for the ““Ca nucleus are chosen to be rC1= 1.24 fm, a = 0.65 fm and V,, = 6.01 MeV, respectively. The angular distribution of the cross sections is almost the same for all cases in table 4.1, so we compare the absolute values at 0”. For the simple [(f7,2)2] 0’ wave function, the cross section calculated with the Tang-Herndon (TH) triton wave function with the hard-core correction is close to that of the RSC.5 triton wave function, but the TH result without the correction overshoots it by about a factor 3. If we include the correction terms the cross sections are reduced by a factor of 112.4-112.9 (the small 4.1.1.
*’We
arc indebted to T. Tdkemasa for kindly making his program available to resolve the disagreement
between our results and theirs.
M. lgarashi et al., Two-nucleon transfer reaction mechanisms
Table 4.1 Comparison of cross sections of the various calculations for the ?a(t, 10.1 MeV Projectile
Nucleus
duldR(0”)
TH without corr. TH with corr. RSCS RSCS RSC5 Exp.
(f,‘$ (f, J2 (LS 4 config. 22 config.
0.2383 0.0981 0.0814 0.1608 0.2321 3.8”’
d’ Experimental
value is taken from ref.
(mb/sr)
43
p)“Ca g.s. reaction at
Relative strength 2.93 1.21 1.00 1.98 2.85 41
[103).
difference comes from a small difference in the angular distribution). Kunz [17] obtained simultaneously almost the same results as in this work. The natural parity strong transitions are particularly sensitive to the pairing correlations in the target nucleus. To take account of this effect, we compare the cross sections obtained with different shell model wave functions of the target nucleus with different truncation spaces. The 4- and 22-configuration wave functions were calculated with the shell model of Sturmian basis [20]. The shell model amplitudes of the four configurations (Of,,,)‘, (l~,,*)~, (Of,,,)2 and (lp,,,)’ are 0.9909, 0.0929, 0.0891 and 0.0386, respectively. Case II in table 3 of ref. [20] was used for the 22-configuration wave function. The pairing correlation enhances the cross section. The enhancement factor corresponding to the extended shell model is about 1.44 compared to the usual shell model with one major shell model space. This factor is important but not enough to explain the difference between the experimental data and the theory as we can see in the last row in table 4.1. The calculated cross sections of the simultaneous transfer processes are still too small by a factor of more than 10. From these results we conclude that this reaction cannot be reproduced by the simultaneous transfer mechanism alone. 4.1.2. Cross section of the one-step (p-t) process in z08Pb(p, tj206Pb reactions The purpose of this work is to investigate the mechanism of (p, t) reactions rather than details of the nuclear structure. Hereafter we concentrate on 208Pb(p,t)206Pb reactions because of the simplicity of the nuclear structure. 208Pbis a doubly closed shell nucleus and the collectivity of the 3- (2.61 MeV) excited state of 208Pbis not so strong. The simple shell model with a core + two-hole configurations gives good predictions on the structure of the 206Pbnucleus [104]. For the unnatural parity (UNPS) 206Pb(3:) transition, the two-step contribution (p-p’-t) via inelastic excitation of the ‘ORPb(3-,2.61 MeV) state is estimated with the Y,-type collective vibration model for the excitation and the microscopic wave function [105] for the transfer. To avoid complications, only spin transfer s = 0 transitions for the (p’-t) step are considered with a finite-range transfer form factor, and the 3- excitation contribution is found to be so small that the final result is not appreciably affected by it. The selection rules are greatly different between natural parity and unnatural parity transitions as mentioned in section 2.3.2. As examples of the two types of transitions, we investigate “‘Pb(p, t)206Pb(01, g.s.) and (3:) E, = 1.34 MeV). The shell model wave functions of the states of the ‘06Pb nucleus used in this work are listed in table 4.2. The two 3’ excited states are assumed to be pure (p,\f,i) and (pli:fy,i) states for simplicity. The radial wave function of the transferred neutron of each transition is calculated by the half
M. lgarashi
44
et al..
Two-nucleon
transfer
reaction
mechanisms
Table 4.2 Shell model wave functions of the states of “‘hPb. Amplitudes of the various neutron hole state configurations State Amplitudes
(MeV)
Ref.
p;:
(i,,.,)’
(fd
0.090 0.122 0.159 0.1794
0.109 0.152 0.204 0.210 0.1736
0.059 0.083 0.082 0.074
I=O’ 0.0 1.167 1.167 1.167 1.167
(PJ 0.822 -0.495 -0.534 -0.396 PO.4138
0.834 0.786 0.851 0.8755
1=3’ 1.34 3.122
(PWf, 2) 1.000 0.000
(P,.zPv) 0.000 I.000
1=2’ 0.80
(PI Zf( 1) 0.724
(P,‘?PI>?) -0.523
(f%P,‘,) PO.176
(il,.J -0.055
(fr.>hq.z) 0.092
(f,,zhv:) 0.012
0.029
(fJ 0.567
(f,,,P, 2) -0.727
(P,.,f,.,) -0.293
(fv2f7.& PO.135
(P,.,f,,,) 0.148
(LA!,)
~O$rv~z)
to;r
(f,.,h,z) 0.019
(i,,.$
1=4’ 1.68
0.077
0.145 0.183 0.210
(fSJ7.2)
(PWf7.Z) 0.185
-0.058
[IO41
h)’ LJ
A2
1~6’ 3.25
(f,,,f,.z) 0.971
(f,‘lhq,?) -0.090
(Pi,z’bz)
(4
0.110
-0.153
-0.102
1=7-
(hzill12) 0.952
(LM 0.250
hhzi,,.,) -0.169
(f7.2i1112) -0.055
0, zLL) 0.023
2.20
[1041 11041 11061 I1071 I1041
hzb,)
-0.050
0.097
(fwh,J
(hw)’
-0.025
PO.051
[IO41
11041 11041
separation energy method. The radius, diffuseness and spin-orbit parameters of the single-particle Woods-Saxon potential for the 20sPb nucleus are Y”= 1.25 fm, a = 0.65 fm and V,, = 6.0 MeV, respectively. The absolute value of the calculated cross section depends on the triton wave function used. The integrated cross sections (00-90”) for the 0: and 3: transitions at E, = 35 MeV are calculated with various triton wave functions and are tabulated in table 4.3. Set A in table 4.4 is used for the optical potential parameters. The calculated cross sections of the 0: ground state transition are about 1/4-l /6 of the experimental data. The TH triton wave function with the hard-core correction gives almost the same result as the other triton wave functions obtained by the Faddeev equations (RSC and URG in table 4.3). The s = 1 transition amplitude increases when the hard-core correction term is taken into account. The reason is that the s = 1 transition amplitude starts from the relative angular momentum of the two transferred
Integrated
Table 4.3 partial cross sections (WYO”) (’m mb) of the “‘XPb(p, t)“‘hPb reactions at 35 MeV
0: (n.s.)
(W = (OW
(110)
TH without corr. TH with corr. RSC3 RSCS URG3 URGS
8.844 x 3.243 x 3.517 x 3.564x 3.912 x 3.954x
2.436 1.461 4.182 4.874 3.988 4.821
10 z lo-’ 10~ 2 IO L lo-’ IO ?
x x x x x x
lo-’ 10 ’ 10 -’ IO-’ lo-‘ lo-‘
3: (303)
Sum of s = 1
Total
3.932 1.563 1.361 9.981 8.681 6.920
3.278 1.374 5.798 4.906 4.647 4.164
3.671 x 10 -’ 1.390x10 ( 7.159 x 10 j 5.904 x 10 1 5.515 x loml 4.856x IO ’
x lo-’ x lo-’ x 10mJ x 10-j x 10 ( x 10 ’
x x x x x x
lO_’ lo-’ lo-” IO-’ lo-’ lo-”
M. Igarashi et al., Two-nucleon transfer reaction mechanism
45
Table 4.4 Distorting potential parameter sets. A: set 1 used in ref. [43] for E, = 35 MeV; B, C: sets 1 and 2 used in ref. [85] for E, = 22 MeV; D: set 2 used in ref. 1681 for E, = 40 MeV; E: used in ref. 11081 for E, = 50.5 MeV Set
Channel
V
w
Wd
r0
a,
rI
a,
Vso
:
47.9 97.8 149.8
0 0 12.0
10.0 14.0 0
1.25 1.25 1.24
0.65 0.65 0.68
1.25 1.25 1.43
0.76 0.78 0.87
0 0 0
51.8 112.0 168.9
0 0 9.9
10.0 19.4 0
1.25 1.25 1.20
0.65 0.682 0.65
1.25 1.25 1.60
0.76 0.783 0.97
6.0 6.0 6.0
1.12 1.12 1.15
0.47 0.47 0.92
1.25 1.25 1.30
57.6 105.8 160.9
2.14 19.68 17.3
1.17 1.15 1.20
0.75 0.81 0.72
1.32 1.34 1.40
0.66 0.68 0.84
6.2 0 2.5
1.01
0.75
1.20
0.72
1.25 1.15 1.30
1.125
0.873
1.026
0.794
1.25
0.75
0.873 0.624 0.75
5.84
1.10
1.125 1.386 1.60
1.168 1.25
0.81 0.72
1.233 1.45
0.78 0.72
5.93
A
t B
P d
t C :
t D
E
P
54.62
8.84 0 0
5.31
t
160.0
20.0
5.60 0
P
47.52 160.0
4.09 20.0
5.51 0
t
rso
as0
rL 1.25 1.25 1.30
1.40 1.13
0.79
1.18 1.40
neutrons A= 1 as mentioned in section 2.3.2 and the hard-core correction term has larger components of higher multipole than the interaction term. For this transition Takemasa et al. [68] reported that the simultaneous transfer mechanism reproduces the observed absolute cross section [109]. But their conclusion must be changed, since a programming error was found in their computer code and they also did not include the hard-core correction terms. For the unnatural parity 3’ transition the TH trition wave function, which is a purely symmetric S-state, gives a very small cross section. Even if we include the hard-core correction terms, the cross section is smaller by more than a factor 10 than the results with other realistic triton wave functions. The main s = 1 transition amplitudes are caused by the hard-core correction terms. As an example of the nuclear interaction with a weak tensor force, we take the URG interaction [60] and the corresponding triton wave function. This force gives a slightly larger binding energy of the triton as shown in table 2.2. From these results we can say that the absolute cross section does not depend so strongly on the triton wave function. It is, however, worth noting that the cross section of the natural parity transition becomes larger with a triton wave function with a larger triton binding energy. In contrast, the cross section of the unnatural parity transition depends more strongly on the wave function and decreases when the binding energy increases. These facts are easily understood as follows. A triton wave function with a larger binding energy has a small volume and therefore, a larger S-state component and better overlap with the nuclear interaction. The higher angular momentum components become small. As shown in table 2.5, the allowed relative angular momenta A of the two transferred neutrons start from h = 0 for the natural parity transition and from A = 1 for the unnatural parity transition. The lowest A-component determines mainly the magnitude of the cross section. Figure 4.1 shows the contributions of individual A components to the cross section of 208Pb(p,t)2”6Pb 0:. The contribution decreases rapidly as h increases, which justifies a truncation in h. In this work we truncate at A= 5. The RSC wave function is used for triton. This wave function has a very small P-state component. Therefore the odd-h components are allowed in s = 0 transitions and even A’s (but h f 0) are allowed in s = 1 transitions as mentioned in section 2.3.2. These components are also taken into account up to A = 5 in this work. But their contributions are negligibly small.
46
M. lgarashi
et al.,
Two-nuclron
transfer
reuction
mechanisms
1 208P~(~,~) *‘+B
0+
EP=~~MEV
10-l
10-Z K 2 10-3 ,E Z 0 10-4 I= G * 1o-5 2 g 10-6
10-7
10-a
10-g 0
10
20
30
0
40
I-+C.M.
50
60
70
80
90
(degree)
Fig. 4.1. One-step DWBA cross section calculations for the 0. g.s.transition. Individual contributions from each relatrve angular momentum A of the transferred two nucleons are shown. The even h’s for (kj) = (Ooo) transfer and the odd h’s for (Isj) = (110) transfer are shown. but contributions from the P-state components are not shown in this figure. The RSCS wave function is used for triton.
For unnatural parity transitions the largest A = 0 component is not allowed by the selection rules. So, the cross section becomes very small compared with natural parity transitions. Figure 4.2 shows the decomposition of the cross section for the 3: and 3; transition into contributions of individual transferred anguiar momentum (Isj) components. We can see that the s = 1 transfers dominate over the s = 0 transfer, and the difference of the combination of single-particle orbits changes the relative strength of the (Isj) components as mentioned in a later section. 4.1.3. Contributions from minor components in the triton wave function In the present calculation the mixed symmetric S’-state and the D-state of the triton wave function are included. In figs. 4.3 and 4.4, we show the contributions of those components to the calculated cross section. For the 0’ ground state transition the S’- and D-states give a small contribution compared to the symmetric S-state, but the S’-state is not negligibly small in the coherent sum. The decomposition into the Derrick-Blatt classification [see eq. (2.2.9)] used here is applied to the product of the interaction potential and the triton wave function instead of the triton wave function alone, because we avoid complexity and this is the only one easily calculated by our method [see eq. (2.3.3)]. On the other hand, in the UNPS transition the contributions from the minor D- and S’-state components of the triton wave function are dominant, because the contribution from the main symmetric S-state component is strongly hindered by the selection rules. Therefore the s = 1 transfer
M. lgarashi et al., Two-nucleon transfer reaction mechanisms
I
20ePb(p- t)206Pb
0
10
20
30
40
50
0HC.M.
Fig. 4.2. Decomposition
60
70
8
1
3*
60
1
1
’
1
70
60
-
Ep = 35 MeV
10
20
(degree)
into transferred
0
47
30
40
0HC.M.
angular momentum (kj) components
50
60
90
(degree)
for the two 3’ transitions.
10-l
lo-"
lod
to-7
b
0
10
20
30
40
0HC.M.
50
60
70
80
90
(degree)
Fig. 4.3. Calculated one-step cross sections for the zOsPb(p, t)Z”bPb g.s. 0’ reaction at E, = 35 MeV, employing the RSC5 triton wave function. ‘The partial cross sections which come from the S-, S’- and D-state components and the total cross section are shown. The same optical potential parameters are used as in ref. [67].
M. lgurashi
er al.,
Two-nucleon
transfer
1
lo-:
208Pb(p,t)206Pb
reactlon
c
mechanisms
/
’
’
3+ Ep=35MeV
loD-STATE
n
s
a
0.
-0.
-1,
Fig. 4.4. Calculated one-step sections and A(0) for the two different 3 ’ excitations. The RSCS triton wave function was used. The partial and total cross sections and A(0) are shown as in fig. 4.3. Set A in table 4.4 was used for the optical potential parameters. A(0) from the S-state component is zero. because the optical potential used here has no spin-orbit distortion term.
amplitudes, which provide a non-zero analyzing power A(8) for the reaction even without spin-orbit distortion, dominate. In addition to this, the A(B)‘s of the two transitions (3;, 1.34 MeV) and (3;, 3.122 MeV) are completely different from each other at forward angles. One-step results are shown in fig. 4.4. These depend strongly on the combination of the single-particle j-shell orbits from which the two neutrons are picked up. We will discuss this point in detail in section 4.3.3. 4.1.4. The -‘“‘Pb(p, t)““Pb reaction at 35, 40 and 50.5 MeV In order to investigate the dependence of the absolute cross section on the transferred angular momentum, we compare the magnitude of the theoretical one-step cross sections with the experimental ones leading to the lowest lying higher-spin states of the residual nucleus in the ““Pb(p, t)‘06Pb reactions at E, = 3.5MeV [l lo], 40 MeV [ill] and 50.5 MeV [ 1081. The normalization constants N(Y) = (T(Y; f&&Q”; e),, are given in table 4.5. We use the shell model amplitudes in table 4.2
M. Igarashi
et al.,
Normalization
Two-nucleon
transfer
reaction
mechanism
49
Table 4.5 constants for the ‘“8Pb(p, t)*06Pb reaction
J”
0; (0.00 MeV)
2,‘(0.80MeV)
4;(1.68MeV)
6;(3.25 MeV)
7;(2.20 MeV)
N(35 MeV) N( 40 MeV) N(50.5 MeV)
6.0 17.3 8.8
3.0 9.0 5.4
2.0 8.0 3.5
2.7 14.7 3.4
3.3 10.5 2.1
and optical potential parameter set A in table 4.4 for 35 MeV incident energy, set D for 40 MeV and set E for 50.5 MeV The normalization factors are smaller for the higher-spin states. Probably this reflects the reliability of the shell model wave functions for those states. For the low-spin states we need a more extended shell model configuration space to take into account the lower-pole correlation [20, 211. The normalization factors for 40 MeV incident energy are larger than those for other energies. The reason for this is not clear. 4.2. The sequential transfer (p-d-t) process First we test the reliability of our projectile form factors for each of the (p-d) and (d-t) steps. 4.2,1. Cross section and polarization observables for one-nucleon transfer reactions 4.2.1.1. (p, d) reactions. The projectile form factors of the (p, d) processes are calculated using the
RSC interaction and the deuteron wave function is obtained with the same RSC interaction. The results are shown in fig. 2.8a. We use these form factors for the exact finite-range analysis of the ‘“*Pb(p, d)*07Pb reactions at 22 MeV [98]. Set B in table 4.4 is used for the distorting potential parameters. The wave functions of the transferred neutron in the target are generated by the conventional separation energy method with parameters rg = 1.25 fm, a = 0.65 fm and V,, = 6.0 MeV. The cross sections and analyzing powers are shown in fig. 4.5. The experimental cross sections and analyzing powers are both reproduced well by the calculation. The 7/2- state is not a pure f,,, single-hole state, so the calculated cross section must be normalized to the experimental data. The L = 2 (D-state) component of the projectile form factor is important to improve the fits to the experimental analyzing power data. 4.2.1.2. (d, t) reactions. The projectile form factors of the (d,t) processes were calculated using the RSC interaction and RSCS triton wave function and the deuteron wave function obtained by the same RSC interaction. The results are shown in fig. 2.8b. We used these form factors for the analysis of the 208Pb(d,t)*07Pb reactions at 17 MeV [99]. The optical model and neutron potential parameters are given in ref. [112], and the binding potential parameters for the transferred neutron are r. = 1.25 fm, a = 0.65 fm and V,, = 6.0 MeV We get good predictions of the angular distribution of cross sections and also vector and tensor analyzing powers. These results are shown in figs. 4.6 and 4.7. The spectroscopic factors for the single-hole states are 2.8 (P~,~), 8.2 (f& 5.3 (P~,~)’ 16.0 (i,,,,) and 8.0 (f,,,), respectively. These were obtained by normalizing the DWBA cross sections to the measured ones, and are expected to be S = 2j + 1 for completely filled neutron shells. The DWBA predictions are smaller than the measured cross sections by about 111.25 on average. The large diffuseness of the real part of the deuteron potential is important
IM. Igarashi
et al..
Two-nucleon
transfer
reucrion
mect~anisrw
(b)
0.898 MeV 3127
:
20epb(p,d)207Pb
1= 3
20
Fig. 4.5.
Experimental
and calculated
show the case of a pure S-state
0.57OMeV
40
8,,
Ep =22MeV 5/2-
-
20
60
(a) cross sectlons and (b) analyzing
40 8,,
(de!& powers for the ““Pb(p,
and the solid lines show the case with the St
d)““Pb
D states in the deuteron
60
(deg) reaction
at 22 MeV. In (b) the dashed lines
wave function.
for reproducing the oscillatory pattern in the angular distributions. It may reflect effects of deuteron break-up in the nuclear field. The D-state component (L = 2, S = 312) of the projectile form factor has very little influence on the cross sections and vector analyzing powers, but produces large changes in the tensor analyzing powers, especially in T2?. The tensor analyzing powers are sensitive primarily to the D-state components in the product of the interaction and triton wave function, rather than to the deuteron D-state alone. But the difference of the tensor analyzing powers for different nuclear interaction potentials or triton wave functions is small compared with the error bars of the experimental data. We cannot choose an interaction potential or a wave function from the fit to the experimental data. 4.2.2. Cross sections of sequential (p-d-t) processes The two-step exact finite-range DWBA calculations were carried out using the computer code TWOFNR [71]. The finite-range projectile form factors for the (p-d) and (d-t) transfer processes mentioned above were used. We must evaluate many different transition amplitudes for each transition,
M. Igarashi et al., Two-nucleon transfer reaction mechanisms
51
Fig. 4.6. Differential cross sections of the 208Pb(d, t)“‘Pb reaction at 17MeV, from top to bottom for the pi12, pi,*, f,,,, f,.? and i,,., transitions and scaled by 1. 114, 114, 118 and 118, respectively.
and there are many intermediate states of the *07Pb nucleus. If we include the D-state (L = 2) component of the projectile form factor, we have many possibilities for the transferred angular momentum (li, si, ii) for each transfer step as shown by eq. (2.4.10). The number of transition amplitudes is doubled because we must calculate the interaction terms and the non-orthogonality terms as mentioned in section 2.4. As an example, in the case of the 3: transition we must evaluate altogether 64 different transition amplitudes and sum them coherently even when only the ground state of the deuteron is considered for the intermediate states. 4.2.2.1. 0; -+o; transition. The results of calculations for the 0,’ -0: transition of the “‘Pb(p, t)*06Pb reaction at E, = 22 MeV are shown in fig. 4.8, together with the results of the calculation of the one-step process and the experimental data [85]. We used the shell model wave function obtained by True [104]. Set C in table 4.4 was used for the distorting potential parameters. The calculated cross sections of the one-step (p-t) processes are smaller by about one order of magnitude than the experimental data. The contribution of the sequential transfer processes through the ground state deuteron channel (curve 2B) is predominant and about 3 times larger than that from the one-step process. Hereafter the contribution of the two-step process means the coherent sum of the interaction and the non-orthogonality terms for that process. The contribution of the unbound states in the intermediate states is rather small. Curve TOTAL shows the coherent sum of all those contributions. The cross section is increased by one order of magnitude by inclusion of the sequential transfer processes. The fit of the angular distribution to the data is reasonably good. The calculated analyzing power functions are shown in fig. 4.8b and compared with the observed data. It is found that the one-step (curve 1) and the two-step [the coherent sum of curves 2B,
52
;\. /\* ,$* *.
M. Igarashi ?I al., Two-nucleon transfer reaction mechanisms
I-9-
\
c
’
I .:
:
: .
i
.1 I
M. Igarashi et al., Two-nucleon transfer reacrion mechanism
53
10-4
10-5
0
10
20
30
0HC. M.
40
50
60
70
80
(degree)
Fig. 4.8. (a) Cross sections and (b) A(0) for the reaction “a Pb(p, t)206Pb0* at E, = 22 MeV, calculated employing the RSC5 triton wave function and the experimental data are shown. Set C in table 4.4 was used for the optical potential parameters. Curve 1 shows the contribution of the one-step process. Curve 2B shows that of the bound deuteron channel, in which only the L = 0 component is taken into account in both (p-d) and (d-t) transfer steps of the (p-d-t) calculation. Curve 2UB(S = 0) shows that from the ‘S, (S = 0) unbound deuteron channel, while curve 2UB(S = 1) shows that from the ‘S, + ‘D, (S = 1) unbound deuteron channels with only the L = 0 component in both transfer steps. The solid curves show the coherent sum of all the processes. The experimental data are from ref. [SS].
2UB(S = 0) and 2UB(S = l)] transfer mechanisms show similar oscillatory angular distributions. The coherent sum of these two (curve 1 + 2B + 2UB) shows almost the same shape as that of the two-step cross section because of its predominance. For the O+ transitions, the L = 2 tensor components of the form factors of the light-ion system are neglected in both (p-d) and (d-t) transfer steps of the two-step process because the contribution of the L = 2 components is very small. Also, we would have to calculate too many transition amplitudes if we include such components, because the O+ states have many particle-pair shell model configurations. Figure 4.9 shows the two-step cross section and analyzing power A(0) calculated with the pure (p,,J2 configuration. Inclusion of the L = 2 tensor component does not affect the cross section, but it does slightly affect the analyzing power. The absolute magnitude of the cross section is uncertain within a factor of about two depending on the choice of distorting potential parameters. The agreement of the calculation including the sequential transfer processes with the analyzing power data becomes rather poor at around 45”. For the 0: transition the A(0) function is also sensitive to the choice of the imaginary part and/or other parameters of the distorting potential for the intermediate deuteron channels. Further study of the distorting potential in the second-order DWBA framework is desirable. 4.2.2.2. 0; +3’
transitions. The results for the two 3’ transitions are shown in fig. 4.10. It is seen
M. lgarashi et al., Two-nucleon transfer reaction mechanisms
54
T
S+
II/ , , , , , , , , 1 0
10
20
30
40
0HC.M.
