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The DWBA for stripping in the shell model approach to reactions
ANNALS
The
OF PHYSICS
DWBA
65. 227-248 (I9711
for Stripping
in the Shell Model
Approach
to Reactions
DAN AGASSI Department
of Nuclear
Ph.vsics,
Weizmann
Institute
of Science,
Rehovot,
Israel
Received July 21, 1970
The DWBA form for the energy-averaged stripping amplitude is derived in the framework of the shell-model approach to reactions. This derivation generalizes for a rearrangement process, a previous work of Mahaux and Weidenmuller which concerns nucleon-nucleus scattering. The present derivation follows a similar line of argumentation. The required statistical assumptions and optical potentials are stated. Antisymmetry is included in the discussion.
I. INTRODUCTION
Stripping is a direct-reaction mechanism for (d, p) scattering, and is therefore analyzed using the DWBA approach [l]. Although much used, this approximation has received little theoretical attention. It has been discussed for a three-body model [2] and to some extent for the general many-body case [3]. It is clear that three-body difficulties form an additional source of complications in any attempt to analyze the (d, p) reaction. A merit of the so-called shell-model approach to reactions is that the many-body complications and the three-body difficulties separate out in a simple way. In fact, once we can describe a deuteron scattering off the average well generated by the target particles, the deuteron-nucleus scattering reduces, effectively, to a nucleon-nucleus problem [4]. Since the DWBA has been already analyzed for nucleon-nucleus scattering [5,6], we are thus motivated to study the DWBA for the stripping within the framework of the shell-model approach. The (d, p) process provides the simplest example of rearrangement collision. In the usual shell-model treatments of reaction [5], rearrangement processes are excluded. In this sense, the present work generalizes the previous ones. Because the underlying extension of the model for a twocontinuum has a similar structure to the one-continuum model [4], the analysis of nucleon-nucleus and deuteron-nucleus direct scattering turn out to be similar. Hence, two related points are demonstrated below, i.e., the similarity between a nonrearrangement and rearrangement direct reaction, and that a stripping picture does emerge from a microscopic model for a direct-rearrangement collision. 227
228
AGASSl
The discussion below may also be viewed as an application of Feshbach’s [3] philosophy to the shell-model approach to reactions in the case of rearrangements. The DWBA is fabricated to apply when a direct-reaction mechanism is dominating the scattering. The cross section resulting from such a mechanism is smoothly energy dependent since it describes a short time-delay process. We adopt [6] the attitude of associating the energy-uueruged T-matrix element with this smoothly energy-dependent process. Hence, “derivation” of the DWBA in a dynamic model amounts to proving that the energy average exact T-matrix element can be recast in the usual DWBA form, on the assumption that the “theoretical” optical potential is properly defined. This kind of analysis cannot justify the common phenomenological means of fitting first elastic scattering data and then computing inelastic transitions. Rather, it has been shown [6] that the DWBA procedure and the optical potential (OP) associated with it, is a consistent approximation wihin the shell-model approach. The relevance of such an analysis is to give some confidence to the legitimacy of the phenomenological procedures. It is this philosophy which underlines the present work. We show that using approximations and qualitative arguments of the type needed for the nucleonscattering case [6], the energy average T-matrix element for the (d, p) process can be expressed in the DWBA form and corresponds to a stripping picture. We are unable, however, to connect the OP defined below with the phenomenological ones, nor can we establish the wellknown relation [3] between the nucleon and deuteron optical potentials. We shali exclude also any complicating factors like doorway states and strongly-coupled channels. In the following, energy averages of the T-matrix are obtained by the common device (T(E))
= T(E + iI),
(I.11
where Z is the averaging interval. The averaging interval is assumed to include a host of compound nucleus resonances. The Z-dependence of the results can usually be neglected if the direct reaction mechanism is correct and there are, for example, no doorway states in the energy region considered. The order of magnitude of Z is 1 MeV for nucleon-nucleus scattering [5]. The DWBA form will be derived using the two-potential scattering formula which holds also for rearrangement collisions [8]. If we consider two decompositions of the complete hamiltonian H, H = Ha + HA = Hb + H;
(1.2)
where the “u” decomposition is appropriate for the incoming channel and the “b” decomposition for the outgoing channel (Ha, Hb include distortion) H&J;*) = EC);*)
H&i*’
= E&*‘,
(1.3)
DWBA FOR STRIPPING
then the T-matrix element for the transition
229
is given by the familiar formula
In order to make the presentation self-contained, we review briefly in Section II the shell-model approach using, however, the second quantization language [4]. We feel it is instructive to consider first the case of nucleon-nucleus scattering [5,6] in order to examine in the simpler situation the line of argumentation for the stripping DWBA. This is carried out in Section III, while in Section IV we proceed to analyze the (d, p) scattering. We make use of the Feshbach projection operator techniques in the formal manipulations.