50
60
70
80
90
0
10
20
40
50
60
70
80
90
(degree>
Fig. 4.9. The contribution
of the L = 2 tensor components to pure (p, JL transfer.
that the sequential transfer process plays the main role in these unnatural parity transitions. The contribution of the one-step (p-t) process to the cross sections is about one to two orders of magnitude smaller than that of the two-step process through the ground state deuteron channel. For these transitions both the one- and two-step cross sections depend strongly on the incident energy near or below the Coulomb barrier as shown in fig. 4.14. At higher energies the one-step process starts to change the angular distribution. The one-step angular distribution is different from the observed shape, and its analyzing power is also quite different from the experimental data, so that the experimental analyzing power can be reproduced only by the dominance of the sequential transfer process. The contribution of the singlet deuteron intermediate channel, which is unbound, is about one order of magnitude smaller than that of the ground state triplet deuteron channel. The fit of the analyzingpower calculation to the data is not much affected by inclusion of the unbound deuteron channels, and it is rather improved. This shows that the analyzing power A(0) does not become very small even when both ground and unbound state deuteron channels are summed over, in contrast to the prediction of the closure approximation [73]. This is because the s = 1 transfer component in the scattering amplitude does not vanish since the cancellation between the singlet and triplet deuteron channels no longer occurs. Consequently A(B) of the two-step process is produced. The origin of A(8) is therefore completely different for the one-step and two-step processes. In the one-step process it is caused by the D-state and S’-state components of the triton wave function. In the two-step process the spin l/2 transfer occurs twice and both s = 0 and s = 1 transfers are possible. The analyzing powers of the one-step process have opposite sign to the experimental data (see fig. 4.10). The predominant two-step process reproduces the experimental A(8) quite well. Thus we see that the analyzing power data are a useful tool for investigating the reaction mechanisms. The analyzing power due to the sequential transfer process also depends strongly on the j-shell
M. lgarashi et al., Two-nucleon transfer reaction mechanism
ONE-STEP
55
(P,T)
Fig. 4.10. Calculated cross sections and A(0) for the two different 3’ excitations in the reaction lo8Pb(p, t)2”6Pb at E, = 22 MeV. The RSC5 trition wave function was used. Set B in table 4.4 was used for the optical potential parameters. The experimental data are from ref. [SS]. Curve 2BS shows the contribution from the bound deuteron channel, in which only the L = 0 (scalar) component is taken into account in both the (p-d) and (d-t) transfer steps, while curve 2BT shows the contribution when the L = 2 (tensor) component is taken into account in one or both transfer steps. Curves 2US0 and 2USl show the contribution from the ‘S, unbound deuteron and the ‘S, + ‘D, unbound deuteron channel. respectively. Curve total [l + 2B t 2UB, where 2UB = 2US0 + 2USl] shows the coherent sum of all processes for the cross section [for A(B)].
structure of the nucleus as in the case of the simultaneous transfer process mentioned above, but the sign of A(B) produced by the j-shell difference is almost opposite for the one-step and two-step processes at forward angles. Compared with the recent experimental A(8) data [87], the predominance of the sequential transfer mechanism is apparently supported, as seen on fig. 4.10. In addition, this conclusion is also consistent with that obtained from differential cross section analyses. We comment on some other details of fig. 4.10. The 2BS curve shows the contribution of the bound deuteron channel, in which only the L = 0 scalar component is taken into account in both (p-d) and (d-t) transfer steps, while the 2BT curve shows the one from the remaining terms in which the L = 2 tensor component contributes in one or both transfer steps. The contribution of the non-orthogonality terms is rather small, but comparable to that of the one-step process. Figure 4.11 shows the same 3: transition at an incident energy of 3.5MeV. The contribution of the one-step process becomes larger than in the 22 MeV case.
M. lgarashi et al.. Two-nucleon
56
rrunsfer reaction mechanrsms
10-5
\ --. ,0-E
i
Cl
\ L..
__L_i-AL
10
20
Fig. 4.11. The 35 MeV calculation
30
40
@C.
M.
50
L1_
60
70
80
‘--I 90
(degree)
for the 3,’ transition
with the distorted
potential
parameter set A in table 4.1. See fig. 1.10 for the notations.
4.3. Some properties of one- and two-step processes 4.3.1. Non-orthogonality terms und the one-srep process The simple sum of the non-orthogonality amplitudes for the intermediate states of both target and projectile nuclei is equal in magnitude and opposite in sign to the one-step amplitude [40, 411. However, we can take into account only the few most important intermediate states in our practical evaluations. Therefore it is interesting to see to what extent this theorem is realized in actual transitions. Figures 4.12 and 4.13 show the contributions from the non-orthogonality terms of each intermediate channel and their coherent sum. For comparison the contributions to the cross section and the analyzing power from the one-step process are also shown. The total cross section of the coherent sum of the non-orthogonality terms is almost comparable to the cross section of the one-step process especially for the unnatural parity transition. We found that the non-orthogonality term almost cancels the one-step process. From these facts we can say that this theorem is fairly well confirmed. Therefore from this point the concept of the number of steps is rather ambiguous. This finding is quite interesting and important. Further studies are in progress and the details will be reported elsewhere. 4.3.2. Incident Figure 4.14 calculation the incident energy
energy dependence of the one- and two-step processes shows the incident energy dependence of the one- and two-step processes. In this optical potential parameters are fixed to set A for both 0: and 3: transfer and only the is changed. For the natural parity 0: transfer the energy dependences of both processes
M. lgarashi et al., Two-nucleon transfer reaction mechanism
1
0
10
Total non-ortho.
20
30
0I+CM.
Fig. 4.12. Cross sections and analyzing E, = 22 MeV.
40
50
60
70
-
Nomortho.
------
One-step
60
(degree) powers of the non-orthogonality
terms and one-step
process for the 208Pb(p, t)*06Pb0;
reaction
at
are almost the same and the two-step contribution is still larger than the one-step contribution even at the highest energy of 80 MeV In contrast, the energy dependence of the one-step process is rather weak at higher energies for the UNPS 3’ transition. Thus, the contribution of the one-step process is larger than the contribution of the two-step process at high incident energies, for example 80 MeV At low incident energies, around 20-30MeV, the two-step process dominates the transition over the one-step process. Both one- and two-step cross sections depend strongly on the incident energy near or below the Coulomb barrier. The reduction of the cross section in this energy region is stronger for the 3: transition than for the 0: transition. The present theoretical predictions have been confirmed experimentally up to 50 MeV incident energy [48]. Therefore it is important experimentally to check the evolution of the one-step cross section to a magnitude comparable with the two-step cross section around 80 MeV. 4.3.3. j-shell structure dependence of the analyzing powers
The differential cross section and the sign of the analyzing power A(8) of one-nucleon transfer reactions are well known to depend on the transferred angular momenta 1 and j, and hence both observables are useful tools to study the single-particle angular momentum state (I, j) to or from which the nucleon is transferred [113]. In the past, information on the j-shell configurations of the transferred pair of nucleons was considered to be unobtainable from the (p, t) reaction.
M. Igarashi er al., Two-nucleon transfer reaction mechanisms
58
104
I
Total nowortho.
i
Fig. 4.13. Cross sections and analyzmg powers of the non-orthogonality
Total non-ortho.
terms and one-step process for the ‘““Pb(p. t)“‘hPh 3; reaction at (a)
22 MeV and (b) 35 MeV.
transitions. It has been found in our work that the two-step analyzing power A(0) 4.3.3.1. 0 ‘-0’ of the (p, t) 0’ -+ 0’ transition sensitively reflects the j-shell difference in the nuclear wave function [45, 461. The two-step analyzing powers of the ( j< = I- 112)’ and ( j-, = I+ 1 /2)2 pick-up reactions show completely out-of-phase angular distributions if the spin-orbit distortions are omitted in the calculation, and even with inclusion of the distortions, the out-of-phase property remains in the forward angular region. The j-shell dependence of the sign of the analyzing powers appears also in the (p-t) one-step mechanism if we use the finite-range calculation employing a realistic triton wave function. Figure 4.15 shows A(8) of the one- and two-step processes (only through the ground state deuteron intermediate channel for simplicity of the calculation) for (pliz)” and (P~/~)’ pair pick-up in the ““Pb(p, t)20hPb 0’ reactions. In this calculation parameter set C in table 4.4 is used and the spin-orbit distortions are omitted. The signs of A(8) for both one-step and two-step processes change depending on the configuration, j:[ = (p, ,,)‘I or j:[= (P~,~)~]. When one-step and two-step processes are compared, the signs are almost opposite to each other and the oscillation of A(8) is weaker for one-step than two-step processes. A(8) for the one-step process disappears completely in the zero-range calculations, because s = 1 spin transfer is not allowed. The major source of s = 1 spin transfer is different in different transfer mechanisms; it is supplied by the minor components (non-S-state) of the
M. Igarashi et al., Two-nucleon transfer reaction mechanisms
59
1
10-l
lo-’ !(b)
mPb(p,t)206Pb
3’ TWO-STEP
Fig. 4.14. The incident energy dependence of the one- and two-step processes. The numbers near the curves show the incident energy, from 15 MeV to 80 MeV. (a) The natural parity 0’ ground state transition, (b) The UNPS 3’ transition of the *“Pb(p, t)“‘Pb reaction.
M. Igarashi et al., Two-nucleon transfer reaction mechanisms
60
im r
1
1
100, 90
“%@-d-
t PO6Pb 0;
80 Ep
70
wi
Fig. 4.15. Shell model orbit dependence
(b)
=
22MeV
TWO- step
of the calculated A(0) of (a) the one-step and (b) the two-step process without spin-orbit
distortion.
triton wave function in the one-step mechanism, whereby the finite-range interaction is essential, and by double spin 112 transfer in the two-step mechanism. This effect remains in the forward angular region even when the spin-orbit distortions are included. Figure 4.16 shows the case with the spin-orbit distortions for the p, d and t channels. At backward angles the spin-orbit distortion effect dominates and violates the simple out-of-phase rule. The largest spin-orbit distortion effects come from the proton channel. The origin of the shell dependences of the one- and two-step analyzing powers is interpreted as follows. When the distorting potential is spin independent and only one set of nuclear shell orbits contributes for the (p, t) reaction, the vector analyzing power eq. (2.6.11) can be (M,) and (MJ expressed as [114, 761 A(8) = -j/2($, + 1)/3s, Im Tll(fJ) =- j/2(& + Q/3& c (-)‘+[+X-J’+J,-Sb+11/(2S+ 1)(2S’ + 1)(2S, + 1)(21+ 1)(21’ + 1) isl’r’ (4.3.la) where
Pkq(h l’s’j’; 0) = ; (-1i’+m-q(lml’q -
m[kq)pI:"(e)p;:;-q(e)*
and p$‘(13) are the same as defined by eqs. (2.6.2) and (2.6.3).
,
(4.3.lb)
M. Igarashi et al., Two-nucleon transfer reaction mechanisms
61
Pb 0; Ep = 22MeV
40
Fig. 4.16. Shell model orbit dependence
50
60
70
of the calculated A(0) of (a) the one-step and (b) the two-step process with spin-orbit
80
distortion.
For O++O+ (p, t) reactions, from the Racah coefficient W(ll’ss’; lj) in eq. (4.3.la), two sets of (kj) transfer are allowed: (000) and (110). The latter transfer is the so-called “unnatural parity” transfer component, which is allowed even in O++ Of transition. This component arises from non-S-state components of wave functions and spin-dependent interactions involved in the calculation. From eq. (4.3.1), we can see that the numerator of the vector analyzing power is expressed in terms of two amplitudes P’“(0) and /I” m-1, of which the m substates differ by one unit. Therefore only the cross form of the two amplitudes corresponding to (Isj) = (000) and (110) appears in the expression for A(B). The reduced amplitude /3:(e) of a one-step process includes a 9-j symbol,
i
4
Sl
il
12
s2
j2
L
%
i
I ,
as eq. (2.3.28) shows. The j-shell dependence of the analyzing power comes from this 9-j symbol. If we neglect the P-state component of the triton wave function, the spin of the transferred two neutrons, SIz, is equal to the transferred spin, s, as mentioned in table 2.4. The s = 1 amplitude is small compared with the s = 0 amplitude, so that we can neglect the contribution of the s = 1 amplitude to the denominator of eq. (4.3.la). A(8) is then proportional to the ratio of two 9-j symbols,
(4.3.2)
M. Igurashi
62
et al..
Two-nucleon transfer reaction
mechanisms
so that the analyzing power for ji = 1, + l/2 and that for jl = I, - 112 are related by
A(H,j,=1,+1/2)=-& A(6, j,=I, I
112).
(4.3.3)
This relation is similar to that found in one-nucleon transfer reactions [114, 1151, in which the transferred (Zsj) is the nuclear shell orbit (I,s,j,) from (or to) which the nucleon is transferred. As for the two-step process, the reduced amplitude /3z(f?) includes the 9-j symbol 1, s1 jl 1, 32 j2 i
I
s
j
I
as eq. (2.4.11) shows if we choose a pure symmetric S-state wave function for the triton. Furthermore the s = 1 amplitude is small compared with the s = 0 amplitude, but not so negligibly small if we choose only the ground state deuteron for the intermediate channel. If we neglect the contribution of the s = 1 amplitude to the denominator of eq. (4.3.la), we have the same relation as eq. (4.3.3). This relation does not hold if the contribution of the s = 1 amplitude cannot be neglected, as we can see in fig. 4.15b. If we take into account the minor components of the triton wave function, the transferred angular momenta (l~j),,~ and (Is~)*“~are not always the same as the angular momenta of the shell model single-particle state of the transferred neutron in the target. Namely, s = 3/2 amplitudes are allowed and the transferred orbital angular momentum I can take on values in the range (j - 3/2] 5 I ‘j + 3/2. These facts also invalidate relation (4.3.3). Figure 4.17 shows the results for the 0: transition of the 2”sPb(p, t)2’)hPbreaction. In this case, the components of the shell model wave function have amplitudes with sign opposite (f5,2)-2 and (P,J2 to each other. The shell model wave functions are shown in table 4.2. The cross section becomes very small because of the cancellation, and the analyzing power differs strongly from that of the 0: transition shown in fig. 4.8b. The analyzing powers of the one- and two-step processes are very different at forward angles. The solid line is the result of a coherent sum of the two processes. Figure 4.18 shows a comparison of the calculated results with the experimental data for the 0: transition [47]. The different shell model configurations give different angular dependences of the analyzing power. The present theoretical resuts are still not conclusive for a consistent understanding of the differential cross section and analyzing power. The nuclear wave function may have more problems. We can, however, say that the analyzing power data are useful to check the shell model configurations. We can use the analyzing power data for a check of the shell model configurations. 4.3.3.2. 0’ -, higher J” transitions. j-shell dependent A(8) have been found in the present work for other 1” transitions. In fig. 4.10 the analyzing power data of the 3’ transitions show almost opposite signs at forward angles for the one-neutron orbits f,,, and f7,2. Figure 4.19 shows A(8) for the one- and two-step processes (only through the ground state deuteron intermediate channel) for four combinations of the pair picked up from the p and f single-particle states in the 208Pb(p,t)*06Pb3’ reaction (see also fig. 4.4). The signs of A(8) for the one- and two-step processes are quite sensitive to a change in j-shell configuration. Furthermore, the signs of A(B) produced by j-shell differences are almost opposite, when one-step and two-step processes are compared at forward angles. The predominance of the two-step process for these transitions determines the final value of A(8).
M. Igarashi et al., Two-nucleon transfer reaction mechanism
10-l
t
1
,j z .‘\ .ii
‘\
I i
i ,_
‘\ \ i 2UB(S=l) \ i ( ,f-~, \il/ ‘\ \il/ ,‘\ 1 ,/-
/ I 0
10
\ \
‘?\
/
1O-6
2UB(S=O) /-\ ; ‘1 ‘\ ,‘. ; ‘\,,j’ ;
\
;,; / /’
1O-5
63
20
30
0I+C.M.
40
50
60
70
80
40
50
60
70
80
(degree)
Fig. 4.17. Calculated (a) cross section and (b) analyzing power of the “‘Pb(p, t)20bPb reaction at 22 MeV leading to the 0; (E, = 1.167 MeV) state. See fig. 4.8 for the notation. True’s shell model wave function 11041 was used.
A clear explanation of the j-shell dependence for the higher 1” transitions is rather difficult, because four different (kj) transfer amplitudes (one for s = 0 and three for s = 1 for fixed 1 and j values, as shown in table 4.6) contribute to these transitions. In general, three self-products of s = 1 amplitudes and five cross-products of s = 0 and s = 1 amplitudes appear in the calculation of the analyzing power. However, the relative strengths and signs of the two-step transition amplitudes are decided mainly by the 9-j symbol
if we assume pure L = 0 (S-state) projectile form factors. The values of the 9-j symbols for the four combinations of (p, f) configurations are listed in table 4.6. We can see that the relative strengths and signs of the different (lsj) amplitudes are drastically changed by the shell model configuration. For the two-step process the contributions of the transition amplitudes caused by the L = 2 projectile form factors are small. These amplitudes include the same 9-j symbol, in which, however, s = 3/2 and 1 is not always the same as the orbital angular momentum of the transferred neutron.
M. lgarashi et al., Two-nucleon transfer reaction mechanism
64
loo
20ePb(p,t)M’Pb 0; Ep =22 MeV
0.8
True
Taka. _---K-H 0.6
(a>
,-\
-
_--
----
T-F(l)
-0.6 -0.8
0.1
’
t
I
20
0
I
I
40
60
I 80
-1.0
s
0
20
BcmMeg)
40
60
80
&-,(deg)
Fig. 4.18. Calculated (a) cross sections and (b) analyzing powers for the *“‘Pb(p, t)IU6Pb reaction at 22MeV leading to the 0; (E, = 1.167 MeV) state. The four curves are the results of calculations using four different shell model wave functions: True: from ref. [104]; Taka: from ref. [106]; K-H: from ref. [107]; T-F(l): from ref. [103]. They are given in table 4.3. The experimental data are from ref. [47].
For the one-step process for unnatural parity transitions the relative strengths and signs of the transition amplitudes are very different from those of the two-step process. The factor
of eq. (2.3.28) decides the (Isj) dependence. When we ignore the P-state component in the triton wave function, then S,, = S, but L runs from 1 - 1 to 1+ 1 when the D-state of the triton (L’ = 2) is taken into account. For’unnatural parity transitions the contribution from the D-state of the triton is the most important. Therefore the (Isj) dependence is very different between one- and two-step processes. The change of the relative strengths of (Isj) transfer in the one-step process is shown in fig. 4.2, and the same for the two-step process is shown in fig. 4.20. These two are very different from each other. The sign change of the amplitudes produces the sign change of the analyzing power through the crossproduct terms. Table 4.6 Values of the 9-j symbol for the four (p, f) configurations
w
(PI,,? f,,,)
(PI,,? f,,,)
(P,,Z? f,!,)
(P,,,. f7,Z)
(303) (213) (313) (413)
lAf&ii -liXfiS% l/V%@ l/V%
-1ivX4 l/VQi 1lQiiB l/XfZZ%
l/V%Y6 -llv%i% l/VQiiZB -liv?SZJ
1/m llV?i% -l/Gi%A -l/V%%6
M. lgarashi et al., Two-nucleon transfer reaction mechanisms
ONE-STEP
[p& f&-j 3+ ---_--
[p;;zf&l
3+
__-----__
[pi;*f$*] 3+
---_-----
[pi;2f&l
3+
-0.4 ONE-STEP+ TWO-STEP -0.8
I
I
0
10
20
30
00-4 C.M. 40 CdegeM 50 60
I 70
80
so
Fig. 4.19. Analyzing powers of the 208Pb(p, t)*06Pb reaction at 22 MeV leading to four 3’ state configurations. (a) The one-step process, (b) the two-step process (the contribution from the ground state deuteron intermediate channel alone is considered) and (c) the coherent sum of both processes.
These results, however, depend somewhat on the incident energy, because the relative strength of different (kj) amplitudes depends also on the incident energy. Figure 4.21 shows the case of incident energy E, = 35 MeV. By comparing this with the case of 22 MeV (fig. 4.19), we can see that A(0) of the transition to the [p$f7/:] configuration state changes strongly with the incident energy. Shown in fig. 4.22 is the calculated 2+ analyzing power with finite-range treatment of the one- and two-step processes. Here the analyzing power A(8) is also sensitive to the nuclear shell configuration. 4.4. Summary and conclusions For the (p, t) reactions, the absolute values of the cross section obtained by one-step DWBA calculations by different authors differ much from one another. For a long time this fact made it difficult to understand the mechanism of this reaction. In this report we made careful studies of the absolute cross sections and conclude that the one-step process alone cannot reproduce the experimental absolute cross section even for strong natural parity transitions. The predominance or comparable importance of the sequential transfer mechanism over the one-step (p-t) mechanism has been demonstrated for low
M. lgarashi ef al., Two-nucleon transfer reaction mechanisms
_. \ 1o-3
208Pb(p-d-t)206Pb ‘\
3; [PA G]
‘\
A ‘\
g $10-4 F
‘\
‘1
‘\
MeV
303
--‘\
Ep ~35
\
Isj
‘\
r\ 6 \
-\
3’
2 1 3
_- --.
313
-_-
413
8 z 0
10-5
5
1O-6 0 Fig, 4.20. Decomposition transitions.
10
20
30
40
0I+ C.M.
50
60
70
60
(degree)
of the calculated cross sections into transferred
IO
20
0HC.M. 30
40
50
60
70
80
90
(degree)
angular momentum (kj) components of the two-step processes for two 3’
Fig. 4.21. Analyzing powers for the ‘““Pb(p, t)*““Pb reaction at 35 MeV leading to four 3’ state configurations.
M. Igarashi et al., Two-nucleon transfer reaction mechanism
67
0.8 0.4 e G a
0.0
-0.4
II
,’
ONE-STEP
.__I
TWO-STEP 1
,,__,;
10-5'
0
’ 10
20
30 @CM.
1 40
50
60
(degree)
’ 70
’
Bo
’
0
10
’ 20
’ 30
0
’ 40 C. M.
50
’ 60
70
80
’
90
(degree)
Fig. 4.22. Calculated analyzing powers and cross sections of the *‘*Pb(p, t)*ObPb reaction at 22 MeV leading to the three 2’ state configurations.
incident energies in the case of both the natural parity and the unnatural parity transitions by accurate and consistent evaluation of the one-step and two-step transfer processes with realistic wave functions for the projectile-ejectile system and the interactions causing the reaction. This predominance of the sequential transfer processes does not affect the customary statement that the (p, t) reactions are good for probing two-particle pair correlations in nuclei, since the two-step amplitude is also enhanced in much the same way as the one-step process by the pair correlation. It was also pointed out that the (p, t) analyzing power is a useful tool to clarify the transfer mechanism, one-step or two-step, and to study nuclear structure, since its shape and sign strongly depend on the difference in the transfer mechanisms and on the nuclear j-shell configurations.
Acknowledgements
The authors are grateful to M. Kawai for his continuous encouragements. Their special thanks are due to S. Ishikawa, T. Sasakawa and T. Sawada for providing us with their triton wave functions. They are indebted to Y. Aoki and other members of the Tandem Accelerator Center, University of Tsukuba.
M. Igarashi et al., Two-nucleon transfer reaction mechanisms
68
They thank P.E. Hodgson for his careful reading of the manuscript. One of the authors (MI) also thanks the members of the theoretical division of the Institute for Nuclear Study, University of Tokyo for their kind hospitality extended to him. The FACOM M380 at the Institute for Nuclear Study was used for the numerical calculations. This work was supported in part by a Grant-in-Aid for Scientific Research from the Japan Ministry of Education (1979-1980). Two of the authors (MI and KIK) thank the Kudo Arts and Sciences Foundation for the financial support needed for this lengthy work.
Appendix A. Hard-core correction to the one-nucleon transfer form factor If the triton wave function is evaluated with a nucleon-nucleon interaction with a hard core, the particle transfer form factors need corrections as mentioned in section 2.3.3. In the case of (d, t) reactions the form factor is given by
(&(r)lVnln2(rnln2) + Vpn2(rpn2) + VA,n2(rn,n2) + ~n2(rpn2)Ilc11(r~ Qr 7
(A4
where the subscript r means integration over the coordinate r, and V(r) is an attractive potential outside the hard core and V’(r,) = S(ra - r,)
$ +-$ c
64.2)
ra .
LX+
The hard-core correction (pseudopotential) terms are evaluated as follows: S(ra - r,) =
6(vR’+ ir’
+ Rr cos 8 - r,) =
j&j w - 0,).
(A.3)
Then
wd(T)/~(r”,“2 - rc)
$ +j+ c
r”*“21 cClt(r, 9,
nl”2+
ca 2?r
=27T
II
rGb (r)
~
rnlnZqbt(r,R)
sin 8 de r2 dr .
64.4)
0 0
Figure A.1 shows the (d, t) form factor thus evaluated with the Tang-Herndon triton wave function and the nucleon-nucleon interaction. The pure S-state deuteron wave function evaluated with the Tang-Herndon interaction was used. The hard-core correction is large inside the triton and has opposite sign to the interaction term. The calculated integrated cross sections for different form factors are compared in table A.1 for the *‘*Pb(d, t)‘07Pb reactions at 17 MeV. The optical potential and the neutron binding potential parameters are the same as in section 4.2.1.2. The Tang-Herndon form factor without the hard-core correction gives cross sections about 3 times larger than with the correction. In the case of RSCS the projectile form factors given in section 4.2.1.2. were used, so they included the L = 2 components. The Tang-Herndon form factor gives cross sections about 39% larger than the RSCS form factor. The RSC3 form factor does not make any difference compared with the RSCS form factor in the integrated cross section.
M. Igarashi et al., Two-nucleon transfer reaction mechanisms
1031
1
1
I
_ Tang-Herndon
I
.
I
69
1
(d, t ) F. F.