II. THE SHELL-MODEL APPROACH OF NUCLEAR REACTIONS The basic idea underlying this approach is to make use of the distinction between the bound and the continuous single-particle orbital of the unperturbed, finite well, shell-model hamiltonian. We then truncate the complete Hilbert space of many-body states to include at most two particles in the continuum, approximate the bound states of the system to have no continuum components, and solve the Schrodinger equation in the model space just as in the usual shellmodel. In the present case, however, the final product is a system of coupled integral equations instead of an eigenvalue problem. Analogous to the situation for bound-state problems, the second quantization language seems to be convenient here. Consider the complete hamiltonian H in the second quantization form
where (c@~k’[$) is the nonantisymmetrized matrix element and the state indices run over bound and continuous distorted states of Ho. For the continuum states of Ho it is convenient to choose the outgoing boundary conditions [4]. Since we use second quantization throughout all derivations, antisymmetry is included automatically in all expressions. Using the distinction between the bound and continuum orbitals, the hamiltonian (2.1) can be decomposed in the following way H = Ho(b) + Ho(c) + V(b, b) + V(c, c) + W(b, c),
cv
230
AGASSI
where
(2.3)
Hqb, c) = v - vyb, b) - V(c, c).
We note from (2.3) that IV(b,c) is that portion of the interaction where at least one orbital is bound. If we denote by “b” and “c” bound and continuum orbitals, W(b, c) can be further decomposed : W(b, c) = V(bc, bb) + V(bb, bc) + V(cc, bb) + V(bb, cc) + V(bc, bc) + V(bc, cc) -t V(cc, bc),
(2.4)
where, for instance, (2.4a) The other terms in (2.4) have obvious meanings. The many-particle basis wavefunctions are chosen to be eigenfunctions of K. = Ho(b) + Ho(c) + V(b, b) + V(c, c) = h(‘)(b) + Ho(c) + V(c, c).
The model Hilbert space consists of three groups of antisymmetrized functions of K. :
(2.5)
eigen-
1. Those where all particles are in bound orbitals, denoted by [a(N + 2)) b space. 2. Those with one particle in a continuum orbital, denoted by jb(N + 1); E) -the c space, 3. Those with a deuteron in the continuum, denoted by [y(N); @g)(s))-the d space. -the
In the b and c subspaces, the V(c, c) interaction vanishes identically whereas in the d space it binds the two continuum particles to form the distorted deuteron. The three-body complications are confined in the construction of the @g)(s) of the d space, which satisfy [Ho(c) + v(cc)]@+q&) = E&q)(&)
(2Sa)
DWBA
FOR
STRIPPING
231
for the deuteron incoming wave. Since the hamiltonian (2Sa) cannot trap any particle in a bound state of the well, it corresponds to a problem which is easier than the usual three-body problem. in particular, if we are below the breakup threshold, (2Sa) is a one-channel scattering problem, and therefore has a Lippmann-Schwinger equivalent. According to the above subdivision of the Hilbert space, the exact solution of H within the model space can be decomposed to its projections on these orthogonal subspaces : ti = tw4 + $(c) + wb
WJ)
It should be realized that in e(b), I& c), and $(d) we lump together all channels with the same outgoing product, namely,
(2.6a)
For deuteron stripping, for instance, $(d) contains an incoming deuteron wave and outgoing scattered deuterons and $(c) contains only outgoing scattered nucleons. For nucleon scattering, on the other hand, we have an incoming nucleon wave in $(c) as well as scattered nucleons, and I&J) contains outgoing scattered deuterons. The T-operator corresponding to the various processes can be deduced easily by comparing (2.6) with the general definition [S]
where SC+) are the (distorted) unperturbed eigenstates of &, . In the following we shall manipulate with the T-operator of (2.7) defined in space of the distorted states. We should therefore bear in mind that in order to obtain the actual T-matrix element, there are additional phase factors [4] and contributions from the unperturbed hamiltonian. These, in turn, do not enter the discussion since they are assumed throughout to be smooth in energy. This assumption is equivalent to the hypotheses of the absence of narrow resonances of KO in the considered energy interval and no thresholds effects. These exclusions are assumed in the rest of the discussion.