-------
Interaction T. _ H.C. corre. T.
Fig. Al. Projectile form factors for the (d, t) reaction with the Tang-Herndon triton and deuteron wave functions. The dashed line shows the interaction term, the dash-dotted line shows the hard-core correction term and the solid line shows the total form factor.
Table A.1 Comparison of cross sections of various calculations for the ‘OaPb(d, t)*O’Pb reactions at 17 MeV. Integrated cross sections (o-180”) in mb for S = 2i + 1 Projectile
PI,2
P3,2
f 512
f712
i,,,z
h 912
TH without corr. TH with corr. RSC3 RSC5
35.65 12.06 8.50 8.50
60.08 20.36 14.44 14.43
29.08 9.71 6.86 6.86
26.36 8.77 6.39 6.37
7.047 2.277 1.753 1.741
2.408 0.790 0.556 0.557
Knutson and Yang [116] calculated the deuteron-triton overlap integrals using several kinds of deuteron-triton overlap functions, but their results should be reinvestigated for the case in which they used an interaction with a hard core since they neglected the hard-core correction terms.
References [l] A. Bohr, Comptes Rendus du Congres Intern. de Physique Nucleaire (Paris, 1964), Vol. I (Centre National de la Recherche Scientifique, Paris, 1964) p, 487. [2] S. Yoshida, Nucl. Phys. 33 (1962) 685. [3] M. El Nadi, Proc. Phys. Sot. A 70 (1957) 62. [4] M. El Nadi and M. El Khishin, Proc. Phys. Sot. 73 (1959) 705. [5] H.C. Newns, Proc. Phys. Sot. A 76 (1960) 489. [6] CL. Lin and S. Yoshida, Prog. Theor. Phys. 32 (1964) 885; C.L. Lin, Prog. Theor. Phys. 36 (1966) 251.
70
M. Igarushi et al.
1Two-nucleon
transfer reactlon mechanisms
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Masamichi IGARASHI Department of Physics, Tokyo Medical College, Shinjuku, Sinjuku-ku, Tokyo 160, Japan Ken-ichi KUBO Department of Physics, Tokyo Metropolitan University, Fukazawa, Setagaya-ku, Tokyo 158, Japan and Kohsuke YAGI Institute of Physics and Tandem Accelerator Center, University of Tsukuba, Tsukuba, Ibaraki 305, Japan
NORTH-HOLLAND
PHYSICS REPORTS (Review Section of Physics Letters) 199, No. 1(1991)1—72. North-Holland
TWO-NUCLEON TRANSFER REACTION MECHANISMS Masamichi IGARASHI* Department of Physics, Tokyo Medical College, Shinjuku, Sinjuku-ku, Tokyo 160, Japan
Ken-ichi KUBO Department of Physics, Tokyo Metropolitan University. Fukazawa. Setagaya-ku, Tokyo 158, Japan
and Kohsuke YAGI Institute of Physics and Tandem Accelerator Center, University of Tsukuba, Tsukuba, Ibaraki 305, Japan Editor: G.E. Brown
Received April 1990
Contents: 1. Introduction .1. Review of previous studies 1.2. Aim of this work 2. Formulation and method of calculation 2.1. The first- and second-order distorted wave Born approximations 2.2. The triton and deuteron wave functions 2.3. The one-step process 2.4. The sequential transfer (p—d—t) process 2.5, The spectroscopic amplitude for the particle transfer reactions 2.6. The differential cross section and analyzing power 2.7. Why analyzing power? 3. Experimental study of (p 1) cross sections and analyzing powers
*
3 3 6 6 6 9 14 25 32 35 37
3.1. The measurement of (p, t) analyzing powers and cross sections 3.2. Experimental results for (p t) analyzing powers and cross sections 3.3. Experimental studies of the (p, d) and (d, t) reactions 4. Results and discussions 4.1. The one-step (p—d) process 4.2. The sequential transfer (p—d—t) process 4.3. Some properties of one- and two-step processes 4.4. Summary and conclusions Appendix A. Hard-core correction to the one-nucleon transfer form factor References
38
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Abstract: The roles of the one-step (p—t) process in which two particles are transferred simultaneously and the two-step (p—d—t) process in which two particles are transferred sequentially, are quantitatively investigated for two-nucleon transfer (p. t) reactions with realistic triton and deuteron wave functions and realistic finite-range nuclear interactions. A detailed exact finite-range formulation is given, and analyses of both the one-step and the 3D two-step process are carried out. For the intermediate channels in the sequential transfer process, ~~and ~ + 1 unbound deuteron states as well as the ground state are taken into account. A summary of experimental studies of cross sections and analyzing powers for (p, t) reactions is given. It is found that the simultaneous transfer process alone can reproduce neither the experimental absolute cross sections 208Pb(p, nort)°6Pbreactions the analyzing powers, at 22 and thatMeV. 35 the These sequential results transfer are definitely process confirmed is dominantbyfor comparing both thethe natural calculated and unnatural analyzingparity powerstransitions with experimental in the data. The nuclear shell structure dependence of the analyzing power for the natural and unnatural parity state transitions is investigated.
1. Introduction 1.1. Review of previous studies Direct two-nucleon transfer (p, t) and (t, p) reactions are regarded as probes of the two-particle pair correlations in nuclei [1, 21, and a great deal of experimental and theoretical work has been done in the past two decades. However, the mechanism of these reactions has not yet been well confirmed because of the complexity of the reaction processes, in which four bodies a proton, two correlated transferred neutrons and a residual (or target) nucleus participate. The one-step (simultaneous two-neutron transfer) process, the two-step (sequential neutron transfer) process and in some cases inelastic channel coupling are of comparable significance. It is not easy to distinguish among those processes as far as the cross sections of the reactions are concerned since the main features of the angular distribution of the cross sections are determined by the selection rules the total transferred orbital angular momentum which are usually the same in these processes. Therefore different conclusions about the reaction mechanism came from different calculations, which were made by employing various approximations to avoid complexity of computation. In the following two subsections, we review this point for previous work, and then clarify the aim of the present work. —
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—
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1.1.1. The one-step DWBA process The (p, t) and (t, p) reactions have first been described by the simultaneous (one-step) two-neutron transfer mechanism. The pioneering studies were made in the plane wave Born approximation (PWBA) [3—5]and reproduced the forward-peaked angular distribution. Later, most analyses were made in the distorted wave Born approximation (DWBA) with zero-range approximation. The latter approximation is based on the assumptions that the interaction which causes the transfer reaction is a short-range central force and that the wave function of the triton is also short range and in a purely space-symmetric S-state, so that the two neutrons are in a singlet spin state. The method of Lin and Yoshida [6] and Glendenning [7] assumes a Gaussian wave function for the triton and a single-particle harmonic oscillator wave function for the neutrons bound in the nucleus. The harmonic oscillator wave functions were transformed to those of the center of mass and of the relative coordinates of the two nucleons using the Moshinsky transformation [8]. The strong absorption of the distorted waves forces the reactions to occur near the surface of the nucleus. Since the harmonic oscillator wave functions decay too rapidly at large distances, the radial functions were connected to Hankel functions of purely imaginary argument with decay constants corresponding to the separation energies of the nucleon pairs from the nucleus. Later, single-particle wave functions in a finite potential well were used. Two methods have been proposed for this purpose. One was the expansion of
4
M. Igarashi et al.. Two-nucleon transfer reaction mechanisms
Woods—Saxon wave functions in terms of harmonic oscillator functions [91,and the other, derived by Bayman and Kallio [101,was a direct numerical method of projection to the relative and the center of mass coordinates. The zero-range DWBA method reproduces the angular distributions and the relative magnitudes of cross sections in most cases. This seemed to support the claim that the reaction mechanism is dominated by the simple direct simultaneous transfer process. The angular distributions show diffraction patterns resulting from strong absorption, with shapes depending on the transferred angular momentum and parity. The angular distributions of the natural parity state transition are little affected by the details of the reaction mechanism, such as inclusion of the two-step mechanism. In many cases, however, the method fails to reproduce the absolute magnitude of the cross sections. The empirical value of the normalization constant of the zero-range DWBA, D~,is about 2—3 x i05 MeV2 fm3 [11, 121. This is an order of magnitude larger than simple theoretical estimates, although the estimates are not entirely reliable. A number of exact finite-range DWBA calculations employing more sophisticated triton wave functions and finite-range interactions have been reported since the work of Bayman [13, 141. He employed the Tang—Herndon triton wave function [15], which was obtained from a variational calculation with a simple state-dependent central two-body interaction potential with a hard core. For the 40’48Ca(t, p)4250Ca(g.s.) reactions, the magnitude of the calculated cross sections was a third of the observed one. In these calculations the pairing enhancement effect was taken into account. They were a great improvement over the zero-range DWBA calculations. However, in most of the calculations including Bayman’s work, triton wave functions generated from two-body interaction potentials with hard cores were used. As has been pointed out by Dobes [161,the use of such singular potentials requires caution and a correction term must be taken into account. This decreases the calculated cross section by a factor of 2.1 to 2.6 [17]. This alters the conclusion that the exact finite-range calculations improve the calculated absolute cross section of the simultaneous transfer mechanism. The wave function of the two transferred nucleons in the target nucleus was usually calculated by the half separation energy method neglecting the correlation between the two nucleons. But this method has no strict justification; the nucleon pair being transferred does not have a finite internal energy near and inside the nucleus, and hence the kinetic energy of the center of mass motion does not have a definite value. This problem was investigated by many authors [18—20],and calculations including the two-body correlation have been carried out. These caculations indicate that the correlation increases the magnitude of the wave function at the nuclear surface, enhancing the cross section especially for transitions to the lowest 0~state. However, the enhancement compared with the wave function with the usual restricted shell model space is several tens of percent and this is not enough to cover the difference between theory and experiment. Transfer reactions to the unnatural parity state of even—even nuclei are completely forbidden in the zero-range one-step DWBA framework. In finite-range DWBA this process is not completely forbidden, but Bayman and Feng [14] got a much smaller cross section than the observed one for the 22Ne(p, t)20Ne 2 reaction. They employed a spatially symmetric S-state triton wave function given by Tang and Herndon. This is questionable since for these reactions relatively large contributions are expected from the minor components such as the mixed symmetry S’- and D-states [21]. According to Nagarajan et al. [21, 22], the finite-range DWBA calculation with the Strayer—Sauer triton wave function can reproduce very well the cross section of 208Pb(p, t)206Pb(3 1.34 MeV) and also (0 g.s.) at 35 MeV, both in absolute magnitude and angular distribution (see also ref. [23]). These works throw doubt on the assumption that the reaction proceeds primarily as a two-step process. This, however, could not be confirmed by the present authors, as described in this paper. ~,
,
M. Igarashi et al., Two-nucleon transfer reaction mechanisms
5
1.1.2. Multi-step processes Many authors have reported that the second- and higher-order multi-step processes play important roles in (p, t) and (t, p) reactions. The effect of inelastic excitation processes on (p, t) reactions has been investigated by Ascuitto and Glendenning [24] and Tamura et al. [25]. If the nuclei involved are deformed (rotational) or have strong collectivity (vibrational), the nuclei are easily excited by inelastic scattering in the incident and/or exit channels. When the parentage of the final state is based more on such an excited state than on the ground state of the target, the transfer reaction to that state is strongly influenced by inelastic excitation. In such cases, the distorted waves are the solutions of the inelastic coupled-channel equations, but the transfer process is treated by the first-order Born term. This is the so-called coupled-channel Born approximation (CCBA). The sign of the interference between the multi-step and the direct one-step amplitudes depends on the details of the structure of the levels [26—28]. The other multi-step process is the sequential nucleon transfer mechanism. The first formulation of this type of transfer has been proposed by Bang and Wollesen [29] and was applied to the t0B(t,p)12B(1~,g.s.)reaction, for which the DWBA calculations had failed to reproduce the cross section. The nucleon transfer multi-step calculation is sometimes referred to as the coupled-reaction-channel (CRC) method. An apparent difference between the shapes of the CCBA and CRC angular distributions was pointed out for the 22Ne(p, t)20Ne(2) transition: the cross section goes to zero or remains finite at zero degrees [30—32](see also ref. [33]). De Takacsy [34, 35] reported that the 208Pb(p, t)206Pb(3~,1.34 MeV) reaction can be well understood by the sequential transfer mechanism. The strict selection rule (S = 0) on the transferred spin for zero-range DWBA is relaxed by spin transfer (S = 1) in two steps. Then, for such unnatural parity state transitions this mechanism gives a larger cross section than the one-step mechanism. The importance of the sequential process in strong natural parity state transitions has also been pointed out by Hashimoto and Kawai [36, 37]. They found in the analysis of the ltSSn(p, t)”4Sn(0, g.s.) transition that the two-step cross section becomes larger than the one-step cross section. Furthermore the phase coherence, existing in the one-step process due to two-nucleon pairing correlation [2], also exists in the two-step amplitudes [36]. The importance of both the CCBA- and CRC-type multi-step processes was reported for the tSO(p t)160(2) transition at 22.4 MeV [32]. Its smooth angular differential cross section can be well understood by the (p—p’—t) inelastic mechanism for the forward angular region and by the (p—d—t) sequential transfer mechanism for the backward region. The general trends of the observed energy dependence of the cross section (E~= 24.4—43.6 MeV) are interpreted as a more rapid decrease of the (p—d—t) mechanism compared with the (p—p’—t) mechanism [38]. The analyzing power data are also consistent with this interpretation [39]. The second-order (p—d—t) amplitude in the prior—prior form representation accompanies the so-called non-orthogonality term, which is due to the non-orthogonality of the d- and t-channels [40, 41]. Hashimoto and Kawai [36, 37] showed by the finite-range calculation that the contribution from this term is relatively small. Kunz and Charlton pointed out the importance of finite-range effects in the multi-step particle transfer amplitude [42]. Charlton [43] and Hashimoto [37]carried out a finite-range two-step calculation using simple S-state wave functions for the triton and deuteron and a simple central interaction, and found that the finite-range effects are small. It has been shown [37, 44] for natural parity transitions that there are important contributions to sequential transfer from the breakup or continuum deuteron intermediate states. However, these calculations used a simple Gaussian triton wave function and simple Gaussian interactions. So the conclusion is to be reinvestigated with amore realistic wave function and interaction.
6
M. Igarashi ci al.. iwo-nucleon transfer reaction mechanisms
Recently, it has been found [45] that the angular distributions of analyzing powers A ~(O)[whichwe abbreviate as A(O)] of (p, t) reactions are sensitive to the interference between simultaneous and sequential transfer amplitudes. Experimental results [46] provide direct evidence of this interference. The variations of the interference as a function of both target mass and bombarding energy, and the dependence on the final states [47, 48] have been observed. 1.2. Aim of this work The aim of this work is to settle this confusing situation between two completely different mechanisms, one-step (summarized in subsection 1.1.1) and two-step (in subsection 1.1.2), by making finite-range calculations of each process to an equally accurate degree and comparing the calculations with the experimental results of (p, t) cross sections and analyzing powers. For this purpose we need to use realistic wave functions for the light particle systems. The triton wave function employed here is the one obtained by solving the three-body Faddeev equation [49, 50]. For the intermediate channels in the sequential transfer process, the 1S0 and 3S1 + 3D, unbound deuteron states as well as the ground state are taken into account. The deuteron unbound state channels are evaluated by the DiscretizedContinuum-Channel Method, where the couplings between the ‘S 3S~+ 3D 0 and 1 channels are neglected [51].The same microscopic two-body interaction used in the calculations of the light particle wave functions is adopted for the transfer interactions. The analyzing power A(O) is also calculated and compared with data, which provides a useful test of the reaction mechanism. The A(O) for the unnatural parity state (UNPS) transition is still large even when the unbound deuteron components are included in the calculation with realistic wave functions (see subsections 2.4.3 and 4.2.2.2). The nuclear structure dependence of the observed analyzing powers for the natural and unnatural parity state transitions is also discussed.
2. Formulation and method of calculation 2.1. The first- and second-order distorted wave Born approximations The transition amplitude for a two-step processes is obtained from the second-order perturbative approximation in the coupled-channels formalism. The Schrödinger equation for the total system is written H~P=E~P,
(2.1.1)
where H=H~=Ha+T~*+Va~ 1I~=~JtIJaXa,
a=i,f,m.
(2.1.2) (2.1.3)
Here Ha is the intrinsic Hamiltonian of channel a, where the subscripts a = i, f and m denote the initial, final and intermediate channels, respectively. Ta is the kinetic energy operator of relative motion and 1’a is the intrinsic V~is the effective interaction. Xa represents the wave function of relative motion and c
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
7
wave function which satisfies (Ha Ea)tIIa = 0. The intrinsic wave function describes both states of the target and projectile nuclei in the channel a. From eq. (2.1.1), the coupled equations for Xa are —
(2.1.4)
(Ta+U~Ea)Xa=~~ (~H—E~X~), /3 +c*
U
= (t/J
V
~/i)
(2.1.5)
,
Ea=E~Ea~
(2.1.6)
If the coupling between the channels is strong, these coupled integro—differential equations must be solved directly under proper boundary conditions for the scattering states. Here we assume that the couplings are not strong so that the perturbative approach is valid. By solving the above equations by iteration up to second order one obtains the transition amplitude T= (t/i~x~~HEjtIIIX~~~)
(one-step)
—
(2.1.7) +~
(two-step).
~
Here the zeroth-order distorted wave y~ and the Green’s function ~ (Tf
+
U~ E~)x~ = 0,
(Tm + Urn
(2.1.8)
—
—
satisfy the equations
Em)G~~~ = —1,
(2.1.9)
with the boundary conditions that ~ is an incident wave plus an incoming wave at infinity and ~ has the asymptotic form of an outgoing wave. The distorted potentials Uf and Urn are the bare potentials defined by not including the coupling effects in the starting equations (2.1.4) [52,53] and are different from the elastic optical potential in principle. However, for simplicity, we usually replace them by the phenomenological elastic optical potential and we know that in the light-ion reactions in which the momentum matching condition is well satisfied this approximation works quite well. The wave function ~ in eq. (2.1.7) is the elastic scattering solution of the coupled-channel equations (2.1.4). Therefore it may be a much better approximation to substitute for it the elastic solution using a phenomenological optical potential [53, 54]. Hereafter we adopt the conventional DWBA method where we ignore the difference between X~and ~ [solution of the homogeneous equation with potential U1 in eq. (2.1.4)]. If we want to use X~ in the following, U1 is the potential reproducing elastic scattering. From eqs. (2.1.2) and (2.1.8) we get the post—prior symmetry of the transition amplitude for the one-step process, 1~ = (i/JfX~~~VI U,Iifi,x~Y~) (prior form) (2.1.10) T~ or —
T~1~ = (/JIX~~~V~—U~Icb,x~~~) (post form).
,
(2.1.11)
8
M. Igarashi
ci
al., Two-nucleon transfer reaction mechanisms
In the particle transfer process with light-ion projectiles, the calculation method has been well established using the prior form for the pick-up process and the post form for the stripping process. It is customary to approximate V~ U by V~,the interaction between the transferred particle and the projectile for the pick-up process, and V~ U1 by V~,the interaction between the transferred particle —
—
and the ejectile for the stripping process. 12~can be expressed in four different forms depending on The channel second-order transition which is referred to for amplitude the two H’sT in eq. (2.1.7); either H~’~ (prior) or H(m) (post) for the first H, and H(m) (prior) or H~1~ (post) for the second H. In the p—d—t process, the choice of H~’1and H(m) (prior and prior) representation is used since the interactions for the smaller particle systems are better known. Hence the p—d—t transition amplitude can be written T~~dI= 1Xt’d’d
(prior-prior)
~
(2.1.12) (prior-non),
K~iX~dd~VPkfrPX~
using eqs. (2.1.2), (2.1.8) and (2.1.9). The first term is the “genuine” two-step term and the second one is the non-orthogonality (NO) correction term. In a rearrangement collision the intrinsic states of different channels are not orthogonal to each other so this second term does not vanish. We should note that the overall minus sign appears in the NO term. If we sum over a complete set of states of both nuclei in the intermediate channel, then the NO term becomes equal and opposite in sign to the one-step amplitude [40, 41]. Therefore the “genuine” two-step term (which should also be summed over the complete set of intermediate states) remains as the lowest-order contribution to the transition. In practice, however, it is not allowed to make a complete sum over all the intermediate states t/Id because the i-I’ in eq. (2.1.3), ~I’= ~I’~+ II~+ ~ in the present case, then encounters the overcompleteness problem. Even if we regard the “genuine” two-step term as only the lowest term, it is not feasible to evaluate all the contributions arising from a complete sum over the intermediate channels, although some of these strengths may be significant in several channels, and others are spread over many channels. If a limited model space is chosen, the tendency remains for the one-step and non-orthogonality terms to interfere destructively. In the independentparticle model limit of the two-nucleon transfer form factor calculation, in which no correlation between the two transferred nucleons is considered, the one-step and NO amplitudes cancel each other in each intermediate state [55, 56]. The concept of a “step” is rather ambiguous, when we employ the presentation that brings about the non-orthogonality term. The practical criterion to choose the intermediate reaction channels m is as follows. We first choose the states which have large parentage coefficients. A plausible approach is thus to find such states that give large cross sections for i m and m f. The theoretical arguments about the intermediate states, however, are not so simple. If states which are not orthogonal are included in the intermediate channels, the problem of “overcounting” possibly occurs. For example, the m = d state in the (p—d—t) process and the m’ = t’ state in the (p—t’—t) process are not orthogonal to each other. Therefore in principle we must not take a simple sum of the two two-step amplitudes, otherwise we overcount. This problem cannot easily be overcome in the framework of the second-order DWBA. One takes care of the overlap between the channels. The problem, however, is automatically overcome by the method of coupled channels for multi-step rearrangement processes [57]. —~
—*
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
9
2.2. The triton and deuteron wave functions
To calculate the absolute magnitude of the cross sections for the transfer reactions, it is essential to employ realistic wave functions for the light ions and interaction potentials which cause the transfer. The consistency between the interaction potential and that used in deriving the light-ion wave function is important, especially in the case of a realistic interaction potential with hard or soft core [16, 17]. The existence of minor components in the triton wave function such as the spatially mixed symmetric S’-state and D-state components has a significant role in the simultaneous two-nucleon transfer process leading to the unnatural-parity states of even—even nuclei, where the contribution from the main component (symmetric S-state) is hindered by the selection rule [22]. 2.2.1. The triton wave function The triton wave function employed here is the one obtained by Sasakawa, Sawada and Ishikawa [49, 50] by solving the three-body Faddeev equation with a realistic nucleon—nucleon interaction. The totally antisymmetric wave function obtained from the three-body Faddeev equation in coordinate space is given by a sum of three functions which are represented by the Jacobi coordinate system (see fig. 2.1) =
t~I(12,3)
+
~(23, 1) + 1(31, 2).
(2.2.1)
The three functions are solutions of the simultaneous equations (E
—
H0) 1(12,3)
=
V,2 ~
(E
—
H0)’P(23, 1)
= V23 ~P,
(E
—
H0) b(31, 2)
= V.~,~P
(2.2.2) where E is the total binding energy, H0 is the free Hamiltonian of the three-body system and V1.1 is the interaction between particles I and j. We can see from eq. (2.2.2) that ~ satisfies the three-body Schrödinger equation, (2.2.3)
(E—H0—V~2—V23—V31)~P~=0.
We write ~(12, 3) as a sum of possible three-body states, ~(12, 3)
=
~ (l~s~)j~t~, (l~1 /2)j~1/2; stttmtni~a(x3,y3),
where the sum runs over several internal states a. The
~
2~
(2.2.4)
~ are spin—angular—isospin functions. An
3
2~3
Fig. 2.1. Jacobi coordinate system of the triton.
10
M. Igarashi et al.. Two-nucleon transfer reaction mechanisms
interacting pair 12 is coupled to form a state [(l~s1)j5t~],where s1 + s2 = s~and t, + t2 = t~,and the spectator particle 3 is in state [(lv 1 !2)j~1/2]. s~and t~are the total spin and isospin of the triton and m1 and n1 are the z-components of the total spin and isospin, respectively. The cba(xi, y3) are radial wave functions of the Jacobi coordinate x3 and y3. The functions P(23, 1) and ‘1(31, 2) are given by analogous forms. The classification of the components of the triton wave function by the symmetry group is carried out as follows. We separate the wave function into spatial and spin—isospin parts, Since the triton has isospin t~= 1/2, we have altogether six mutually orthogonal spin—isospin functions, Z~(12,3)
=
(ST) ~ (12, 3); S0t1M~n1~
(2.2.5)
,
where the interacting pair 12 has the intrinsic spin s~= S and isospin t~= T. S0 and t~(= 1/2) are, respectively, the total intrinsic spin and isospin of the triton. The superscript n specifies the various sets of S, T and S0 as
Z~’~(12,3)for 12~(12,3) for Z 31(12,3) for Z~ 141(12,3) for Z 51(12,3) for Z~ Z~61(12,3) for
T=1, S=0, S0=1/2, T=0, S=1, S, 1=1/2, T=0, S=l, S 0=3!2, T=1, S=l, S 03/2, T=0, S=0, S~ 1/2,
(2.2.6)
T=1, S=1, S 01/2.
We can rewrite eq. (2.2.4) in the form 1’~1(12,3),
(2.2.7)
~(12, 3) = ~ g(tl)(xs, y3)Z where the g1’~(x 3,y3) are the spatial parts corresponding to Z~(12, 3). We take appropriate linear combinations of these six functions to form the following six orthogonalized spin—isospin functions of proper symmetries: 5~(12, 3) + Z~61(12,3)1,
Ws(12, 3) = (1/~)[Z~
WA(12, 3) = —(1 /~)[Z”~(12,3)
—
5~(12, 3)
W~
Z~2~(12, 3)],
Z16~(12,3)],
—
12(12,3) = —(1 /~)[Z~
1(12,3)
W~
2(12,3)
W
=
(1 I~)[Z”
141(12, 3),
172(12, 3) = —Z
+
Z~2~(12, 3)],
W~
31(12,3).
2(12,3) = Z~
(2.2.8)
M. Igarashi et al., Two-nucleon transfer reaction mechanisms
11
Ws is totally symmetric, WA is totally antisymmetric, W~12is of a mixed symmetry with (12) symmetric, W~ is of a mixed symmetry with (12) antisymmetric, W~ is symmetric in spin space but of mixed symmetry in isospin space with (12) isospin symmetric, and finally W~,is symmetric in spin space but of a mixed symmetry in isospin space with (12) isospin antisymmetric. If we decompose the wave function ‘1 into these six spin—isospin components, we obtain the Derrick—Blatt classification [58], ~(12, 3) =
~AWS
+ ~SWA
+
+
+
+ ~12W~12.
(2.2.9)
Here, the cP are the spatial parts in the Derrick—Blatt classification and can be rewritten in terms of the g~(x,y) as follows: 5~(x, y) + g16~(x,~)] = _(1/~)[gW(x, y) g~2~(x, ~)] (1!~)[g~ —
1/2
~3/2
= —(1 I~)[g~5~(x, ~) =
g~4~(x, ~)‘
—
~3/2
g~(x~ ~)]~
~
1/2
= (1 /~)[g( ‘~(x,y)
+
g~2~(x, y)],
(2.2.10)
= g13~(x,y).
We truncate the function space at either 3 or 5 channels. In the case of a three-channel wave function, the spectator is in S-states and the interacting pair is in ‘S 3S, + 3D, states. In the case of 0 and and the interacting pair is in a five-channel wave function, the spectator can also be in a D-state, 3~, + 3D, states. The channels are further classified by the total spatial angular momentum L 0 and the total intrinsic spin S,~.These are represented by the index4~(x, k y), in table 2.1. g~5~(x, y) and g~6~(x, y) vanish, and so do Since we restrict l~and 1~to even values, the terms g~ 113A ~ and ~~2• Bui the total wave function 1I~of eq. (2.2.1) has non-zero values of and ~‘3I2’ because the other partitions ‘13(23, 1) and P(31, 2) contribute. However, there is no contribution to ~pA, since the contributions from ‘1(23, 1) and ‘13(31,2) cancel each other. We define a probability ~‘t/2
P(L0, r) =
ff
~r~2
dx dy
(2.2.11)
for each Derrick—Blatt classification symmetry r [which specifies six different components in eq.
Table 2.1 Classification of states cP~(i2,3) with five channels a 32 4 5 6 7 8
channel k
2~*
32
3D,
4 5 5 5 5
3S
‘S 3S,0
3D, 3D, 3D, 3D,
pair(12) (1,, j,)
particle 3 (L0, S~)
Z”
(0, 1/2) (0,1/2) (0,1/2) (2, 3/2) (2,3/2) (2,3/2) (2,3/2) (2, 3/2)
(0,1/2) (2,3/2) (2, 3/2) (0,1/2) (1,1/2) (1,3/2) (2, 3/2)
Z’3’ Z~’
t(12, 3)
12
M. Igarashi et al.. Two-nucleon transfer reaction mechanisms Table 2.2 Binding energies (B) and state probabilities (P) of the triton (on the left-hand side) and the deuteron (on the right-hand side) wave functions for different nucleon—nucleon forces triton B,
RSC3 RSC5 URG3 URGS UG3 UG5 TRS3 TRS5 TH
P
(MeV)
0 (%)
P0 (%)
6.389 7.031 6.945 7.480 6.844 7.395 6.829 7.464 7.42
90.08 88.90 92.30 91.25 91.59 90.48 91.09 89.87 1001)0
1.91 1.67 1.52 1.38 1.36 1.23 1.67 1.49
deuteron P~ (%) 0.09 0.05 0.07 0.06
P,, (%)
Bd (MeV)
(%)
8.01 9.34 6.19 7.32 7.05 8.22 723 8.53
2.225 2.225 2.150 2.150 2.082 2.082 2.226 2.226 2.207
6.47 6.47 4.95 4.95 5.51 5.51 5.92 5.92 0.00
(2.2.10)], and the spatial total angular momentum L0. The probabilities of admixture of the S, S’, P and D-states in table 2.2 are defined by 0J~)~
Ps.P(0,r—1/2~)+P(0,r-lI2),
~s~(
P~=P(1,r=S)+P(1,r=1/2~)+P(1,r=1/2)+P(1,r=3/2~)+P(1,r=3I2), (2.2.12) =
P(2, r3/2*)
+
P(2, r3!2).
Here we can show that P(L 0, r = 1/2 ~) = P(L0, r = 1/2 ) and P(L0, r = 3 /2k) = P(L0, r = 3/2) for all L,,. The P-state admixture appears in the case of the five-channel wave function but this is very small. To investigate the dependence of the absolute cross section of the one-step (p, t) process on the triton wave function, we tested various wave functions obtained with different nucleon—nucleon interaction potentials and different truncation of the spin—angular momentum space. The corresponding binding energies and the probabilities of admixture of S, 5’, P and D-states in the triton bound state wave function used in this work are listed in table 2.2. RSC, URG, UG and TRS represent the wave function for the Reid soft-core [59], Ueda—Riewe—Green model II [60], Ueda—Green model I [61]and Tourreil—Rouben—Sprung [62] nucleon—nucleon interaction, respectively. The numbers 3 and 5 mean the three-channel and five-channel wave functions, respectively. TH stands for the variational wave function with Tang—Herndon [15] interaction, which is also listed in the table for comparison. All calculated binding energies of the triton, B~,are smaller than the experimental value 8.482 MeV. Various additional sources of binding energy have been suggested such as three-body forces [50],but we do not discuss this further. This insufficiency of the binding energy decreases the magnitude of the transfer reaction cross section, since it indicates that the overlap between interaction and wave function is insufficient. The xy~~(x,y)’s of the RSC5 triton wave function are shown in figs. 2.2 and 2.3. The functions are contracted in the x-direction and surprisinglywith long-ranged in athe Thesoft-core ‘S~andradius the 3~, 3S, component 1,, = 2 have nodey-direction. at around the as components with l~ = 0 and the a function of x, which is caused by the strong repulsive core of the RSC potential coupled with a particle exchange effect of the Faddeev equation.
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
13
0.15
‘N
y=1.98fm
/ /
0.10
\\ \\
~
/~
~0.05
,1/
3S 1
_su2N..
—
“~
~
f: : ~ —0.10
0
1
2
3
4
5
6
x ( fm ) Fig. 2.2. The RSC5 triton wave function. The xy~(x,y)’s of eq. (2.2.4) as a function of x at y
1.98fm.
2.2.2. The deuteron wave function The sequential transfer (p—d—t) process passes through the intermediate deuteron unbound state as well as the ground state. The wave functions used are evaluated with the same realistic nucleon— nucleon interactions with non-central components that we used for the triton wave function calculation. The binding energies and the D-state admixture probabilities in the ground state are listed in table 2.2 for various interaction potentials. 0.15
x=1.5fm /
\
/
0.10 /
/ >,0.05
~:
\
I,---., I, //3/
~i
~1 N
-‘
D, —S,,2 D1-d312
/,
3c,
S~1~
\
N .
~:::~ —0.10 0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15
y ( Im ) Fig. 2.3. The RSC5 triton wave function. The xyç&jx, y)’s as a function of y at x = 1.5 fm.
14
M. Igarashi et al. - Two-nucleon transfer reaction mechanisms
The unbound ‘S0 and 3~,+ 3D1 states are calculated with the same realistic nucleon—nucleon interactions at appropriate incident energies. The coupled-channel equations (with a tensor force) are solved exactly for the bound and the scattering states. For the case of a momentum dependent force the method of numerical calculation given by Sawada and Sasakawa [63] is used. These deuteron wave functions are used for the calculation of the projectile form factors connected with the one-nucleon transfer process and the method will be described in sections 2.4.1 and 2.4.4. 2.3. The one-step process The transition amplitude for the simultaneous two-nucleon transfer process [one-step (p, t) process] is given by
~(~
T10 = KxH(k,, r,)~~( r~,rB(~B)~V~ ~,
~)
r~,r2)~~(k0, ~
(2.3.1)
Here x ~ (x is the proton (triton) distorted wave, and r~(ri) is the relative coordinate of the proton (triton) with respect to the target (residual) nucleus. The ~(i’sare the internal wave functions of the proton, the triton, and the target and residual nuclei. ~ r, r) and tI~A(~B,r,, r2) describe the internal motion of the transferred two-nucleon pair in the triton and the target nucleus. V is the residual interaction causing the two-nucleon transfer, V= + V2,, + VBP UAP in the prior form representation, where V,,~and V2,, are the interactions between the proton and transferred nucleons, VBP is the one between the proton and the residual nucleus, and Ui,,., is the optical potential used to generate the distorted wave x In the DWBA method, the approximation (xit/Jlt/JB~VBP UAP~tIJPt/JAXP)=0 is commonly used. The adequacy of this approximation has been argued on both theoretical and practical grounds [64] (other references are cited in ref. [52]). With this approximation the form factor of the projectile—ejectile system is given by —
—
(~t(~p’
r~,~
+
~
(2.3.2)
2.3.1. The one-step (p, t) form factor When the same interactions V,~and V2~,are used in both the evaluation of the triton wave function by the Faddeev equation and the calculation of the reaction form factor, the product (V1~+ ~ can be evaluated easily with the aid of the Faddeev equation (2.2.2), (V,3
+
V,3)~I-’~ = (E
—
H0)[13(13, 2)
+
‘13(23, 1)] .
(2.3.3)
On the left-hand side of this equation, ~ consists of three components, and both the wave function and the interaction potential have angular momenta, if the interaction potential has the tensor term. Many terms must therefore be evaluated, and the treatment of the angular momentum coupling is tediously complicated. On the other hand, the right-hand side can be evaluated simply, because E Hf) is a scalar operator. The method, however, requires a better accuracy of the wave function for the evaluation of H013 than is the case using the left-hand side of eq. (2.3.3). The coordinates used to describe the reaction are shown in fig. 2.4. There are three independent coordinate vectors, which describe the relative positions of the four particles participating in the reaction. In order to eliminate one coordinate vector and express the form factors as a function of the —
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
15
r
Fig. 2.4. Radial coordinates describing the one-step two-nucleon transfer process A(p, t)B.
channel vectors r~and r,, we use the coordinate transformation twice; once here and again just after eq. (2.3.28). First we transform them to the relative and center of mass coordinates of the two transferred neutrons. For the triton system we take the transformations (x,, y,)
~s
(R’, r)
,
R’ = ~y,
+
~x,
,
r=y,
(x2,
~‘
(R’,r)
,
R’ = ~y2
—
~x2
,
r= —y2—
y2)
~x,
—
(2.3.4a)
,
,
(2.3.4b)
and for the target system (r,, r2)
~
(R, r)
R = ~(r, + r2)
,
,
r = r,
—
r2
(2.3.5)
.
Corresponding transformations of the wave function are performed in the generalized Bayman—Kallio method [10, 65]. First we consider the transformation of the form factor of the triton system, eq. (2.3.2). According to eq. (2.3.3), the products of the interaction and the triton wave function are replaced by the sum of two functions, [V31(x2)+ V23(x,)]W = (E— H0)[t13(x2, y2)+ cP(x,, y,)].
(2.3.6)
We transform the second term on the right-hand side, (E H0)~(x,,y1), to the (R’, r) coordinate system, where the pair 1 and 2 are coupled first and then particle 3 is coupled to them in the angular—spin—isospin space. We separate the orbital and intrinsic spin parts of the wave function by transforming it to the L—S coupling scheme, 2)J~1/2; s~t im mi) a4~a(xi, y,) 13(23,1) = ~ (l~s~) j~t1,(l~lI ~x 5x ~x = ~ l~ 1/2 j~ (s 1/2)5’M~.}(23,1)~I(t~1/2)t,n~}(23,1)~ aL’S’Ms. L’ S’ st —
x (L’ m~ M —
5. S’MsIsim~)4 ~
y1),
(2.3.7)
16
M. Igarashi et a!.. Two-nucleon transfer reaction mechanisms
where the LS—jj transformation bracket is related to the 9j symbol by 11
~! 11
12
S
L
S
2
li
=
~,
________________________________________ 11~ Si + 1)(2j, + 1)(2L + 1)(25 + 1)1 12 ~2 12
.
(2.3.8)
J
~L S J 3, 1)) is the three-body spin function in which pair 23 is coupled to the intrinsic spin state Ks~1 s~ /2)S’Mç.}(2 and totally to the state S’. (t~1/2)t~n~}(23, 1)) is the analogous isospin function. The intrinsic spin and isospin functions are transformed from the (23, 1) coupling scheme to the (12, 3) coupling scheme using Racah algebra, (s~1/2)S’M~.}(23,1))~(ç1/2)t 1n,}(23, 1)) =
~ (s,s2(s)s3S’~s,s3(s~)s1S’)~(s~ 1 I2)S’M~.}(12,3)) x t,t2(t~)t3t,It2t3(t~)t,t,) (t 1 /2)t5n,}(12, 3)).
The spatial part is transformed to the form (E
—
)
~
13(~,
H0)~’~’~’~’(x1, y~)[y’oI(~ x y
=
(2.3.9)
A~.
g~~’(r, R~)[YA(~)x YA’(~l)]~’,, (2.3.10)
where lX 1~ 1),, A and A’ are the orbital angular momenta associated with coordinates x,, y,, rand R’, respectively, and L’ is the total one. The summation over A and A’ runs in principle up to infinity, but the contributions from the terms decrease rapidly as A and A’ increase. To obtain (r, R’) we set the z-axis along the ~ direction as shown in fig. 2.5 and multiply both sides of eq. (2.3.10) by (—)‘ +M (L’ _MfAlM~~A0)Y~,(R~)* and integrate over R’( 0, and then sum over M’. This choice of z-axis is different from those in other works [10, 65]. The merit of this choice is discussed later. Since
4)
g~~,f31)
E~I e
e
r
X2 R 2
Fig. 2.5. The integration of eq. (2.3.11) is performed with I=z and
~1
andx, in the x—z plane.
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
17
Y~,(~~)V(2A+1)/4ir8(m,0), we obtain R’)
g~Yt)L(r
xf (E
—
=
E (—)~M’(L’_M’A’M’~A0)\/2L 1~’(I,) x
H
1
y’Yl(fl]L’yL’(~l)*
y,)[Y
0)’’~’~(x,,
di’.
(2.3.11)
The explicit expression for the spherical harmonics is
~)etm4d~o(0)V(21
Y~(0,
=
+
1)/4~,
(2.3.12)
where d~~(0) is the reduced rotation matrix. The vectors x,, y 2 and R’ are now in a plane containing the z-axis. We can choose ~ = = = 0 without loss of generality. Finally, then, expression eq. (2.3.11) becomes (l~,1~)L’
g~~ (r, R ~=
1)(2l~+ 1)(2A + 1)(2A + 1) 2(2L’ + 1)
\/(2l~
+
x ~
(AOA’M’IL’M’)(lxmxly M’
x
f
—
mx~L’M’)
t~~’(x,,y,)d~
(E
—
0(~jd~m0(00)dZ0(0) d cos 0,
H0)
(2.3.13)
in which 2 + R’2 R’rcos 0, y, = ~\/~r2 + R’2 + 3R’rcos 0 x, ~~r cos0~=(—~r+R’cos0)/x,, cos0~=~(~r+R’cos0)Iy,. —
(2.3.14)
This implies the selection rule lx+ly+A+A’=even,
(2.3.15)
which ensures parity conservation. We can show that the second term (E
—
H
0)c13(x2, y2) of eq. (2.3.6) has exactly the same contribution as (E H0)11(x1, y1). We can express the Jacobi coordinate system (x2, y2) in terms of r, R’ and 0 as follows: 2 + R’2 + R’r cos 0, y 2 3R’r cos 0, x2 = ~~r 2= ~ + R’ (2.3.16) cos = —(~r+R’ cos 0)/x 2 cos = —~(~rR’ cos 0)/y2 —
—
,
—
18
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
According to eqs. (2.3.14) and (2.3.16), replacement of 0 by
ir
—
0 interchanges (x2, 0.,~,,y2, 0~.)and
(x,, 0~,y,,ir—00). Because d~1(7r 0) —
= (....)t_md/
-
~(0)
-
(2.3.17a)
,
we have and
4~,=0.
(2.3.17b)
We get the same type of integrand for the second term as on the right-hand side of eq. (2.3.13), differing only by a phase factor (_)bx~, where we put l~= l~= l~.Furthermore, the transformation coefficients for the intrinsic spin and isospin spaces have the following relations: (s,s2(s~)s3S’~s3s,(s5)s2S’) =
(2.3.18a)
s,s2(s~)s3S’~S2s3(s~)s1S’) ,
(_)50
(t1t1(t~)t3t~lt3t,(t~)t2t,) = (—)‘~‘~( t,t2(t~)t3t,~t2t3(t~)t1t,) .
(2.3.18b)
In the full space, the two terms of eq. (2.3.6) differ only by a factor (_)1x+A+so+so+tx+to~ This factor equals + I because of the antisymmetric nature of the pair of fermion particles. As a result the two terms give an identical contribution. As the final result the projectile form factor becomes lx
(~i(~p’ r, ~
+
~
=
~x
Ix
l~ s~ j~, I.~J0~~’S’M5S~2T12L’ 5’ s~
x (s,s2(S,2)s3S’Is2s3(s~)s,S’)~t,t2(T,2)t3t~~t2t3(t5)t,t,) 5~~5’V(25’ + 1)(2s + 1)W(s~S, x ~ (—) 2s~L’; S’s) x (L’ m,
—
M5.
M5. — m~Ism, — m~)(s~m0s m~— m~js~m~)(T,2 n~ n,, t~n~It~n,)
~12
—
5’(”,)X X~2(h1
x [x
2)]~,_mp[Xu1(Tt) X t~’(r,
x ~
R’)[YA(~) x Y 4’(~’)]~-Mç..
g~
(2.3.19)
For the target system we expand the wave function and the two-nucleon wave functions ~2N12(r,,
I/llaMa T8N~
~jAMA,TANAB’i’2)
~
in terms of the wave function r2) in the j—j coupling scheme,
t/JJAMATANA
SBA(ala2 jT,2)
a1a2jT12
x (JBMBI
MA
—
M81 JAMA)(TBNBT,2
~JBMB,TBNB~~i~2
NA — NB~TANA)
Ti2NA_NB(r,, r2)
,
(2.3.20)
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
where the
19
a2 jT,2)’s are the parentage coefficients of the two-nucleon configurations in the target nucleus IA MA.TANA• The normalized two-nucleon wave function I/r1mT12~’~2(r,, r2) is transformed ai a2 into the L—S coupling scheme, SBA(a,
I/’•
r2)
,j,JmT12NI2(
a1a2
1’
x
1/2(1 +1 ôaa)
=
(r,)I/i~ (r
[~.
2)
j1m1
—
1/2(1 + ôai.a2) x (L m
[
(_)i_T12~fjimi(T2)t~ri2m2(Tl)1[Xii(Ti)X
—
~
iI2II
12 1/2 L ~i2
LS12M51,
Ms S,2Msi2~jm)[~S1(u,)X
x {[~fri) X
12(T2)JN1~i2 1 1.
x
1/2 1,1
1,
1 =
~ (j,m1j2m2~jm) m1(m2)
I/’I2(r2)]~M5
()si2+
—
X52(~T2)]~[Xu1(Ti) X Xt2(T2)].~~
T12[I/~ (r2) X
I/~2(r,)]~M5}.
(2.3.21)
We transform the radial part to the form [I/i,(r1)X I/i~(r2)]rn —Ms ~
—
J l.A
= An f~1”2~(r,
x I/t2(r~)]rn — M5
()5i2 + Ti2[ t/i()
2)
X ~‘A(1~)]rn_M~
(2.3.22)
.
R)[Y5(i
As before, we set the z-axis along the ~ direction, and get f(t~.l2)L(r
R)
=
1
—
(_Y~2+Ti2+l
2
V(2l~+ 1)(2l2 + 1)(2A + 1)(2A + 1) 2L+1
x ~ (AOAMjLM)(1
1m,12M—mJLM)
Mm1
x
f
ut(rt)ut(r2)d~1o(0,)di2 M—m1,0’,‘02)d~0(0)d cos 0,
(2.3.23)
where 2+R2+rRcos0, r, =\I~r cos0 1=(~r+Rcos0)/r1,
r
2+R2 rRcos0, —
2=~~r
(2.3.24)
cos02=(—~r+Rcos0)/r2.
This also brings the selection rule 11+12+A+A=even.
(2.3.25)
We integrate over the spin and isospin functions of the transferred two-nucleon pair 12 and also over the vector i~to eliminate the dependence of the form factor on it, and we obtain
20
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
J
1t”t~~(r,R)[YA(~) x (t~,t )L’ {g~ m,—M 4,~(r, R’)[YA(~)x y(~~)]L* ~ 5.J AA =~
(
)..i+AAy(2L
)A+mt~Ms.~
+
AMB+mPMSJ
1)(2L’ + 1)W(LL’AA’; IA)
A
x
(L MA
x
g(~l~)L~ (r,
—
MB +
mp
—
M~.L’ —m,
+
Ms.~lMA
—
MB + m0 —
m~)
R’)f~,,”2~~(r, R)[Y4(1~)x YA(k’)]~M÷
(2.3.26)
We can sum easily over A, because the spatial parts of the functions g and f do not depend on A. Then, ~
W(LL ‘AA’; lA)g~5’.~Y)L (r, Rl)f
(_)l.
t2)L(r
R)
A
=
—L
1
x
21
x
+
+
1
—
1)(212
(
2
)S17+Tis+ti+t2+L
—
2
(.,~Si2+Ti7+tr+tv+L’
1)(2A + 1)(2A’ 1)(2L’ + 1)
+ 1)(21~+ 1)(2l~+
1
~
1
(2L
+
+
1)
(LML’M’IlM+M’)(AMA’M’llM+M’)
MM’ m1 m,
x (l,m,12 M
J J
X
—
mjLM)(l1m~l~ M’
u~(r,)u~(r 2 )dtim1.0 (01)d
—
m~lL’M’)
m10(02)d~0(0)dcos 0
—1
x
(E
—
H0)
txty~’(X,, Yi)~,o(0xi)~i~mx.0(0yi)~,o(0’)
dcos 0’.
(2.3.27)
—1
We get the final expression of the transfer form factor, which is a function of the coordinates R and R’, as follows:
~ =
r~,r~)t/l8(~8)IV,~ + V2P p(~p)~A(~B, T~, t(yP~mP(JMi MA — MBIJAMA)(l MA ~
—
MB +
mp — m~s m~ m —
211 MB
i
tsjAfl’
x (s~m~s m~ mPIs~m~)(TBNB T,2 NA —
—
NBI
TANA)(T,2 n1
—
n~t0n~It~n1)
jtl +t2~tx~ty~t
~
X
sBA(aia2
jT,2) 1/2(1
aia2j~]~ LL’S,2S
X
[1, 112 [L
+ 6a1,a2)
1/2 j,][ l~ Sx 1x1 1/2 12 [l~ 1/2 j~W(s~S,2s,L’;S’s)W(LL’is; IS,2) ~i2 ~ L’ 5’ s,j
I
x (SiS2(Si2)515’1S253(Sx)StS’) (t,t2(T,2)t3t~It2t3(t~)t,t~)
—
MA)
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
x (_)~P~5~t(_)S5i2(25+ 1)1/(21 + 1)(2L —
x
1
()1~~+t~L’
—
+
1)(2L’
+
21
1)(25’ + 1)
S
()
12+Ti2+1,+12+L
—
1
S,2+T,2+t~+t~+L’
() —
2 2 1 /(21, +1)(2l2+1)(2/~+1)(2l~—1)(2A+1)(2A+1) x21+1 ~ (2L+1)(2L’+1) x
~
(LML’M’IlM
+
M’)(AMA’M’IlM
+
M’)
MM’mim.1
x (l,m,l2 M — m1~LM)(l1m~l~ M’ m~IL’M’) —
x
x
f J
r2(J ui(ri)ut(r2)d~ 0(0)d~m 0(02)d~0(0)d cos 0
(E
—
H0)
x [Y~(’~)x
x.ty)L’(x,,
yi)d’~o(0i)4~rn o(0yi)~o(0’)d cos o’) dr
}‘A~(R)]MA_MB+mp_mt~
(2.3.28)
This transfer form factor, which is represented in the coordinate system (R, R’), is transformed once more to the channel vector (re, r~)coordinate system by the ordinary method of calculation of finite-range DWBA form factors [66]. 2.3.2. The selection rules for the one-step process From eqs. (2.3.27) and (2.3.28) we can get the relations of the angular momentum coupling as shown in fig. 2.6. An additional restriction is that A + ~12 is even as a consequence of the antisymmetry of the two transferred neutrons. The relative orbital angular momentum A is even for ~i2 = 0 or odd for ~12 = 1. The parity conservation imposes the restrictions that 1~~~1y = (_)A+A’ = +1 (2.3.29) = () ,
Fig. 2.6. The angular momentum coupling illustrating the selection rules.
M. Igarashi ci a!., Two-nucleon transfer reaction mechanisms
22
.
(2.3.30) 7rB designate the intrinsic parities of the proton, triton and the states of the target Here ~ ~“A and and residual nuclei A and B, respectively. The natural parity transitions satisfy the condition ~A
~B
= ~l1+12
= ()A+A
= ~IT
~,
= ~A~B
=
(~)‘~
(2.3.31)
where j is the transferred total angular momentum, while unnatural parity transitions satisfy the condition (2.3.32)
~A~B().
The component in which A equals zero contributes only to the natural parity transition. If A and ~12 are zero, then A = L and j = L, so that the parity change is given by the relation = ~A~B
= ()A+,.t
= ~
=
(—)‘.
(2.3.33)
However, if A $0, z~ir= (—)‘~ is allowed and unnatural parity transitions can occur. The A = 0 component is the main component of the triton wave function [thedefinition of A is accord with eq. (2.3.10)]. Therefore unnatural parity transitions are much weaker than natural parity transitions. The selection rules for each component of the triton wave function with a particular symmetry type is given as follows. Since we use eq. (2.3.3) in this analysis, we need to classify the symmetry type of (E H 0)[13(13, 2) + 13(23, 1)]. This classification is therefore not purely about the triton wave function, but rather about the product of the transfer interaction and the triton wave function. The interaction potentials have a dependence on the spin and angular momentum. We consider only the term (E H0)13(23, 1), since the two terms of eq. (2.3.3) obey completely the same selection rules. The recoupling coefficients (eq. 2.3.9) from the spin—isospin function Wr(23, 1) of eq. (2.2.8) to the z~~~(12, 3) are given in table 2.3. In the (p, t) reactions the isospin of the two transferred particles is 12~(12, 3), and Z~3~(12, 3) and Z15~(12, 3) are allowed to be[see onlyeq. in triplet states 1). Therefore not allowed (2.2.6)]. For(T,2 this=reason only S Z 12 = 0 transitions are allowed for the spatially symmetric state and only S,2 = 1 transitions are allowed for the mixed symmetry state with total spin —
—
S0=3/2.
The selection rules for each particular symmetry type are summarized in table 2.4. The spatial parts of each triton state are also given in the same table, where the ~k’~ represent the spatial part of the function indexed by k, which appears in table 2.1. In the case of the triton wave function with the three-channel configuration the transferred spin s in the reaction is always equal to the total spin ~i2 of the two transferred neutrons. Hence, A = even for s = 0 and A = odd for s = 1. The P-states are allowed in the case of the five-channel wave function, but the probabilities of the P-state are very small. For the P-states the s = 1 transition is allowed in the case of ~i2 = 0. Namely, both even and odd A are allowed
WA(23,1) W~ 2(23,1) W~,,,(23,1)
1(12,3) Z~’ —1/V~ —~/~/4 0
Table 2.3 The recoupling coefficients of spin—isospin functions Zt2~(12,3) Z’31(12. 3) Z’41(12, 3)
Z’”(12, 3)
Z16(12, 3)
1/V~
0
0
0
0
—‘~ñ/4 0
0 —t/2
0
\ro/4 0
0
v’~/2
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
23
Table 2.4 Selection rules for particular symmetry types of the triton wave function state
(L
S sym. S mix.
(0,1/2) (0,1/2)
Psym. P mix. (S0=1/2)
(1,1/2) (1,1/2)
P mix. (S0=3/2) D mix.
(1,3/2) (2, 3/2)
0, S~)
S,2
spatial part
—(1/’v’~)(~,—~2—~5) 0 (1/V’~)(~, + + f~) 0 1 ~is/~ 0 ~6/v’~ 0 1 ~ I ~ +~+ 1
A
s
even even odd even, 0 even, 0 odd odd odd
0 0 1 1 1 0 and/or 1 0 and/or 1 1
Table 2.5 Allowed values of the relative angular momentum of the transferred two nucleons 0~—s0~
0*3*
(!sj)
A
(!sj)
A
(000)
0,2,4,... (1,3,...)
(303)
2,4,6,... (1,3,...)
(110)
1,3,5,... (2,4,...)
(213) (313) (413)
1,3,5,... (2,4,...)
for the transferred spin s = 1 transition, but A = 0 is not allowed. For the P-states both s = 0 and s = 1 transitions are allowed in the case of ~12 = 1. The allowed values of the relative angular momentum A are listed in table 2.5, where the set (lsj) stands for the transferred orbital, spin and total angular momenta of the transfer form factor. The odd A’s for the s = 0 transition, and the even A’s for the s = 1 transition are allowed in the case of the five-channel triton wave function, but not allowed in the case of the three-channel wave function. 2.3.3. The hard-core correction to the form factor for particle transfer processes Several exact finite-range DWBA calculations [13, 14, 67, 68] with various sophisticated triton wave functions and finite-range interactions have been reported. The resulting cross sections of all these calculations are smaller than the observed ones, but the differences among them are not very large. Most of these calculations, however, employ two-body interactions with a hard core like the Tang— Herndon interaction [15]. As has been pointed out by Dobes [16], the use of such a singular potential for rearrangement collisions requires a significant correction. The form factor for rearrangement collisions contains an overlap integral of the type (I/tuncorrL1”k//corr), in which the interaction V is put between the wave function çt’corr, which correlates with the interaction in question, and the uncorrelated wave function I/’uncorr~An overlap integral of this type has a finite contribution from the inner region of the hard core, which is shown by the hatched area in fig. 2.7. If we consider a square potential with finite V~,and thickness r, and increase V~to infinity, the potential times the correlated S-state scattering wave function inside the potential becomes
M. Igarashi et a!.. Two-nucleon transfer reaction mechanisms
24
r~
r
Fig. 2.7. Hard-core effect.
lirn V(r)I/i(r) = lirn
k’ sec(kr~)e’~,
(2.3.34)
where E = h2k2/2~and V0 E = h2k’2/2/L, where E and ~sare the scattering energy and the reduced mass of the scattering particle, respectively. This may be represented as a discontinuity of the slope of the wave function at r = r~,and is described by the following additional term, the so-called pseudopotential term: —
V’(r)=
)
-~-- 8(r—r —~-— r. 2~sr~ c ar..
-~—
(2.3.35)
This hard-core correction term has opposite sign compared with the usual overlap integral from the region of attractive potential and reduces the total magnitude of the overlap integral. This term is familiar in Brueckner theory [69], but it has not been used in transfer reactions. In the case of (p, t) reactions the overlap integral with hard-core corrections is given by +
V~~(r~~) + V~~(r~~) + ~
(2.3.36)
These correction terms are calculated exactly for the case of the triton wave function obtained by Tang and Herndon (TH). Its spatial part is assumed to be totally symmetric, and expressed as ~~=N~f(Ir,_r 2I)xf(Ir2_T3I)xf(H_r,~),
(2.3.37)
where N, is the normalization constant. The two-body interaction used by TH is V(r)
=
P0V0(r)
+
P,V,(r)
(2.3.38)
,
where P0 (P,) is the spin singlet (triplet) projection operator, and V1(r) (i = 0, 1) is given by ‘71(r)
=
(r ~ d)
,
V,(r)
=
—V~exp[— K~(r— re)]
(r
r~).
(2.3.39)
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
The parameters used are r~= 0.45 fm, V~= 277.07 MeV, V? 2.735
=
549.26 MeV,
25 K
0 = 2.211 fm’ and K1 = fm’. The radial function f(r) in eq. (2.3.37) has the form 2[exp(—ar)+ B~exp(—f3r)] (r d), (2.3.40) f(r) = u(r)/r (r~d), f(r) = A~r~’
and u(r) is obtained as a solution of the differential equation Fl 22du ——
—i
+[~(V (2.3.41)
0+1/~)—e]u(r)=0.
The constants A and B, are determined so as to match smoothly the two expressions at r = d. The other parameters are determined to minimize the binding energy of the triton in a variational sense [15]. TH gave a few sets of parameters and we here choose the following set: a = 0.265 fm’, /3 = 3.60 fm’, d = 1.20 fm and e = —10.0 MeV. In this case the pseudopotential is given by V’(r) = 150.435 MeV fm x ~5(r re).
(2.3.42)
—
The importance of the hard-core correction terms in the (d, t) form factors is illustrated in appendix A. 2.4. The sequential transfer (p—d—t) process The second-order DWBA amplitude for the sequential transfer (p—d—t) process is given by TIdP
=
~
~
“~~~)~frB(~B)IV,2 + V,P~d(~P, ~
X G~(r~, T~)(~d(~P, T4~C(~B, TI)I V~p -
Kx~(k,,~
x
(I/~(~,
+
exchange terms
r~,r
V2~
T~hc(~B, T,)~
(1.E-32),
T))
p(~p)~A(~B’ TI,
B(~B)k~d(~P, ~c(~B~ P(~P)~A(~B,Ti,
T2)X~~~(k~, re))
r))
r2)x~(k~, Tn)) (2.4.1)
where G~(rd,r~)is the distorted wave Green function of the intermediate deuteron channel and the second term is the channel non-orthogonality term. For the calculation of these amplitudes we need three kinds of form factors of the projectile system: (a) (~PjV23JI/Jd(r)),
(b) (çll~(T)~V~2 + V,3~I/t,(r,P))r’ (c) (I/id(r)kit(T, p))r’ where the subscript T stands for an integration over the coordinate T. We evaluate these form factors employing the same realistic triton and deuteron wave functions as well as the realistic transfer interaction as for the one-step process. Our purpose is to calculate the one and two-step processes with the same degree of accuracy. 2.4.1. The one-particle transfer form factors The RSC5 triton wave function, and the deuteron ground state and unbound state wave functions
26
M. Igarashi ci al.. Two-nucleon transfer reaction mechanisms
obtained with the Reid soft-core potential are used in the present calculation. Since these wave functions include the D-state and the interaction includes L—S and tensor components, the transfer form factors of the projectile—ejectile system consist of I = 0 and I = 2 angular momentum components. The projectile form factor of the (p, d) process can be calculated directly as a product of the interaction and the deuteron wave function [70]. The form factors between the deuteron and triton are evaluated using the relations (I/i0~V,2+ V3jI/,)
(I/’~~ I/’1)
=
(I/i0(12)~E— H0~’1(12,3) +
(I/.~~(12)~ 13(12,3)
=
13(23, 1)
+
+
cP(31, 2))
(2.4.2)
,
13(31, 2)) .
(2.4.3)
2.4.1.1. The projectile form factor for the (p, d) process. We define the deuteron wave function, which includes the D-state component, by Isdtdmdnd)
=
(lms~m~
~
—
(~~)
mlsdmd)[X,,
x
X
~(~n)]~m[~,(Tp)
X,(T~)1~i’Y~(r)u1(r),
ims~~
(2.4.4) where u1(r) is the radial wave function of each component. The nuclear interaction has central, L~S and tensor force components, V~~(r) =
1’~(r)+ VLS(r)L S
+
‘
VT(r)S,. .
(2.4.5)
The projectile form factor is given by (s0t~m~n0 ~
(Im~md — mp
~
= I!
Isdtdmdnd)
x (t0n~t5n0 5P
x
—
m~Jm0
—
m~)(s~m0Jm~— mP~Sdmd)
‘.,
-I
—
flp tdnd )i
m a, —,n Y~ (r)~2,’
n’Pn(_)5P~J’dV(2s
+
—m
(o~~ )x,~ P(r~) 1)(2j + 1) W(ISflSdsP; ~
-
(2.4.6)
(_)
We sum over I’ and s~and get
E
(IsPflsdmd
~
0m0)
V~(r)u0(r)— =
I ‘S~~s
\/~
—V’s VT(r)uO(r) + V~(r)u0(r)
(I = 0, ~
VT(r)u2(r) 3VLS(r) [V~(r)
—
—
2VT(r)]u 2(r)
=
1),
(I = 2, s,,,, = 1), (1 = 0, s~= 0).
(2.4.7)
Substituting (2.4.7) into (2.4.6) we get two form factors with (1= 0, j= 1/2) and (1= 2, j=312). 2.4.1.2. The projectile form factor for the (d, t) process. The numerical calculation method is
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
27
analogous to the case of the one-step (p, t) form factor. The final expression is given by
(~~(12)J V23 + V~jI/~) =
~ (Ims~m~jm + ~1
m~)(J0m~ m —
.1
x (T,2 n1
—
m
“
—
m
m~jm
+
m~~s~m~)
n
n~t,,n~t~n~)i Y, ~
x ~ \/~(2Jx+1)(2Jy+1)(2lx+1)(2ly+1)(2l+1)(2j+1)(2J0+1)(2L’+1) tXSxIr LL’S’
x (2S’ + 1)K5i52(Si2)53S’~5253(5x)5iS’)(t1t2(T,2)t3tjt2t3(ç)t,t1) 1x ~ j~ L 1 L’ 1 x 1~ s~ j~ ~12 ~ 2 jtx+Iy~/~L L’ S Si ~d .1 5i —
~‘
x ~ (—)t~M(lml~ M
x
fJ
(E
—
—
m~~L’M)(L’ —MlM~L0)
H 0)
Y)d~O(0x)4~mO(0y)d~M ~(0) d cos
tty)L’(x
This form factor also has two components (1 = 0, j = 1/2) and (I
=
2, j
0 uL(r)r dr.
=
(2.4.8)
3 /2).
2.4.1.3. The non-orthogonality overlap form factor for the (d, t)process. The direct overlap integral appears in the overlap form factor in the non-orthogonality term of the two-step transition amplitude. This direct overlap integral is easily calculated as
j m + mfl)(Jd m~ m — m~j m + m~js1m~)
(c~d(12)~(12, 3)) = ~ (ims~
—
tjL
x
(T,2 n~ n~~
The other terms (~d(12)~(23, 1)
—
+
f
2 dr. (2.4.9) ~LI(
p)uL(r)r
~(31, 2)) are calculated in an analogous way as in subsection
2.4.1.2.
The calculated form factors of the projectile—ejectile system for the ground state (~d= 1, t 0
=
0)
deuteron channel are shown in fig. 2.8. 2.4.2. The finite-range form factor for the one-particle transfer process The finite-range form factor of each one-particle transfer step in the two-step transition amplitudes is calculated by the code TWOFNR [71]. This code uses the method of coordinate transformation given by Austern et a!. [66] for the calculation of the finite-range form factors. Let the orbital and total angular momenta of the transferred nucleon in the target nucleus and the projectile nucleus be (1,, and (12, 12)’ respectively, as shown in fig. 2.9. The allowed transferred
1~)
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
28 10~ ~
1
~2
(b)
~a) 102
,,,
101
.
~p IVnP I~d)
~l
-
\,
~
‘V
~ ~
-
L=2
(~d~Vnn~V~pNI’t)
‘~s\
I0l
-
L=0
/
.
“S
N
~
102
N
-
/‘
-
10~ I
L—2
3
<~dI’4’t)
/
‘~
io10~ ~ 0
1
2
3
4
5
6
7
8
0
1
2
R(FM)
3
4
5
6
7
8
R(FM)
Fig. 2.8. Form factors of the projectile—ejectile system evaluated with the RSC potential, the RSC ground state deuteron and the RSC5 triton wave function. (a) Projectile form factors of the (p. d) process. (b) Ejectile form factors of the (d. t) process. (c) Overlap form factor of the (d, t) process.
angular momenta, usually denoted by 1, s and j, are given by
11
1—I2H
ii
121
<—~I+i2,
~I2,
~
(2.4.10)
The transferred angular momenta (Is]) corresponding to the first and second steps are referred to as (lsj),0, and (Isj)2fl0, respectively. The inclusion of the (12 = 2,12 = 3/2) component of the projectile form factor compels us to calculate many transition amplitudes. Such transition amplitudes arising from minor admixtures are not important for the absolute cross section at low incident energy, but they play important roles in the tensor polarizations and analyzing powers [72].
Fig. 2.9. The coordinates and angular momenta involved in the one-particle pick-up reaction A(a, c)C.
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
29
2.4.3. The spin—isospin closure approximation The deuteron is a loosely bound particle and it is easily broken up. Furthermore, if we use the spin—isospin closure approximation in the intermediate deuteron channels, the selection rule for the transferred spin s is quite different from the case in which only the ground state deuteron channel is taken into account, as shown below. Therefore, for the intermediate channels of sequential transfer processes, we take into account the ‘S0 and ~ + 3D, unbound deuteron states as well as the ground state. As Pinkston and Satchler [73] pointed out, ‘S 0 channels in the spin—isospin closure approximation drastically change the spin transfer selection rule in the sequential transfer process from that when only the ground state channel is taken into account, namely the s = 1 transfer component disappears. The assumptions implicit in this approximation are the following: (a) The triplet (s~= 1, t~= 0) and singlet (s~= 0, t,~= 1) deuteron states are degenerate. (b) The triton and deuteron are purely in the symmetric S-state and the interactions which cause the transfers are purely central forces. (c) The optical potential of the intermediate deuteron channel does not have a spin—orbit coupling term. Under these assumptions, the transition amplitude through the singlet (s~= 0, t, = 1) deuteron intermediate channels has the same radial overlap factor as the one through the triplet (s~= 1, t, = 0) deuteron channels and we can easily sum the spin—isospin kinematical factor of these two deuteron channels. This kinematical factor is given by I, s, j, __________________ 12 ~2 12 V(2s~+ 1)(2t~+ 1)W(sas,s8s2 5c5Y~4’(tat1t8t2 tat) Is] X
\/n~Ksctc{l(sata);s,t,)-\/7c~sbtb{I(sCtC);s2t2)
(2.4.11)
,
where Ii’ 5, and], for i = 1, 2 are the angular momenta transferred in the first (i = 1) and second (i = 2) steps, respectively, and 1, s and j are the transferred angular momenta through the whole sequential process. The spin and angular momentum relations are shown in the fig. 2.10. The values without the 9-] symbols are listed in table 2.6. If we choose only the triplet deuteron ground state channel, both s = 0 and s = 1 transferred spin transition amplitudes are allowed. But if we sum the transition amplitudes from both triplet and singlet intermediate channels, the s = 1 transferred spin transition amplitudes cancel. Therefore the two-step amplitude has no s = 1 transfer amplitude and no distinctive selection tc
~c
S6
intermediate (lsj~-~ ~a
~a
“A
incident
“N~(ls~)2nd lsjt
~
8b
tb
final
Fig. 2.10. The spin and angular momentum scheme of the two-step process.
~B
30
M. Igarashi ci a!.. Two-nucleon transfer reaction mechanisms Table 2.6 spin—isospin kinematical factors Deuteron channel Transferred s, t
(s
(s = 0. (s = 1.
-3v~!4 -V3/4
t
=
t
=
1) 1)
1.
t
0)
(s =0, ç
=
1)
(s
=
1.
t
=0)+ (s =0, ç
=
I)
.\/~i4
\‘~/4
0
rule from the one-step process. This fact is very important for polarization phenomena. In the case of transitions to the unnatural parity states, the cross section of a sequential transfer calculation under these assumptions is strongly reduced and the analyzing power vanishes exactly if no spin—orbit term is included in any of the distorting potentials. So we must check the validity of these three assumptions of the spin—isospin closure approximation. The reality of the closure approximation will be checked by using realistic unbound deuteron wave functions and realistic transfer interactions. In our calculation the three assumptions mentioned above are found to break down. The analyzing power data of the 205Pb(p, t)206Pb 3 transitions are useful for this check, as discussed in subsection 4.2.2.2. 2.4.4. The unbound intermediate deuteron channels The intermediate channels of the unbound deuteron states are treated in the following way. Let an unbound deuteron wave function be $(k, T). It is a positive-energy solution of the Schrodinger equation for the p—n system, [T~
2/2,a~)k2]4.(k, r) = 0 (2.4.12) 2+ V~,,(r) (h where r, T~ 2,V~and k are the relative coordinate vector, the kinetic energy, the interaction potential and the asymptotic wave number of relative motion between p and n, respectively. Here we use a realistic nucleon—nucleon interaction for Vu,,. namely the Reid soft-core potential. We truncate the continuous spectrum of 4(k, r) at a certain value kmax, and divide the range [0, kmax] into a finite number, N, of bins with equal widths ~k = k, k, . We take the average of cb(k, r) in each bin, ,
—
—
=
~
J
~(k,
T)
dk,
.-
(2.4.13)
4
to2P~/2~R, representis equal in the bin,average and assume the c.m. to the energythat in each bin. energy of the p—n pair corresponding to h Thus, 112k2
E
0=
112P2
(2.4.14) 2~R 2 + ~(~k 2. (2.4.15) = ~(k~+ k,1) 1) The “wave packet” ~ 1(r)has a faster decay of amplitude along the relative distance r than the original —i- +
2,a~
—i-
.
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
31
scattering wave ~(k, T). We calculate the projectile form factor of each one-particle transfer step using the wave packet function. Namely (a) (~~IV23l~~(r)), (b) (4~(T)lV,2+ V131I/i1(r, p)),. and (c) (~~(r)l ~f,(T, P))r are used for each discretized unbound intermediate deuteron channel. The value of kmax depends on the spin state of the p—n system and also on whether one calculates the interaction term or the non-orthogonality term of the T-matrix element. We used theinteraction following values: 3D, state for the terms, kmax= 1.Ofm’ for the ‘S0 state and kmax=2.0fm~’ for the 35~ + and kmax = 1.Ofm for the S 0 state and kmax =3.Ofm for the 5, + D, state for the nonorthogonality terms. The criterion for these choices is the convergence of the transition amplitudes. In the present calculation, ~k is chosen to be kmax!8 and the k-space truncations and discretizations are shown in fig. 2.11. Figure 2.12 shows the contributions of individual bins (1—8 corresponding to the range 0.1—1.0fm~) in the singlet unbound deuteron (‘S0) channel to the interaction term. The convergence with respect to a change of kmax is quite rapid in this case, but depends on the spin state of the deuteron channels and on whether the amplitude is the interaction term or the non-orthogonality term. The choice of the distorted potential for the c.m. scattering wave calculation in the discretized unbound channels has not been well established. Since the coupling among the continuum channels is strong, the folding potential of the proton and neutron optical potentials with the “wave packet” functions may not be reliable for the second-order DWBA calculation [51]. We should evaluate the coupling effects between the continuum channels with the distorted potential. For simplicity, however, we use the elastic scattering optical potential at the energy corresponding to the ground state deuteron channel for all unbound state channels. A theoretical approach for the excited intermediate deuteron states has also been proposed by Austern and Kawai [74] based on the closure property (the adiabatic method).
Interaction
Non-orthogoncility
term
term
1 frrc
Li ~ is
0
3D 1 Fig. 2.11. Truncation and discretization of k-space.
32
M. igarashi et a!.. Two-nucleon transfer reaction mechanisms
~
1~
~ -deutert TOtQ~-~....jnterQctbfl
—-.-------
i:
I~:
Si::
iO~
~ax~°fm1
0°
20°
N
40°
60°
80°
~9c.m. 10°Pb(p—d—t)205Pb3’ process at E~= 35MeV. The Fig. 2.12. The contributions to the interaction terms from individual bins for the case of the intermediate deuteron state was spin singlet and k,,.~,= 1.Ofm’. The first neutron is picked up from the 1,,, state and the second one from the p.. state.
2.5.
The spectrscopic amplitude for the particle transfer reactions
2.5.1. The antisymmetrization of the DWBA amplitude The completelyantisymmetrized wave function of the total system I/i~,A can be expanded as a linear combination of products of the separately antisymmetrized projectile I/la and target I/’A wave functions. Let P’A(i) represent the operator that interchanges 2m particles between the two groups a and A, i.e., PaA(l) consists of m transpositions. The number of independent m interchange operators Pa~(1) is N’A = (~)( ~,‘), where i = 1, 2, . . - , ~ Then the total number of possible interchanges is given by (2.5.1) where we assume a ~ A. One obtains for the completely antisymmetric wave function with the proper normalization / I/IA
=
)
+A~~2 1
a
a
m N~’A
m=0
m
-
PaA(1)I/II/1AXaA
()
=1
,
(2.5.2)
where XaA is the wave function of relative motion between a and A. The nuclear matrix element of the particle transfer reaction, which is represented by the prior form, is reduced to
M. Igarashi et al., Two-nuc!eon transfer reaction mechanisms —1/2
(~/ib,BIVaA
(a +A)
UaA~aA~
-
Na”A
~
)
(a a
a
33
~b,BIVaA
~
-
UaAlP~(~)~a~AXaA)
(2.5.3)
K~BUaAXaA).
The last equation may be obtained as follows. The potential VaA — UaA can be replaced by H E as already mentioned in section 2.1. Then the operator ~~A(~) commutes with the full Hamiltonian H, and we also use the fact that PamA(~)t/IbB= ()mIfrb,B. 4’b,B is expanded similarly, —
N~
b
\t/2
/
,, ,.~ U
=
)
(b + B)
5
v
z1 (—) ~ n0 i=t b
—1/2 ~
,, P8~(t)ç&~ç1J8x~fi
b—a ~ (_)(n-P)(b-a) p=O
(_)fl
0
N~PN~~ ~ i=t 1=t
~ P~(i)P~B(])~BxbB,
(2.5.4)
n=
where b > a and P~B(]) stands for p interchanges~ofparticles between the group ä~=(b — a), i.e., the particles belong to b but not to a, and B [74, 75]. The phase factor ()(1~P)(l~~) results from the antisymmetry of the wave function of nucleus b when the (n — p) particles pass through group e~. Inserting eq. (2.5.4) in eq. (2.5.3), 1/2 ~/ib,BIVaA
-
UaAl~aA) =
(a +A)
1/2
(b
+
B)
b
b—a
~
~
~
~(n-P)(b-a+1)
K
x XbBI/1bh/1Bl1~a; (1)(VaA — UaA)IcllaclJAxaA~, where we have used
P~B(j)4’A = (—YI/’A.
K~/ib,BIVaA- UaA~aA)
=
1/2
~
x
~
(2.5.5)
Writing m = n —p, then
(~)(~) 1/2
(b; a)(B)
1/2
a (_)m(b-a+t)(b
(XbBI/lbt/IBIPaB(l)(VaA
—
~
-
a+ m)
.
(2.5.6)
Thus the nuclear matrix element is seen to have (a + 1) terms; the m = 0 term is the direct term; the second one corresponds to one particle in a being exchanged with one in core B; the third corresponds to two particles being exchanged, and so on. The exchange terms are in general expected to be considerably smaller than the direct term, although this fact has not been carefully examined. Recently a theoretical consideration of exchange effects in DWBA was proposed by Kozak and Levin [64]. In this work only the direct term will be considered. For the case of the direct term the spectroscopic amplitude is given by b 1/2 1/2 St12 = ()(b~a)B() (a(sa, ta), b — a(s, t)I}bs 8, tb)(B) KB(JB, TB), A — B(], t)I}AJ~,TA~. (2.5.7)
34
M. Igarashi ci a!., Two-nucleon transfer reaction mechanisms
The phase factor ()(ha)B is required for expressing S 1/2 in terms of the c.f.p., because the particles in A need be reordered from a + 1, ~, b,. . . , b + B to b + 1,. . - b + B, a + 1, ~, b, which requires (b a)B transpositions. The phase factor is positive for the one-step two-nucleon transfer process, but is negative for the sequential two-nucleon transfer process, because the same kind of spectroscopic amplitudes appear twice. The phase factors of one of them is negative and the other is positive, because one of the nuclei is odd and the other is even. This phase difference is important for the coherent sum of the one- and two-step processes. ,
—
2.5.2. The spectroscopic amplitudes of each process
The spectroscopic amplitudes of the projectile system are listed in table 2.7. The c.f.p. for the decomposition of the projectile system is included in the radial part of the form factor in this work. Here only the Clebsch—Gordan coefficient for the isospin transformation and the statistical factor are tabulated. The spectroscopic amplitude of the target system is given by 51/2(] t)= (~)‘2(B(J8, TB), A
—
B(j, t)l}AJA, TA).
(2.5.8)
For the case of (p, t) pick-up reactions from a doubly closed shell nucleus the spectroscopic amplitudes in the J representation (separate treatment of protons and neutrons) are given as follows. (i) A 0~ J transition of the (p, t) process, in which two neutrons are transferred from the same shell 11’ —+
2(j~, ~B)
S~
=
(2]i+
1)12(
]~J1i(JB);
i~(JB)l}J~’~’ (a))
=
1/~J~ +1.
(2.5.9)
This factor is independent of whether 11 is an outermost or inner shell. (ii) A 0~ J transition of the (p, t) process in which one neutron is transferred from an outer shell 12’ and another one from an inner shell —*
],,
S~!~2(jj2;~B)
= (_)212(211+
1 )12(2i 2+
1)1 /2(
.21i(.)
i~ I)
j~”~’(o))
]~]~ 0
=
.212 - -2j,+i (0)) 12‘ X 12 (12); 121)12
12
0
lB
~B
0
—1/2J~+ 1.
(2.5.10)
Table 2.7 The spectroscopic amplitudes of the projectile system Reaction (p.1) (p,d) (sd = 1, (p,d*) (Sd =0. (dl) (sd = 1, (d*.t) (sd =0,
(çn~tn~t~n~) Id
td td td
=0) = 1) =0) = I)
-V~7~ —iiV~
(1 )‘~
—1
1/V’~ 1
2 S~ -v~
1 \/~ —l
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
35
(iii) A one-particle transfer from an outermost shell 12’ 5T1/2 (12)
=
v~Ki-n,—l 2 (u2); 121}122(a2)) U(a, U2JAJ2 fl
(2.5.11)
JBa2),
where a1 is the core spin of target A; ~A= at x ]~~(a2), and the recoupling coefficient of angular momenta U is given by
U(abcd; ef)=~(2e+ 1)(2f+
1)
W(abcd; ef).
(2.5.12)
(iv) A one-particle transfer from an inner shell j,, 2(j~) = ()JAJ
ai
],~}j~(a,))U(a
1(_)n2~Kifi’(u);
S~/
(2.5.13)
2~,J~j,; jBa,),
where n2 is the neutron number of the outermost shell and this shell is not always fully filled and has spin a2. The spectroscopic amplitude of the (p—d—t) process used for the pure configuration [j,, j,]J is given
by 2’1’\/2(2J + 1). S~/~(j~) x 5~2(],) = (—) The spectroscopic amplitude of the (p—d—t) process used for the pure configuration
(2.5.14)
=
by
1!2
=51/2(J)
x
S~/2(j2) = (_)11~2~V2J +
1~
— V2J
+
1
[],,j
2]J is given
ôj.J7~
(2.5.15)
S
where j’ denotes the total angular momentum of the intermediate channel. In actual calculations we construct the transition amplitudes using these pure configuration spectroscopic amplitudes and the configuration mixing coefficients given in table 4.2. 2.6. The differential cross section and analyzing power The transition amplitudes of the one-step process (eq. 2.3.1) can be written in the form ~
=
~ (2] + 1)”2(JAMA1 MB — MAlJBMB)B’~’°(0),
(2.6.1)
tsj
B°~”°(o)= ~ S2f3;rlia(kb,k),
(2.6.2)
shell
where m = MB — MA + mb — ma is the possible magnetic substate quantum number of the transferred orbital angular momentum 1. The spectroscopic amplitude S1t2 and the reduced amplitude f.3 depend on the nuclear shell configurations. The sequential transition amplitude TIdP (eq. 2.4.1) is written in the same form but the function B(0) is now B~a(0)=
5~2X$~m~(k 8,k). shett,(tsj) is,.(tsi)2fld
(2.6.3)
M. Igarashi et a!., Two-nucleon transfer reaction mechanisms
36
The angular differential cross section is defined by du.
=
U~hf
~
k Tr(rtTp’T~), ~a
j~ 2 2
(21Tt1
)
(2.6.4)
where T is the sum of ~ and ~ and the is’s are the reduced masses. In the case of equal probability of observation of all final spin substates for equally prepared initial spin substates, the efficiency and density matrix elements take the following forms: rnbMB,mhMB =
(2.6.5)
mhmbMBMB~
P~nMA.m~M~ = 25a + 1
2~A
+
1
(2.6.6)
3mam~6MAM~’
Hence the cross section is given by do
V
2jB+l
11
kb
P~aPb
=
(2~h2)2 ka (25a + 1)(2JA + 1)
jmmm~
lB 1
where B. is the sum of ~
2
mmm
b
a(O)l
and BtdP for all possible l and
(2.6.7a)
,
s,
m”m”(o)= ~[B B7
1~°(0)+B~1(0)].
(2.6.7b)
tst s
In the case of vector polarized projectiles, the density matrix is given by P
2Sa+1 2JA+1 (i+
+1
(2.6.8)
~~MAM~’
and hence the differential cross section is da~— dUf1 dfl — ~Ifl
P~aPb
+ (21Th2)2
_____
1T’P
k,, 1 T ka (25a+1)(2JA +1) 5a+1 r~ k
a
.
~ T ~1 Sal
2 69 .
~
where ~a is the vector polarization of the projectiles. The analyzing power Ab is defined by P •A
— do~Id~1 —1do~1/dQ1 d~1/dQ1 —
3
—
+
If
1
a
the polarization
b)
~a
—
is prepared
~a
Tr[T(Pa Sa)T~1 +1 Tr(TT~) -
2 6 10 ( . . )
along the y-axis, which is perpendicular to the scattering plane zx,
A~(0)becomes [114] ~1mmbma
A~(0)°°
V(~a
—ma)(sa
+
ma + 1) ~
Bmmbma(OY
5 a
Jmm5m,,
j
\ I
.
(2.6.11)
M. Igarashi et a!., Two-nuc!eon transfer reaction mechanisms
37
2.7. Why analyzing power? For the reaction A(a, b)B, let us denote the transition amplitudes corresponding to each different path of the reaction process by ~ ~ Let m = MB MA + mb — ma, as in eqs. (2.6.2) and (2.6.3), and let the superscripts indicate the paths of the different processes. Figure 2.13 shows the four different paths as an example of two-nucleon pick-up reaction mechanisms. We express the differential cross section (2.6.7) and the analyzing power (2.6.11) symbolically by the forms —
r(0)
=
~
~ T~
A(0)=Im ~ m,~m’
=
~
(~
T~2 + ~ T~T~’)*),
(2.7.1)
(~
(2.7.2)
T~T~*)/cr(0).
pp
As the m = 0 component is usually predominant, we explicitly write o~and A in the forms ff(0) = I T~I2+ I T~2~2 + 2 Re[T~ T~2)*]+ . A(0)
=
. .
(2.7.3)
,
Im(T~’)T~’)* + T~T~*+ T~,l)T~2)* + . . .)/~r(O)
(2.7.4)
Suppose that process (1) is the main contribution and process (2) is one of the correction terms such as higher-order contributions. Then we see that the correction (2) is more sensitively involved in A(0). The main term is the cross term of the m = 0 (spin-nonflip) and m = 1 (spin-flip) components in process (1); the interference terms between processes (1) and (2) therefore give rise to more sensitive contributions to A(0). This is particularly so for unnatural parity state transitions in (p, t) reactions, where the one-step m = 0 transition is almost forbidden, and the sequential transfer m = 0 transition is allowed. Hence the interference term T~!)T~* makes the main contribution to A(0). By utilizing this characteristic of analyzing powers, we have extensively studied nuclear reaction mechanisms as well as microscopic structures of nuclei and effective nuclear forces using the 22 MeV polarized proton beams accelerated by the University of Tsukuba 12 MV Tandem Accelerator.
X Fig. 2.13. Various competing processes in the reaction XN+ 2(p, t)YN. The direct process (1), the sequential transfer (p—d—t) two-step processes (3), (4), and the inelastic two-step process (2) are indicated.
38
M. Igarashi et al.. Two-nucleon
3. Experimental
transfer reaction mechamms
study of (p, t) cross sections and analyzing powers
3.1. The measurement of (p, t) analyzing powers and cross sections We explain the measurement of the analyzing power for the reaction 2”8Pb(p, t)‘OhPb as a typical example of a (p, t) analyzing power experiment. Polarized proton beams were produced by a Lamb-shift-type polarized ion source with nuclear-spin filter [77]. The 22.0 * 0.05 MeV polarized proton beams were obtained using the University of Tsukuba Tandem Accelerator. The beams were analyzed by the 90” analyzer magnet and focused on a *“‘Pb target by a switching magnet and a pair of quadrupole magnets. The beam intensity on the target was 50-100 nA with a spot size diameter of less than 1 mm. The energy spread of the beam was 0.5 keV. We used a 208Pb target, 0.48 -t 0.02 mgicm’ thick, backed by a 3 pm aluminum foil. The enrichment of the ‘08Pb target was 98.69% ‘08Pb (0.26% for *06Pb and 1.03% for *“‘Pb). The thickness of the ““Pb target was measured by weighing and checked by comparing the proton elastic scattering data with the optical model prediction. The experimental setup of the detection systems is shown in fig. 3.1. We measured and monitored the beam polarization P by using the reaction ‘Li(p, a)4He at 0,,,, = 130”, where its analyzing power A(0) is known to be 0.80 t 0.01 [78]. The polarization monitor target LiF was placed upstream of the 20sPb target. The monitor target was 60 kg/cm2 thick LiF evaporated onto a carbon foil. We measured the left-right asymmetry of the alpha particle yield of the reaction ‘Li(p, cy)4He using two solid-state detectors at 0,,,, = 130” (see fig. 3.1). The up and down spin direction of the incident protons was flipped every 100 nC of beam current integration by the fast spin state interchange system (FASSICS) [79]. This system controls the spin-flipping pulses of the polarized ion source and supplies gate signals to the particle counting system for each spin state. The typical beam polarization was P = 0.80, and the error in the beam polarization was kO.02. The degree of beam polarization thus obtained was consistent with that obtained by the quench-ratio method [77]. Tritons from the (p, t) were momentum analyzed by a magnetic spectrograph [80] and detected by a single-wire position-sensitive proportional counter (SWPC) system [81]. The horizontal acceptance of Magnetic
spectrograph
\
Bcm Fig. 3.1. Experimental
setup for the measurement
=130°
of analyzing powers and cross sections for the reaction “‘Pb(p, t)2”hPb at E, = 22 MeV,
M. Igarashi et al., Two-nucleon transfer reaction mechanisms
39
the magnetic spectrograph was A8 = 2.0”, which corresponds to a solid angle of 2.0 msr. The SWPC system has a sensitive length of 30 cm and is composed of a position counter and a AE counter. This system covers an excitation energy range up to about 4 MeV in “‘Pb for a given magnetic field. The coincidence requirement between the position and AE signals lowered the background level in the triton spectrum. The vector analyzing power A(8) for the (p, t) reaction is obtained from the relation
N+(8) - N_(8) F N+(8) + N_(8) ’ 1
A(@) =
(3.1.1)
where N+(8) [N_(8)] is the number of tritons at the reaction angle 8 produced by the spin-up [spin-down] proton beams with degree of polarization P. The differential cross section ~(0) with the unpolarized proton beam is given by g(0) =d@)ldfl=
&[N+(e) + N_(0)],
(3.1.2)
where the factor (Yis obtained from the target thickness, the number of incident protons and the solid angles of the spectrometer. We measured the angular distributions of the analyzing power A(B) and the cross section g(0) from 19~~~ = 5” to 90” in 5” steps. 3.2. Experimental results for (p, t) analyzing powers and cross sections We show in fig. 3.2 a typical momentum spectrum of tritons from the reaction 208Pb(p,t)206Pbwith a = 25”. The full width at half maximum (FWHM) of the triton 22 MeV spin-up proton beam at 19,~~ spectrum is 30 keV, which is mostly determined by the energy losses in the 208Pb target. All the observed triton peaks were identified: the excitation energies of the levels of 206Pbobtained from the peak positions in the triton spectrum coincide with those obtained by previous experimental work [82] with the experimental error of t10 keV In the following sections, the statistical errors in the observed analyzing powers and cross sections are always indicated by error bars in the figures of their angular distributions. Usually the error bars for the cross section data do not exceed the size of the data points (circles etc.). The absolute errors of the analyzing powers (cross sections) are estimated to be about 4% (15%). We list our (p, t) experiments in table 3.1. Most experiments have been done at an incident proton energy E, = 22 MeV, but some at energies of 17.0, 18.5, 20.0, 21.0, 21.5, 22.5,28.0,35.0, and 50.0 MeV in order to investigate the effect of the energy dependence of the (p, t) transitions. 3.3. Experimental studies of the (p, d) and (d, t) reactions In order to evaluate the contribution of sequential transfer (p-d-t) two-step processes to the (p, t) reaction, we made (p, d) and (d, t) experiments. In the study of 208Pb(p,t)206Pb(3:) at E, = 22 MeV, for example, we measured the cross sections ~(0) and the vector analyzing powers A(B) for the transitions 208Pb(p,d)207Pb(p,,,, p 312, f 512, f 712, h 9/2, i 1312 ) at E, = 22 MeV, and for the transition 207Pb(d,t)206Pb(3:) at E, = 17 MeV. In addition, we measured (~(0) and the vector and tensor analyzing powers iT,,(8), T20(0), T2,(0) and T22(0) for the transitions 208Pb(d,t)207Pb(pl/2, p 312, f S/2, f712, i 1312 ) at E, = 17 MeV so as to study single-particle transitions. We measured the tensor analyzing powers
M. Igarashi et al., Two-nucleon transfer reactlon mechanisms
40
208Pb(p,t)206Pb 8 l&,=25’
Ep=22.0
MeV
SPIN UP
800 CHANNEL
1200
1600
2000
NUMBER
Fig. 3.2. The triton momentum spectrum of the 2”8Pb(fi, t)*O’Pb reaction at E, = 22 MeV, O,,, = 25”, with spin-up protons.
T2,(0) to obtain precise nuclear structure information concerning the deuterons and tritons involved in the reactions; the tensor analyzing powers TZq(0), especially T2r(0), are sensitive to the non-S components such as the D component of the deuteron and/or triton. Our (p, d) and (d, t) experiments are listed in table 3.2. The experimental cross section and analyzing-power data have all been analyzed in terms of the first-order DWBA method in the references cited and they provide fundamental information for the analysis of the (p-d-t) two-step processes. In addition to the (p, d) and the (d, t) experiments, we carried out systematic measurements of the elastic and inelastic scattering of polarized protons from various nuclei [83] and also of the elastic scattering of vector- and tensor-polarized deuterons [84]. The aim of the elastic scattering experiments is to obtain the optical potential parameters for protons and deuterons while that of the proton inelastic scattering experiments is to fix the strength of the inelastic scattering part of the (p-p’-t) two-step process. For the triton channels, we use the Los Alamos triton elastic scattering data [117].
4. Results and discussions 4.1. The one-step (p-d)
process
The absolute values of the calculated cross section of one-step (p, t) processes have been investigated by many authors. The results, however, scatter over a wide range, which obscures our understanding of
M. Igarashi et al., Two-nucleon transfer reaction mechanisms
41
Table 3.1 The (p, t) reactions for which analyzing powers and cross sections have been measured. Analyses in terms of one- and two-step processes are given in the cited references, E, is in the lab. system, Qp is the Q-value for the ground state transition and .I: the final state spin and parity of the nth excited state of excitation energy E, Target nucleus
Final nucleus
EP (MeV)
*“Pb
‘“‘Pb
22.0
J:(K) -
$eV) -5.63
Ref.
(MW
45, 47
O;, O’(1.167) 3;(1.341),
3;(3.12)
2;(0.803),
2;(1.467),
4;(1.684),
4;(1.998),
7,(2.200),
7;(2.865)
48, 85-89 2;(1.784), 4;(2.928),
2:(2.149), 5;(2.783),
2;(2.424),
87
5,(3.014),
35.0
3;(1.341)
48
50.0
3l(1.341)
48 47
“‘Pb
‘04Pb
22.0
-6.34
0;
“‘Pb
2”‘Pb
22.0
-6.70
06
47
14’Nd
‘42Nd
22.0
-5.47
0:
45, 90
“‘Ba
““Ba
18.5
-7.035
0;
20.0
0;
21.0
0: 0; , 2; (0.816)
22.0
91 91 91 91, 92 91
28.5 ‘Ye
‘Ye
22.0
-6.02
0; , 2; (0.743)
“‘Te
“‘Te
22.0
-6.59
O;, 2;(0.666)
28, 45, 92-94
rZ2Te
12”Te
22.0
-8.55
0;) 2;(0.560)
92, 93
““Sn
““Sn
22.0
-7.19
O;, 2;(1.294)
92, 93
l16Cd
““Cd “‘Cd
22.0
-6.36
0;) 2; (0.558)
92, 93
22.0
-7.10
O;, 2;(0.617)
92
“‘Cd
““Cd
22.0
-7.89
06, 2; (0.374)
92
““Pd
‘“‘Pd
22.0
-6.48
0;) 2; (0.434)
28, 45, 92-94
0;) 2;(0.434)
92
“kd
55.2
92
‘““Pd
22.0
-7.28
O;, 2;(0.512)
92
‘04Pd
22.0
-8.16
0;) 2;(0.556)
92
rU2Pd
22.0
-9.15
0;) 2;(0.557)
45, 92, 93
O;, 2;(0.557)
92
55.2 lU2Ru
22.0
-6.65
O;, 2;(0.475)
92
“mRu
22.0
-7.54
0;) 2;(0.540)
92
“Ru
22.0
-8.70
O;, 2;(0.653)
45, 92, 93
9hM~
22.0
-6.98
0;) 2; (0.778)
92
y4Mo
22.0
-8.05
O;, 2;(0.871)
92
=Mo
22.0
-9.26
0;. 2;(1.509)
46, 92, 95, 96
“Zr
17.0
-7.35
0;
45, 91, 95, 96
18.5
91
20.1
91
21.5
91
22.0
91
22.5
91 91
28.5 @Ni
“*Ni
“Ni
‘“Ni
22.0
-8.02
18.0
-9.94
97 97 97
20.0
97
22.0 ‘“Ni
‘“Ni
22.0
-11.90
97
58Ni
‘“Ni
22.0
- 13.97
97
42
M. igurashi
et (II.,
Two-nucleon
transfer
reuctum
mechanisms
Table 3.2 (p. d) and (d, I) reactions for which analyzing powers and cross sections have been measured. The definitions of E, or E,, Q,. Jz(E,) are the same as in table 3.1. Analyses in terms of DWBA are given in the cited references E, or E,
Q,
J:(k)
Reaction
(Me”)
(MeV)
(MeV)
“‘HPb(p, d)“,~Pb
22.0
-5.14
-‘“XPb(d,t)““Pb
17.0
-1.11
li2-(0), 3i2 j/2 (0.570), 13/2’(1.633). I K(U), 3:2 512 (0.570). o;, 0;(1.167)
X?Pb(d, t)?“hPh lJ’Nd(p. d)‘“Nd lJ’Nd(p. d)li’Nd “hSn(p. d)“‘Sn “‘Cd@. d)“‘Cd “‘Cd(p, d)“‘Cd “lXPd(p, d)““Pd
17.0 22.0 22.0 22.0 22.0 22.1) 22.0
-0.48 -5.59 -7.59 -7.11 ~6.82 -7.18 m-7.00
““Pd(p, d)‘“‘Pd “‘2R~(p, d)“” Ru “Zr(p, d)“Zr
22.0 22.0 22.0
-7.32 Ph.99 -6.42
Ref. (0.898) 7’2 (2.340) 9K (3.409) ~(0.898) 712 (2.340), 13/2’(1.1633)
2;(0.803), 21’(1.467). 7i2 (0). 3/2 ~(0.742) 3i2’(0), 112’(0.194). Ii?‘(O), 3/2 (0.16), l/?‘(O) Ii2 (0) 5/2’(O). 1/2’(0.12) lli2~(0.21) Si?‘(O) 5/2’(O) 5/2’(O), 5!2’(1.47),
3,‘(1.304) 1112 (0.754) 1112 (0.32)
l/2-(1.21).
7/2’(1.88)
98 98 98 99 99 100 87 98 101 101 IO1 101 101 101 101 101 101
the reaction mechanism of (p, t) reactions. In this section we carefully study the absolute values of calculated cross sections, in particular their dependence on different triton wave functions. Test of the computer code and comparison with the results obtained with the Tang-Herndon triton wave function As a test of our computer code PTFFITWOFNR [102, 711, we repeated the same calculations as the ones undertaken by Bayman and Feng [14] employing direct integration by the Monte Carlo method. We obtained perfect agreement with their results. We compared the results with the code by Takemasa [68]*’ in detail employing an expansion in harmonic oscillator wave functions. After correction of an error in his code, we got good agreement. Previous calculations [13, 14, 67, 681 do not include the hard-core correction terms mentioned in section 2.3.3. For a detailed study of the hard-core correction, we investigate the ground state transition of the “)Ca(t, p)42Ca reaction at E, = 10.1 MeV as an example. In table 4.1 cross sections of simultaneous transfer calculated under different assumptions are compared with the measured one. The optical potential parameter set II of ref. [14] is used. The radius, diffuseness parameter and the spin-orbit strength of the single-particle binding potential for the ““Ca nucleus are chosen to be rC1= 1.24 fm, a = 0.65 fm and V,, = 6.01 MeV, respectively. The angular distribution of the cross sections is almost the same for all cases in table 4.1, so we compare the absolute values at 0”. For the simple [(f7,2)2] 0’ wave function, the cross section calculated with the Tang-Herndon (TH) triton wave function with the hard-core correction is close to that of the RSC.5 triton wave function, but the TH result without the correction overshoots it by about a factor 3. If we include the correction terms the cross sections are reduced by a factor of 112.4-112.9 (the small 4.1.1.
*’We
arc indebted to T. Tdkemasa for kindly making his program available to resolve the disagreement
between our results and theirs.
M. lgarashi et al., Two-nucleon transfer reaction mechanisms
Table 4.1 Comparison of cross sections of the various calculations for the ?a(t, 10.1 MeV Projectile
Nucleus
duldR(0”)
TH without corr. TH with corr. RSCS RSCS RSC5 Exp.
(f,‘$ (f, J2 (LS 4 config. 22 config.
0.2383 0.0981 0.0814 0.1608 0.2321 3.8”’
d’ Experimental
value is taken from ref.
(mb/sr)
43
p)“Ca g.s. reaction at
Relative strength 2.93 1.21 1.00 1.98 2.85 41
[103).
difference comes from a small difference in the angular distribution). Kunz [17] obtained simultaneously almost the same results as in this work. The natural parity strong transitions are particularly sensitive to the pairing correlations in the target nucleus. To take account of this effect, we compare the cross sections obtained with different shell model wave functions of the target nucleus with different truncation spaces. The 4- and 22-configuration wave functions were calculated with the shell model of Sturmian basis [20]. The shell model amplitudes of the four configurations (Of,,,)‘, (l~,,*)~, (Of,,,)2 and (lp,,,)’ are 0.9909, 0.0929, 0.0891 and 0.0386, respectively. Case II in table 3 of ref. [20] was used for the 22-configuration wave function. The pairing correlation enhances the cross section. The enhancement factor corresponding to the extended shell model is about 1.44 compared to the usual shell model with one major shell model space. This factor is important but not enough to explain the difference between the experimental data and the theory as we can see in the last row in table 4.1. The calculated cross sections of the simultaneous transfer processes are still too small by a factor of more than 10. From these results we conclude that this reaction cannot be reproduced by the simultaneous transfer mechanism alone. 4.1.2. Cross section of the one-step (p-t) process in z08Pb(p, tj206Pb reactions The purpose of this work is to investigate the mechanism of (p, t) reactions rather than details of the nuclear structure. Hereafter we concentrate on 208Pb(p,t)206Pb reactions because of the simplicity of the nuclear structure. 208Pbis a doubly closed shell nucleus and the collectivity of the 3- (2.61 MeV) excited state of 208Pbis not so strong. The simple shell model with a core + two-hole configurations gives good predictions on the structure of the 206Pbnucleus [104]. For the unnatural parity (UNPS) 206Pb(3:) transition, the two-step contribution (p-p’-t) via inelastic excitation of the ‘ORPb(3-,2.61 MeV) state is estimated with the Y,-type collective vibration model for the excitation and the microscopic wave function [105] for the transfer. To avoid complications, only spin transfer s = 0 transitions for the (p’-t) step are considered with a finite-range transfer form factor, and the 3- excitation contribution is found to be so small that the final result is not appreciably affected by it. The selection rules are greatly different between natural parity and unnatural parity transitions as mentioned in section 2.3.2. As examples of the two types of transitions, we investigate “‘Pb(p, t)206Pb(01, g.s.) and (3:) E, = 1.34 MeV). The shell model wave functions of the states of the ‘06Pb nucleus used in this work are listed in table 4.2. The two 3’ excited states are assumed to be pure (p,\f,i) and (pli:fy,i) states for simplicity. The radial wave function of the transferred neutron of each transition is calculated by the half
M. lgarashi
44
et al..
Two-nucleon
transfer
reaction
mechanisms
Table 4.2 Shell model wave functions of the states of “‘hPb. Amplitudes of the various neutron hole state configurations State Amplitudes
(MeV)
Ref.
p;:
(i,,.,)’
(fd
0.090 0.122 0.159 0.1794
0.109 0.152 0.204 0.210 0.1736
0.059 0.083 0.082 0.074
I=O’ 0.0 1.167 1.167 1.167 1.167
(PJ 0.822 -0.495 -0.534 -0.396 PO.4138
0.834 0.786 0.851 0.8755
1=3’ 1.34 3.122
(PWf, 2) 1.000 0.000
(P,.zPv) 0.000 I.000
1=2’ 0.80
(PI Zf( 1) 0.724
(P,‘?PI>?) -0.523
(f%P,‘,) PO.176
(il,.J -0.055
(fr.>hq.z) 0.092
(f,,zhv:) 0.012
0.029
(fJ 0.567
(f,,,P, 2) -0.727
(P,.,f,.,) -0.293
(fv2f7.& PO.135
(P,.,f,,,) 0.148
(LA!,)
~O$rv~z)
to;r
(f,.,h,z) 0.019
(i,,.$
1=4’ 1.68
0.077
0.145 0.183 0.210
(fSJ7.2)
(PWf7.Z) 0.185
-0.058
[IO41
h)’ LJ
A2
1~6’ 3.25
(f,,,f,.z) 0.971
(f,‘lhq,?) -0.090
(Pi,z’bz)
(4
0.110
-0.153
-0.102
1=7-
(hzill12) 0.952
(LM 0.250
hhzi,,.,) -0.169
(f7.2i1112) -0.055
0, zLL) 0.023
2.20
[1041 11041 11061 I1071 I1041
hzb,)
-0.050
0.097
(fwh,J
(hw)’
-0.025
PO.051
[IO41
11041 11041
separation energy method. The radius, diffuseness and spin-orbit parameters of the single-particle Woods-Saxon potential for the 20sPb nucleus are Y”= 1.25 fm, a = 0.65 fm and V,, = 6.0 MeV, respectively. The absolute value of the calculated cross section depends on the triton wave function used. The integrated cross sections (00-90”) for the 0: and 3: transitions at E, = 35 MeV are calculated with various triton wave functions and are tabulated in table 4.3. Set A in table 4.4 is used for the optical potential parameters. The calculated cross sections of the 0: ground state transition are about 1/4-l /6 of the experimental data. The TH triton wave function with the hard-core correction gives almost the same result as the other triton wave functions obtained by the Faddeev equations (RSC and URG in table 4.3). The s = 1 transition amplitude increases when the hard-core correction term is taken into account. The reason is that the s = 1 transition amplitude starts from the relative angular momentum of the two transferred
Integrated
Table 4.3 partial cross sections (WYO”) (’m mb) of the “‘XPb(p, t)“‘hPb reactions at 35 MeV
0: (n.s.)
(W = (OW
(110)
TH without corr. TH with corr. RSC3 RSCS URG3 URGS
8.844 x 3.243 x 3.517 x 3.564x 3.912 x 3.954x
2.436 1.461 4.182 4.874 3.988 4.821
10 z lo-’ 10~ 2 IO L lo-’ IO ?
x x x x x x
lo-’ 10 ’ 10 -’ IO-’ lo-‘ lo-‘
3: (303)
Sum of s = 1
Total
3.932 1.563 1.361 9.981 8.681 6.920
3.278 1.374 5.798 4.906 4.647 4.164
3.671 x 10 -’ 1.390x10 ( 7.159 x 10 j 5.904 x 10 1 5.515 x loml 4.856x IO ’
x lo-’ x lo-’ x 10mJ x 10-j x 10 ( x 10 ’
x x x x x x
lO_’ lo-’ lo-” IO-’ lo-’ lo-”
M. Igarashi et al., Two-nucleon transfer reaction mechanism
45
Table 4.4 Distorting potential parameter sets. A: set 1 used in ref. [43] for E, = 35 MeV; B, C: sets 1 and 2 used in ref. [85] for E, = 22 MeV; D: set 2 used in ref. 1681 for E, = 40 MeV; E: used in ref. 11081 for E, = 50.5 MeV Set
Channel
V
w
Wd
r0
a,
rI
a,
Vso
:
47.9 97.8 149.8
0 0 12.0
10.0 14.0 0
1.25 1.25 1.24
0.65 0.65 0.68
1.25 1.25 1.43
0.76 0.78 0.87
0 0 0
51.8 112.0 168.9
0 0 9.9
10.0 19.4 0
1.25 1.25 1.20
0.65 0.682 0.65
1.25 1.25 1.60
0.76 0.783 0.97
6.0 6.0 6.0
1.12 1.12 1.15
0.47 0.47 0.92
1.25 1.25 1.30
57.6 105.8 160.9
2.14 19.68 17.3
1.17 1.15 1.20
0.75 0.81 0.72
1.32 1.34 1.40
0.66 0.68 0.84
6.2 0 2.5
1.01
0.75
1.20
0.72
1.25 1.15 1.30
1.125
0.873
1.026
0.794
1.25
0.75
0.873 0.624 0.75
5.84
1.10
1.125 1.386 1.60
1.168 1.25
0.81 0.72
1.233 1.45
0.78 0.72
5.93
A
t B
P d
t C :
t D
E
P
54.62
8.84 0 0
5.31
t
160.0
20.0
5.60 0
P
47.52 160.0
4.09 20.0
5.51 0
t
rso
as0
rL 1.25 1.25 1.30
1.40 1.13
0.79
1.18 1.40
neutrons A= 1 as mentioned in section 2.3.2 and the hard-core correction term has larger components of higher multipole than the interaction term. For this transition Takemasa et al. [68] reported that the simultaneous transfer mechanism reproduces the observed absolute cross section [109]. But their conclusion must be changed, since a programming error was found in their computer code and they also did not include the hard-core correction terms. For the unnatural parity 3’ transition the TH trition wave function, which is a purely symmetric S-state, gives a very small cross section. Even if we include the hard-core correction terms, the cross section is smaller by more than a factor 10 than the results with other realistic triton wave functions. The main s = 1 transition amplitudes are caused by the hard-core correction terms. As an example of the nuclear interaction with a weak tensor force, we take the URG interaction [60] and the corresponding triton wave function. This force gives a slightly larger binding energy of the triton as shown in table 2.2. From these results we can say that the absolute cross section does not depend so strongly on the triton wave function. It is, however, worth noting that the cross section of the natural parity transition becomes larger with a triton wave function with a larger triton binding energy. In contrast, the cross section of the unnatural parity transition depends more strongly on the wave function and decreases when the binding energy increases. These facts are easily understood as follows. A triton wave function with a larger binding energy has a small volume and therefore, a larger S-state component and better overlap with the nuclear interaction. The higher angular momentum components become small. As shown in table 2.5, the allowed relative angular momenta A of the two transferred neutrons start from h = 0 for the natural parity transition and from A = 1 for the unnatural parity transition. The lowest A-component determines mainly the magnitude of the cross section. Figure 4.1 shows the contributions of individual A components to the cross section of 208Pb(p,t)2”6Pb 0:. The contribution decreases rapidly as h increases, which justifies a truncation in h. In this work we truncate at A= 5. The RSC wave function is used for triton. This wave function has a very small P-state component. Therefore the odd-h components are allowed in s = 0 transitions and even A’s (but h f 0) are allowed in s = 1 transitions as mentioned in section 2.3.2. These components are also taken into account up to A = 5 in this work. But their contributions are negligibly small.
46
M. lgarashi
et al.,
Two-nuclron
transfer
reuction
mechanisms
1 208P~(~,~) *‘+B
0+
EP=~~MEV
10-l
10-Z K 2 10-3 ,E Z 0 10-4 I= G * 1o-5 2 g 10-6
10-7
10-a
10-g 0
10
20
30
0
40
I-+C.M.
50
60
70
80
90
(degree)
Fig. 4.1. One-step DWBA cross section calculations for the 0. g.s.transition. Individual contributions from each relatrve angular momentum A of the transferred two nucleons are shown. The even h’s for (kj) = (Ooo) transfer and the odd h’s for (Isj) = (110) transfer are shown. but contributions from the P-state components are not shown in this figure. The RSCS wave function is used for triton.
For unnatural parity transitions the largest A = 0 component is not allowed by the selection rules. So, the cross section becomes very small compared with natural parity transitions. Figure 4.2 shows the decomposition of the cross section for the 3: and 3; transition into contributions of individual transferred anguiar momentum (Isj) components. We can see that the s = 1 transfers dominate over the s = 0 transfer, and the difference of the combination of single-particle orbits changes the relative strength of the (Isj) components as mentioned in a later section. 4.1.3. Contributions from minor components in the triton wave function In the present calculation the mixed symmetric S’-state and the D-state of the triton wave function are included. In figs. 4.3 and 4.4, we show the contributions of those components to the calculated cross section. For the 0’ ground state transition the S’- and D-states give a small contribution compared to the symmetric S-state, but the S’-state is not negligibly small in the coherent sum. The decomposition into the Derrick-Blatt classification [see eq. (2.2.9)] used here is applied to the product of the interaction potential and the triton wave function instead of the triton wave function alone, because we avoid complexity and this is the only one easily calculated by our method [see eq. (2.3.3)]. On the other hand, in the UNPS transition the contributions from the minor D- and S’-state components of the triton wave function are dominant, because the contribution from the main symmetric S-state component is strongly hindered by the selection rules. Therefore the s = 1 transfer
M. lgarashi et al., Two-nucleon transfer reaction mechanisms
I
20ePb(p- t)206Pb
0
10
20
30
40
50
0HC.M.
Fig. 4.2. Decomposition
60
70
8
1
3*
60
1
1
’
1
70
60
-
Ep = 35 MeV
10
20
(degree)
into transferred
0
47
30
40
0HC.M.
angular momentum (kj) components
50
60
90
(degree)
for the two 3’ transitions.
10-l
lo-"
lod
to-7
b
0
10
20
30
40
0HC.M.
50
60
70
80
90
(degree)
Fig. 4.3. Calculated one-step cross sections for the zOsPb(p, t)Z”bPb g.s. 0’ reaction at E, = 35 MeV, employing the RSC5 triton wave function. ‘The partial cross sections which come from the S-, S’- and D-state components and the total cross section are shown. The same optical potential parameters are used as in ref. [67].
M. lgurashi
er al.,
Two-nucleon
transfer
1
lo-:
208Pb(p,t)206Pb
reactlon
c
mechanisms
/
’
’
3+ Ep=35MeV
loD-STATE
n
s
a
0.
-0.
-1,
Fig. 4.4. Calculated one-step sections and A(0) for the two different 3 ’ excitations. The RSCS triton wave function was used. The partial and total cross sections and A(0) are shown as in fig. 4.3. Set A in table 4.4 was used for the optical potential parameters. A(0) from the S-state component is zero. because the optical potential used here has no spin-orbit distortion term.
amplitudes, which provide a non-zero analyzing power A(8) for the reaction even without spin-orbit distortion, dominate. In addition to this, the A(B)‘s of the two transitions (3;, 1.34 MeV) and (3;, 3.122 MeV) are completely different from each other at forward angles. One-step results are shown in fig. 4.4. These depend strongly on the combination of the single-particle j-shell orbits from which the two neutrons are picked up. We will discuss this point in detail in section 4.3.3. 4.1.4. The -‘“‘Pb(p, t)““Pb reaction at 35, 40 and 50.5 MeV In order to investigate the dependence of the absolute cross section on the transferred angular momentum, we compare the magnitude of the theoretical one-step cross sections with the experimental ones leading to the lowest lying higher-spin states of the residual nucleus in the ““Pb(p, t)‘06Pb reactions at E, = 3.5MeV [l lo], 40 MeV [ill] and 50.5 MeV [ 1081. The normalization constants N(Y) = (T(Y; f&&Q”; e),, are given in table 4.5. We use the shell model amplitudes in table 4.2
M. Igarashi
et al.,
Normalization
Two-nucleon
transfer
reaction
mechanism
49
Table 4.5 constants for the ‘“8Pb(p, t)*06Pb reaction
J”
0; (0.00 MeV)
2,‘(0.80MeV)
4;(1.68MeV)
6;(3.25 MeV)
7;(2.20 MeV)
N(35 MeV) N( 40 MeV) N(50.5 MeV)
6.0 17.3 8.8
3.0 9.0 5.4
2.0 8.0 3.5
2.7 14.7 3.4
3.3 10.5 2.1
and optical potential parameter set A in table 4.4 for 35 MeV incident energy, set D for 40 MeV and set E for 50.5 MeV The normalization factors are smaller for the higher-spin states. Probably this reflects the reliability of the shell model wave functions for those states. For the low-spin states we need a more extended shell model configuration space to take into account the lower-pole correlation [20, 211. The normalization factors for 40 MeV incident energy are larger than those for other energies. The reason for this is not clear. 4.2. The sequential transfer (p-d-t) process First we test the reliability of our projectile form factors for each of the (p-d) and (d-t) steps. 4.2,1. Cross section and polarization observables for one-nucleon transfer reactions 4.2.1.1. (p, d) reactions. The projectile form factors of the (p, d) processes are calculated using the
RSC interaction and the deuteron wave function is obtained with the same RSC interaction. The results are shown in fig. 2.8a. We use these form factors for the exact finite-range analysis of the ‘“*Pb(p, d)*07Pb reactions at 22 MeV [98]. Set B in table 4.4 is used for the distorting potential parameters. The wave functions of the transferred neutron in the target are generated by the conventional separation energy method with parameters rg = 1.25 fm, a = 0.65 fm and V,, = 6.0 MeV. The cross sections and analyzing powers are shown in fig. 4.5. The experimental cross sections and analyzing powers are both reproduced well by the calculation. The 7/2- state is not a pure f,,, single-hole state, so the calculated cross section must be normalized to the experimental data. The L = 2 (D-state) component of the projectile form factor is important to improve the fits to the experimental analyzing power data. 4.2.1.2. (d, t) reactions. The projectile form factors of the (d,t) processes were calculated using the RSC interaction and RSCS triton wave function and the deuteron wave function obtained by the same RSC interaction. The results are shown in fig. 2.8b. We used these form factors for the analysis of the 208Pb(d,t)*07Pb reactions at 17 MeV [99]. The optical model and neutron potential parameters are given in ref. [112], and the binding potential parameters for the transferred neutron are r. = 1.25 fm, a = 0.65 fm and V,, = 6.0 MeV We get good predictions of the angular distribution of cross sections and also vector and tensor analyzing powers. These results are shown in figs. 4.6 and 4.7. The spectroscopic factors for the single-hole states are 2.8 (P~,~), 8.2 (f& 5.3 (P~,~)’ 16.0 (i,,,,) and 8.0 (f,,,), respectively. These were obtained by normalizing the DWBA cross sections to the measured ones, and are expected to be S = 2j + 1 for completely filled neutron shells. The DWBA predictions are smaller than the measured cross sections by about 111.25 on average. The large diffuseness of the real part of the deuteron potential is important
IM. Igarashi
et al..
Two-nucleon
transfer
reucrion
mect~anisrw
(b)
0.898 MeV 3127
:
20epb(p,d)207Pb
1= 3
20
Fig. 4.5.
Experimental
and calculated
show the case of a pure S-state
0.57OMeV
40
8,,
Ep =22MeV 5/2-
-
20
60
(a) cross sectlons and (b) analyzing
40 8,,
(de!& powers for the ““Pb(p,
and the solid lines show the case with the St
d)““Pb
D states in the deuteron
60
(deg) reaction
at 22 MeV. In (b) the dashed lines
wave function.
for reproducing the oscillatory pattern in the angular distributions. It may reflect effects of deuteron break-up in the nuclear field. The D-state component (L = 2, S = 312) of the projectile form factor has very little influence on the cross sections and vector analyzing powers, but produces large changes in the tensor analyzing powers, especially in T2?. The tensor analyzing powers are sensitive primarily to the D-state components in the product of the interaction and triton wave function, rather than to the deuteron D-state alone. But the difference of the tensor analyzing powers for different nuclear interaction potentials or triton wave functions is small compared with the error bars of the experimental data. We cannot choose an interaction potential or a wave function from the fit to the experimental data. 4.2.2. Cross sections of sequential (p-d-t) processes The two-step exact finite-range DWBA calculations were carried out using the computer code TWOFNR [71]. The finite-range projectile form factors for the (p-d) and (d-t) transfer processes mentioned above were used. We must evaluate many different transition amplitudes for each transition,
M. Igarashi et al., Two-nucleon transfer reaction mechanisms
51
Fig. 4.6. Differential cross sections of the 208Pb(d, t)“‘Pb reaction at 17MeV, from top to bottom for the pi12, pi,*, f,,,, f,.? and i,,., transitions and scaled by 1. 114, 114, 118 and 118, respectively.
and there are many intermediate states of the *07Pb nucleus. If we include the D-state (L = 2) component of the projectile form factor, we have many possibilities for the transferred angular momentum (li, si, ii) for each transfer step as shown by eq. (2.4.10). The number of transition amplitudes is doubled because we must calculate the interaction terms and the non-orthogonality terms as mentioned in section 2.4. As an example, in the case of the 3: transition we must evaluate altogether 64 different transition amplitudes and sum them coherently even when only the ground state of the deuteron is considered for the intermediate states. 4.2.2.1. 0; -+o; transition. The results of calculations for the 0,’ -0: transition of the “‘Pb(p, t)*06Pb reaction at E, = 22 MeV are shown in fig. 4.8, together with the results of the calculation of the one-step process and the experimental data [85]. We used the shell model wave function obtained by True [104]. Set C in table 4.4 was used for the distorting potential parameters. The calculated cross sections of the one-step (p-t) processes are smaller by about one order of magnitude than the experimental data. The contribution of the sequential transfer processes through the ground state deuteron channel (curve 2B) is predominant and about 3 times larger than that from the one-step process. Hereafter the contribution of the two-step process means the coherent sum of the interaction and the non-orthogonality terms for that process. The contribution of the unbound states in the intermediate states is rather small. Curve TOTAL shows the coherent sum of all those contributions. The cross section is increased by one order of magnitude by inclusion of the sequential transfer processes. The fit of the angular distribution to the data is reasonably good. The calculated analyzing power functions are shown in fig. 4.8b and compared with the observed data. It is found that the one-step (curve 1) and the two-step [the coherent sum of curves 2B,
52
;\. /\* ,$* *.
M. Igarashi ?I al., Two-nucleon transfer reaction mechanisms
I-9-
\
c
’
I .:
:
: .
i
.1 I
M. Igarashi et al., Two-nucleon transfer reacrion mechanism
53
10-4
10-5
0
10
20
30
0HC. M.
40
50
60
70
80
(degree)
Fig. 4.8. (a) Cross sections and (b) A(0) for the reaction “a Pb(p, t)206Pb0* at E, = 22 MeV, calculated employing the RSC5 triton wave function and the experimental data are shown. Set C in table 4.4 was used for the optical potential parameters. Curve 1 shows the contribution of the one-step process. Curve 2B shows that of the bound deuteron channel, in which only the L = 0 component is taken into account in both (p-d) and (d-t) transfer steps of the (p-d-t) calculation. Curve 2UB(S = 0) shows that from the ‘S, (S = 0) unbound deuteron channel, while curve 2UB(S = 1) shows that from the ‘S, + ‘D, (S = 1) unbound deuteron channels with only the L = 0 component in both transfer steps. The solid curves show the coherent sum of all the processes. The experimental data are from ref. [SS].
2UB(S = 0) and 2UB(S = l)] transfer mechanisms show similar oscillatory angular distributions. The coherent sum of these two (curve 1 + 2B + 2UB) shows almost the same shape as that of the two-step cross section because of its predominance. For the O+ transitions, the L = 2 tensor components of the form factors of the light-ion system are neglected in both (p-d) and (d-t) transfer steps of the two-step process because the contribution of the L = 2 components is very small. Also, we would have to calculate too many transition amplitudes if we include such components, because the O+ states have many particle-pair shell model configurations. Figure 4.9 shows the two-step cross section and analyzing power A(0) calculated with the pure (p,,J2 configuration. Inclusion of the L = 2 tensor component does not affect the cross section, but it does slightly affect the analyzing power. The absolute magnitude of the cross section is uncertain within a factor of about two depending on the choice of distorting potential parameters. The agreement of the calculation including the sequential transfer processes with the analyzing power data becomes rather poor at around 45”. For the 0: transition the A(0) function is also sensitive to the choice of the imaginary part and/or other parameters of the distorting potential for the intermediate deuteron channels. Further study of the distorting potential in the second-order DWBA framework is desirable. 4.2.2.2. 0; +3’
transitions. The results for the two 3’ transitions are shown in fig. 4.10. It is seen
M. lgarashi et al., Two-nucleon transfer reaction mechanisms
54
T
S+
II/ , , , , , , , , 1 0
10
20
30
40
0HC.M.
50
60
70
80
90
0
10
20
40
50
60
70
80
90
(degree>
Fig. 4.9. The contribution
of the L = 2 tensor components to pure (p, JL transfer.
that the sequential transfer process plays the main role in these unnatural parity transitions. The contribution of the one-step (p-t) process to the cross sections is about one to two orders of magnitude smaller than that of the two-step process through the ground state deuteron channel. For these transitions both the one- and two-step cross sections depend strongly on the incident energy near or below the Coulomb barrier as shown in fig. 4.14. At higher energies the one-step process starts to change the angular distribution. The one-step angular distribution is different from the observed shape, and its analyzing power is also quite different from the experimental data, so that the experimental analyzing power can be reproduced only by the dominance of the sequential transfer process. The contribution of the singlet deuteron intermediate channel, which is unbound, is about one order of magnitude smaller than that of the ground state triplet deuteron channel. The fit of the analyzingpower calculation to the data is not much affected by inclusion of the unbound deuteron channels, and it is rather improved. This shows that the analyzing power A(0) does not become very small even when both ground and unbound state deuteron channels are summed over, in contrast to the prediction of the closure approximation [73]. This is because the s = 1 transfer component in the scattering amplitude does not vanish since the cancellation between the singlet and triplet deuteron channels no longer occurs. Consequently A(B) of the two-step process is produced. The origin of A(8) is therefore completely different for the one-step and two-step processes. In the one-step process it is caused by the D-state and S’-state components of the triton wave function. In the two-step process the spin l/2 transfer occurs twice and both s = 0 and s = 1 transfers are possible. The analyzing powers of the one-step process have opposite sign to the experimental data (see fig. 4.10). The predominant two-step process reproduces the experimental A(8) quite well. Thus we see that the analyzing power data are a useful tool for investigating the reaction mechanisms. The analyzing power due to the sequential transfer process also depends strongly on the j-shell
M. lgarashi et al., Two-nucleon transfer reaction mechanism
ONE-STEP
55
(P,T)
Fig. 4.10. Calculated cross sections and A(0) for the two different 3’ excitations in the reaction lo8Pb(p, t)2”6Pb at E, = 22 MeV. The RSC5 trition wave function was used. Set B in table 4.4 was used for the optical potential parameters. The experimental data are from ref. [SS]. Curve 2BS shows the contribution from the bound deuteron channel, in which only the L = 0 (scalar) component is taken into account in both the (p-d) and (d-t) transfer steps, while curve 2BT shows the contribution when the L = 2 (tensor) component is taken into account in one or both transfer steps. Curves 2US0 and 2USl show the contribution from the ‘S, unbound deuteron and the ‘S, + ‘D, unbound deuteron channel. respectively. Curve total [l + 2B t 2UB, where 2UB = 2US0 + 2USl] shows the coherent sum of all processes for the cross section [for A(B)].
structure of the nucleus as in the case of the simultaneous transfer process mentioned above, but the sign of A(B) produced by the j-shell difference is almost opposite for the one-step and two-step processes at forward angles. Compared with the recent experimental A(8) data [87], the predominance of the sequential transfer mechanism is apparently supported, as seen on fig. 4.10. In addition, this conclusion is also consistent with that obtained from differential cross section analyses. We comment on some other details of fig. 4.10. The 2BS curve shows the contribution of the bound deuteron channel, in which only the L = 0 scalar component is taken into account in both (p-d) and (d-t) transfer steps, while the 2BT curve shows the one from the remaining terms in which the L = 2 tensor component contributes in one or both transfer steps. The contribution of the non-orthogonality terms is rather small, but comparable to that of the one-step process. Figure 4.11 shows the same 3: transition at an incident energy of 3.5MeV. The contribution of the one-step process becomes larger than in the 22 MeV case.
M. lgarashi et al.. Two-nucleon
56
rrunsfer reaction mechanrsms
10-5
\ --. ,0-E
i
Cl
\ L..
__L_i-AL
10
20
Fig. 4.11. The 35 MeV calculation
30
40
@C.
M.
50
L1_
60
70
80
‘--I 90
(degree)
for the 3,’ transition
with the distorted
potential
parameter set A in table 4.1. See fig. 1.10 for the notations.
4.3. Some properties of one- and two-step processes 4.3.1. Non-orthogonality terms und the one-srep process The simple sum of the non-orthogonality amplitudes for the intermediate states of both target and projectile nuclei is equal in magnitude and opposite in sign to the one-step amplitude [40, 411. However, we can take into account only the few most important intermediate states in our practical evaluations. Therefore it is interesting to see to what extent this theorem is realized in actual transitions. Figures 4.12 and 4.13 show the contributions from the non-orthogonality terms of each intermediate channel and their coherent sum. For comparison the contributions to the cross section and the analyzing power from the one-step process are also shown. The total cross section of the coherent sum of the non-orthogonality terms is almost comparable to the cross section of the one-step process especially for the unnatural parity transition. We found that the non-orthogonality term almost cancels the one-step process. From these facts we can say that this theorem is fairly well confirmed. Therefore from this point the concept of the number of steps is rather ambiguous. This finding is quite interesting and important. Further studies are in progress and the details will be reported elsewhere. 4.3.2. Incident Figure 4.14 calculation the incident energy
energy dependence of the one- and two-step processes shows the incident energy dependence of the one- and two-step processes. In this optical potential parameters are fixed to set A for both 0: and 3: transfer and only the is changed. For the natural parity 0: transfer the energy dependences of both processes
M. lgarashi et al., Two-nucleon transfer reaction mechanism
1
0
10
Total non-ortho.
20
30
0I+CM.
Fig. 4.12. Cross sections and analyzing E, = 22 MeV.
40
50
60
70
-
Nomortho.
------
One-step
60
(degree) powers of the non-orthogonality
terms and one-step
process for the 208Pb(p, t)*06Pb0;
reaction
at
are almost the same and the two-step contribution is still larger than the one-step contribution even at the highest energy of 80 MeV In contrast, the energy dependence of the one-step process is rather weak at higher energies for the UNPS 3’ transition. Thus, the contribution of the one-step process is larger than the contribution of the two-step process at high incident energies, for example 80 MeV At low incident energies, around 20-30MeV, the two-step process dominates the transition over the one-step process. Both one- and two-step cross sections depend strongly on the incident energy near or below the Coulomb barrier. The reduction of the cross section in this energy region is stronger for the 3: transition than for the 0: transition. The present theoretical predictions have been confirmed experimentally up to 50 MeV incident energy [48]. Therefore it is important experimentally to check the evolution of the one-step cross section to a magnitude comparable with the two-step cross section around 80 MeV. 4.3.3. j-shell structure dependence of the analyzing powers
The differential cross section and the sign of the analyzing power A(8) of one-nucleon transfer reactions are well known to depend on the transferred angular momenta 1 and j, and hence both observables are useful tools to study the single-particle angular momentum state (I, j) to or from which the nucleon is transferred [113]. In the past, information on the j-shell configurations of the transferred pair of nucleons was considered to be unobtainable from the (p, t) reaction.
M. Igarashi er al., Two-nucleon transfer reaction mechanisms
58
104
I
Total nowortho.
i
Fig. 4.13. Cross sections and analyzmg powers of the non-orthogonality
Total non-ortho.
terms and one-step process for the ‘““Pb(p. t)“‘hPh 3; reaction at (a)
22 MeV and (b) 35 MeV.
transitions. It has been found in our work that the two-step analyzing power A(0) 4.3.3.1. 0 ‘-0’ of the (p, t) 0’ -+ 0’ transition sensitively reflects the j-shell difference in the nuclear wave function [45, 461. The two-step analyzing powers of the ( j< = I- 112)’ and ( j-, = I+ 1 /2)2 pick-up reactions show completely out-of-phase angular distributions if the spin-orbit distortions are omitted in the calculation, and even with inclusion of the distortions, the out-of-phase property remains in the forward angular region. The j-shell dependence of the sign of the analyzing powers appears also in the (p-t) one-step mechanism if we use the finite-range calculation employing a realistic triton wave function. Figure 4.15 shows A(8) of the one- and two-step processes (only through the ground state deuteron intermediate channel for simplicity of the calculation) for (pliz)” and (P~/~)’ pair pick-up in the ““Pb(p, t)20hPb 0’ reactions. In this calculation parameter set C in table 4.4 is used and the spin-orbit distortions are omitted. The signs of A(8) for both one-step and two-step processes change depending on the configuration, j:[ = (p, ,,)‘I or j:[= (P~,~)~]. When one-step and two-step processes are compared, the signs are almost opposite to each other and the oscillation of A(8) is weaker for one-step than two-step processes. A(8) for the one-step process disappears completely in the zero-range calculations, because s = 1 spin transfer is not allowed. The major source of s = 1 spin transfer is different in different transfer mechanisms; it is supplied by the minor components (non-S-state) of the
M. Igarashi et al., Two-nucleon transfer reaction mechanisms
59
1
10-l
lo-’ !(b)
mPb(p,t)206Pb
3’ TWO-STEP
Fig. 4.14. The incident energy dependence of the one- and two-step processes. The numbers near the curves show the incident energy, from 15 MeV to 80 MeV. (a) The natural parity 0’ ground state transition, (b) The UNPS 3’ transition of the *“Pb(p, t)“‘Pb reaction.
M. Igarashi et al., Two-nucleon transfer reaction mechanisms
60
im r
1
1
100, 90
“%@-d-
t PO6Pb 0;
80 Ep
70
wi
Fig. 4.15. Shell model orbit dependence
(b)
=
22MeV
TWO- step
of the calculated A(0) of (a) the one-step and (b) the two-step process without spin-orbit
distortion.
triton wave function in the one-step mechanism, whereby the finite-range interaction is essential, and by double spin 112 transfer in the two-step mechanism. This effect remains in the forward angular region even when the spin-orbit distortions are included. Figure 4.16 shows the case with the spin-orbit distortions for the p, d and t channels. At backward angles the spin-orbit distortion effect dominates and violates the simple out-of-phase rule. The largest spin-orbit distortion effects come from the proton channel. The origin of the shell dependences of the one- and two-step analyzing powers is interpreted as follows. When the distorting potential is spin independent and only one set of nuclear shell orbits contributes for the (p, t) reaction, the vector analyzing power eq. (2.6.11) can be (M,) and (MJ expressed as [114, 761 A(8) = -j/2($, + 1)/3s, Im Tll(fJ) =- j/2(& + Q/3& c (-)‘+[+X-J’+J,-Sb+11/(2S+ 1)(2S’ + 1)(2S, + 1)(21+ 1)(21’ + 1) isl’r’ (4.3.la) where
Pkq(h l’s’j’; 0) = ; (-1i’+m-q(lml’q -
m[kq)pI:"(e)p;:;-q(e)*
and p$‘(13) are the same as defined by eqs. (2.6.2) and (2.6.3).
,
(4.3.lb)
M. Igarashi et al., Two-nucleon transfer reaction mechanisms
61
Pb 0; Ep = 22MeV
40
Fig. 4.16. Shell model orbit dependence
50
60
70
of the calculated A(0) of (a) the one-step and (b) the two-step process with spin-orbit
80
distortion.
For O++O+ (p, t) reactions, from the Racah coefficient W(ll’ss’; lj) in eq. (4.3.la), two sets of (kj) transfer are allowed: (000) and (110). The latter transfer is the so-called “unnatural parity” transfer component, which is allowed even in O++ Of transition. This component arises from non-S-state components of wave functions and spin-dependent interactions involved in the calculation. From eq. (4.3.1), we can see that the numerator of the vector analyzing power is expressed in terms of two amplitudes P’“(0) and /I” m-1, of which the m substates differ by one unit. Therefore only the cross form of the two amplitudes corresponding to (Isj) = (000) and (110) appears in the expression for A(B). The reduced amplitude /3:(e) of a one-step process includes a 9-j symbol,
i
4
Sl
il
12
s2
j2
L
%
i
I ,
as eq. (2.3.28) shows. The j-shell dependence of the analyzing power comes from this 9-j symbol. If we neglect the P-state component of the triton wave function, the spin of the transferred two neutrons, SIz, is equal to the transferred spin, s, as mentioned in table 2.4. The s = 1 amplitude is small compared with the s = 0 amplitude, so that we can neglect the contribution of the s = 1 amplitude to the denominator of eq. (4.3.la). A(8) is then proportional to the ratio of two 9-j symbols,
(4.3.2)
M. Igurashi
62
et al..
Two-nucleon transfer reaction
mechanisms
so that the analyzing power for ji = 1, + l/2 and that for jl = I, - 112 are related by
A(H,j,=1,+1/2)=-& A(6, j,=I, I
112).
(4.3.3)
This relation is similar to that found in one-nucleon transfer reactions [114, 1151, in which the transferred (Zsj) is the nuclear shell orbit (I,s,j,) from (or to) which the nucleon is transferred. As for the two-step process, the reduced amplitude /3z(f?) includes the 9-j symbol 1, s1 jl 1, 32 j2 i
I
s
j
I
as eq. (2.4.11) shows if we choose a pure symmetric S-state wave function for the triton. Furthermore the s = 1 amplitude is small compared with the s = 0 amplitude, but not so negligibly small if we choose only the ground state deuteron for the intermediate channel. If we neglect the contribution of the s = 1 amplitude to the denominator of eq. (4.3.la), we have the same relation as eq. (4.3.3). This relation does not hold if the contribution of the s = 1 amplitude cannot be neglected, as we can see in fig. 4.15b. If we take into account the minor components of the triton wave function, the transferred angular momenta (l~j),,~ and (Is~)*“~are not always the same as the angular momenta of the shell model single-particle state of the transferred neutron in the target. Namely, s = 3/2 amplitudes are allowed and the transferred orbital angular momentum I can take on values in the range (j - 3/2] 5 I ‘j + 3/2. These facts also invalidate relation (4.3.3). Figure 4.17 shows the results for the 0: transition of the 2”sPb(p, t)2’)hPbreaction. In this case, the components of the shell model wave function have amplitudes with sign opposite (f5,2)-2 and (P,J2 to each other. The shell model wave functions are shown in table 4.2. The cross section becomes very small because of the cancellation, and the analyzing power differs strongly from that of the 0: transition shown in fig. 4.8b. The analyzing powers of the one- and two-step processes are very different at forward angles. The solid line is the result of a coherent sum of the two processes. Figure 4.18 shows a comparison of the calculated results with the experimental data for the 0: transition [47]. The different shell model configurations give different angular dependences of the analyzing power. The present theoretical resuts are still not conclusive for a consistent understanding of the differential cross section and analyzing power. The nuclear wave function may have more problems. We can, however, say that the analyzing power data are useful to check the shell model configurations. We can use the analyzing power data for a check of the shell model configurations. 4.3.3.2. 0’ -, higher J” transitions. j-shell dependent A(8) have been found in the present work for other 1” transitions. In fig. 4.10 the analyzing power data of the 3’ transitions show almost opposite signs at forward angles for the one-neutron orbits f,,, and f7,2. Figure 4.19 shows A(8) for the one- and two-step processes (only through the ground state deuteron intermediate channel) for four combinations of the pair picked up from the p and f single-particle states in the 208Pb(p,t)*06Pb3’ reaction (see also fig. 4.4). The signs of A(8) for the one- and two-step processes are quite sensitive to a change in j-shell configuration. Furthermore, the signs of A(B) produced by j-shell differences are almost opposite, when one-step and two-step processes are compared at forward angles. The predominance of the two-step process for these transitions determines the final value of A(8).
M. Igarashi et al., Two-nucleon transfer reaction mechanism
10-l
t
1
,j z .‘\ .ii
‘\
I i
i ,_
‘\ \ i 2UB(S=l) \ i ( ,f-~, \il/ ‘\ \il/ ,‘\ 1 ,/-
/ I 0
10
\ \
‘?\
/
1O-6
2UB(S=O) /-\ ; ‘1 ‘\ ,‘. ; ‘\,,j’ ;
\
;,; / /’
1O-5
63
20
30
0I+C.M.
40
50
60
70
80
40
50
60
70
80
(degree)
Fig. 4.17. Calculated (a) cross section and (b) analyzing power of the “‘Pb(p, t)20bPb reaction at 22 MeV leading to the 0; (E, = 1.167 MeV) state. See fig. 4.8 for the notation. True’s shell model wave function 11041 was used.
A clear explanation of the j-shell dependence for the higher 1” transitions is rather difficult, because four different (kj) transfer amplitudes (one for s = 0 and three for s = 1 for fixed 1 and j values, as shown in table 4.6) contribute to these transitions. In general, three self-products of s = 1 amplitudes and five cross-products of s = 0 and s = 1 amplitudes appear in the calculation of the analyzing power. However, the relative strengths and signs of the two-step transition amplitudes are decided mainly by the 9-j symbol
if we assume pure L = 0 (S-state) projectile form factors. The values of the 9-j symbols for the four combinations of (p, f) configurations are listed in table 4.6. We can see that the relative strengths and signs of the different (lsj) amplitudes are drastically changed by the shell model configuration. For the two-step process the contributions of the transition amplitudes caused by the L = 2 projectile form factors are small. These amplitudes include the same 9-j symbol, in which, however, s = 3/2 and 1 is not always the same as the orbital angular momentum of the transferred neutron.
M. lgarashi et al., Two-nucleon transfer reaction mechanism
64
loo
20ePb(p,t)M’Pb 0; Ep =22 MeV
0.8
True
Taka. _---K-H 0.6
(a>
,-\
-
_--
----
T-F(l)
-0.6 -0.8
0.1
’
t
I
20
0
I
I
40
60
I 80
-1.0
s
0
20
BcmMeg)
40
60
80
&-,(deg)
Fig. 4.18. Calculated (a) cross sections and (b) analyzing powers for the *“‘Pb(p, t)IU6Pb reaction at 22MeV leading to the 0; (E, = 1.167 MeV) state. The four curves are the results of calculations using four different shell model wave functions: True: from ref. [104]; Taka: from ref. [106]; K-H: from ref. [107]; T-F(l): from ref. [103]. They are given in table 4.3. The experimental data are from ref. [47].
For the one-step process for unnatural parity transitions the relative strengths and signs of the transition amplitudes are very different from those of the two-step process. The factor
of eq. (2.3.28) decides the (Isj) dependence. When we ignore the P-state component in the triton wave function, then S,, = S, but L runs from 1 - 1 to 1+ 1 when the D-state of the triton (L’ = 2) is taken into account. For’unnatural parity transitions the contribution from the D-state of the triton is the most important. Therefore the (Isj) dependence is very different between one- and two-step processes. The change of the relative strengths of (Isj) transfer in the one-step process is shown in fig. 4.2, and the same for the two-step process is shown in fig. 4.20. These two are very different from each other. The sign change of the amplitudes produces the sign change of the analyzing power through the crossproduct terms. Table 4.6 Values of the 9-j symbol for the four (p, f) configurations
w
(PI,,? f,,,)
(PI,,? f,,,)
(P,,Z? f,!,)
(P,,,. f7,Z)
(303) (213) (313) (413)
lAf&ii -liXfiS% l/V%@ l/V%
-1ivX4 l/VQi 1lQiiB l/XfZZ%
l/V%Y6 -llv%i% l/VQiiZB -liv?SZJ
1/m llV?i% -l/Gi%A -l/V%%6
M. lgarashi et al., Two-nucleon transfer reaction mechanisms
ONE-STEP
[p& f&-j 3+ ---_--
[p;;zf&l
3+
__-----__
[pi;*f$*] 3+
---_-----
[pi;2f&l
3+
-0.4 ONE-STEP+ TWO-STEP -0.8
I
I
0
10
20
30
00-4 C.M. 40 CdegeM 50 60
I 70
80
so
Fig. 4.19. Analyzing powers of the 208Pb(p, t)*06Pb reaction at 22 MeV leading to four 3’ state configurations. (a) The one-step process, (b) the two-step process (the contribution from the ground state deuteron intermediate channel alone is considered) and (c) the coherent sum of both processes.
These results, however, depend somewhat on the incident energy, because the relative strength of different (kj) amplitudes depends also on the incident energy. Figure 4.21 shows the case of incident energy E, = 35 MeV. By comparing this with the case of 22 MeV (fig. 4.19), we can see that A(0) of the transition to the [p$f7/:] configuration state changes strongly with the incident energy. Shown in fig. 4.22 is the calculated 2+ analyzing power with finite-range treatment of the one- and two-step processes. Here the analyzing power A(8) is also sensitive to the nuclear shell configuration. 4.4. Summary and conclusions For the (p, t) reactions, the absolute values of the cross section obtained by one-step DWBA calculations by different authors differ much from one another. For a long time this fact made it difficult to understand the mechanism of this reaction. In this report we made careful studies of the absolute cross sections and conclude that the one-step process alone cannot reproduce the experimental absolute cross section even for strong natural parity transitions. The predominance or comparable importance of the sequential transfer mechanism over the one-step (p-t) mechanism has been demonstrated for low
M. lgarashi ef al., Two-nucleon transfer reaction mechanisms
_. \ 1o-3
208Pb(p-d-t)206Pb ‘\
3; [PA G]
‘\
A ‘\
g $10-4 F
‘\
‘1
‘\
MeV
303
--‘\
Ep ~35
\
Isj
‘\
r\ 6 \
-\
3’
2 1 3
_- --.
313
-_-
413
8 z 0
10-5
5
1O-6 0 Fig, 4.20. Decomposition transitions.
10
20
30
40
0I+ C.M.
50
60
70
60
(degree)
of the calculated cross sections into transferred
IO
20
0HC.M. 30
40
50
60
70
80
90
(degree)
angular momentum (kj) components of the two-step processes for two 3’
Fig. 4.21. Analyzing powers for the ‘““Pb(p, t)*““Pb reaction at 35 MeV leading to four 3’ state configurations.
M. Igarashi et al., Two-nucleon transfer reaction mechanism
67
0.8 0.4 e G a
0.0
-0.4
II
,’
ONE-STEP
.__I
TWO-STEP 1
,,__,;
10-5'
0
’ 10
20
30 @CM.
1 40
50
60
(degree)
’ 70
’
Bo
’
0
10
’ 20
’ 30
0
’ 40 C. M.
50
’ 60
70
80
’
90
(degree)
Fig. 4.22. Calculated analyzing powers and cross sections of the *‘*Pb(p, t)*ObPb reaction at 22 MeV leading to the three 2’ state configurations.
incident energies in the case of both the natural parity and the unnatural parity transitions by accurate and consistent evaluation of the one-step and two-step transfer processes with realistic wave functions for the projectile-ejectile system and the interactions causing the reaction. This predominance of the sequential transfer processes does not affect the customary statement that the (p, t) reactions are good for probing two-particle pair correlations in nuclei, since the two-step amplitude is also enhanced in much the same way as the one-step process by the pair correlation. It was also pointed out that the (p, t) analyzing power is a useful tool to clarify the transfer mechanism, one-step or two-step, and to study nuclear structure, since its shape and sign strongly depend on the difference in the transfer mechanisms and on the nuclear j-shell configurations.
Acknowledgements
The authors are grateful to M. Kawai for his continuous encouragements. Their special thanks are due to S. Ishikawa, T. Sasakawa and T. Sawada for providing us with their triton wave functions. They are indebted to Y. Aoki and other members of the Tandem Accelerator Center, University of Tsukuba.
M. Igarashi et al., Two-nucleon transfer reaction mechanisms
68
They thank P.E. Hodgson for his careful reading of the manuscript. One of the authors (MI) also thanks the members of the theoretical division of the Institute for Nuclear Study, University of Tokyo for their kind hospitality extended to him. The FACOM M380 at the Institute for Nuclear Study was used for the numerical calculations. This work was supported in part by a Grant-in-Aid for Scientific Research from the Japan Ministry of Education (1979-1980). Two of the authors (MI and KIK) thank the Kudo Arts and Sciences Foundation for the financial support needed for this lengthy work.
Appendix A. Hard-core correction to the one-nucleon transfer form factor If the triton wave function is evaluated with a nucleon-nucleon interaction with a hard core, the particle transfer form factors need corrections as mentioned in section 2.3.3. In the case of (d, t) reactions the form factor is given by
(&(r)lVnln2(rnln2) + Vpn2(rpn2) + VA,n2(rn,n2) + ~n2(rpn2)Ilc11(r~ Qr 7
(A4
where the subscript r means integration over the coordinate r, and V(r) is an attractive potential outside the hard core and V’(r,) = S(ra - r,)
$ +-$ c
64.2)
ra .
LX+
The hard-core correction (pseudopotential) terms are evaluated as follows: S(ra - r,) =
6(vR’+ ir’
+ Rr cos 8 - r,) =
j&j w - 0,).
(A.3)
Then
wd(T)/~(r”,“2 - rc)
$ +j+ c
r”*“21 cClt(r, 9,
nl”2+
ca 2?r
=27T
II
rGb (r)
~
rnlnZqbt(r,R)
sin 8 de r2 dr .
64.4)
0 0
Figure A.1 shows the (d, t) form factor thus evaluated with the Tang-Herndon triton wave function and the nucleon-nucleon interaction. The pure S-state deuteron wave function evaluated with the Tang-Herndon interaction was used. The hard-core correction is large inside the triton and has opposite sign to the interaction term. The calculated integrated cross sections for different form factors are compared in table A.1 for the *‘*Pb(d, t)‘07Pb reactions at 17 MeV. The optical potential and the neutron binding potential parameters are the same as in section 4.2.1.2. The Tang-Herndon form factor without the hard-core correction gives cross sections about 3 times larger than with the correction. In the case of RSCS the projectile form factors given in section 4.2.1.2. were used, so they included the L = 2 components. The Tang-Herndon form factor gives cross sections about 39% larger than the RSCS form factor. The RSC3 form factor does not make any difference compared with the RSCS form factor in the integrated cross section.
M. Igarashi et al., Two-nucleon transfer reaction mechanisms
1031
1
1
I
_ Tang-Herndon
I
.
I
69
1
(d, t ) F. F.
-------
Interaction T. _ H.C. corre. T.
Fig. Al. Projectile form factors for the (d, t) reaction with the Tang-Herndon triton and deuteron wave functions. The dashed line shows the interaction term, the dash-dotted line shows the hard-core correction term and the solid line shows the total form factor.
Table A.1 Comparison of cross sections of various calculations for the ‘OaPb(d, t)*O’Pb reactions at 17 MeV. Integrated cross sections (o-180”) in mb for S = 2i + 1 Projectile
PI,2
P3,2
f 512
f712
i,,,z
h 912
TH without corr. TH with corr. RSC3 RSC5
35.65 12.06 8.50 8.50
60.08 20.36 14.44 14.43
29.08 9.71 6.86 6.86
26.36 8.77 6.39 6.37
7.047 2.277 1.753 1.741
2.408 0.790 0.556 0.557
Knutson and Yang [116] calculated the deuteron-triton overlap integrals using several kinds of deuteron-triton overlap functions, but their results should be reinvestigated for the case in which they used an interaction with a hard core since they neglected the hard-core correction terms.
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M. lgarashi et al., Two-nucleon transfer reaction mechanisms
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