232
AGASSI
In the context of the present work, the energy E of the system is assumed to be in the region where the spectrum of the compound nucleus hamiltonian /r(‘)(b) (Eq. 2.5) is very dense. We shall further assume that the “sea” of eigenstates of h(‘)(b) is “random,” i.e., there is no state or group of states that couple in a particular fashion with the continuum. In other words, we exclude doorway states or giant resonances in the energy interval considered [5,6]. The different components of the interaction (2.4) have a simple physical interpretation. The V(& M) term (and its conjugate) scatter a bound particle to the continuum or vice-versa. These matrix elements, therefore, generate processes where the incoming projectile is trapped in a bound orbital, so the whole system is temporarily bound and forms the compound system. Later, one particle leaks back to the continuum via the conjugate matrix elements. It is therefore expected, as we shall see, that they are important constituents of the OP [5] for nucleon scattering. In passing, we should note that the elements V(bc, M) couple also the d and c spaces, simply because one particle is added to the continuum. It will prove convenient to decompose these two functions of these matrix elements, Hence, we deime V’(bc, bb) = V(d, c) + V(c, b),
Gw
where, for instance, V(d, c) couples the d and c spaces. The decomposition (2.8) can be carried out by the use of projection operators of the d and c spaces. The V(cc, M) terms in (2.4) play for deuterons a similar role that the V(& M) play for nucleons. They are thus V(d, b) terms. They give rise to compoundnucleus scattering processes, and therefore will play a role in the construction of the deuteron OP. The V@c, bc) matrix elements scatter directly, elastically or inelastically, a one-continuum particle. Consequently these matrix elements are responsible for a direct reaction mechanism. We note that they can scatter directly either nucleons or deuterons and, as before, it will be convenient to disentangle these two functions. Hence we define J’(bc, bc) = V-(c,c) + V(d, d)
(2.9)
with obvious notations. We would like to further distinguish between the elastic and inelastic components in (2.9) using, again, the c and d projection operators. This separation will be needed since the OP usually fits the elastic scattering on the ground state, and elastic scattering, in turn, has some direct reaction portion in it. Denoting by (V(c, c)) the elastic scattering portion of V(c, c) and by A[V(c, c)] the inelastic part, we have
DWBA FOR STRIPPING
233
The last group of matrix elements, V(cc, k), in (2.4) scatter directly one particle in the continuum and one particle bound to the continuum and enter, therefore, the category V(d, c) without the need for projectors. They correspond, in particular, to a stripping process where one particle is believed to scatter directly while the other is trapped in the bound orbital. The standard equations in the shell-model approach are obtained when the $ of (2.6) is inserted in the Schrodinger equation (2.11)
E$ = E$.
In (2.11) we neglect those terms that couple with portions outside the model Hilbert space, and finally we project (2.11) into the various channels in the b, c and d spaces. In the following we shall apply this procedure repeatedly.
III. NUCLEON-NUCLEUS
SCATTERING
We review in this section the results obtained previously [5,6], recast in the notation of Section II, and with some slight modification-this in order to outline the generalization in the next section. For the one-continuum problem, the model space is restricted to the b and c spaces. Consequently the complete hamiltonian to be considered, in the notation of Eq. (2.4), is I-l = KO + V(bc, bb) + V(bb, bc) + V(bc, bc), = KO + V(c, b) + V(b, c) + V(bc, bc),
(3.1)
with the exact solution decomposition $ = WI
+ $64.
(3.la)
In the present case, there is no need for projection operators since only one particle is in the continuum. A. Modelfor
Pure Compound-Nucleus
Suppose for the moment deleted, i.e., we consider
Scattering Mechanism
that the V(bc, bc) matrix elements in (3.1) are
H’ = KO + V(c, b) + V(b, c).
(3.2)
The resulting hamiltonian will correspond to a model of purely compoundnucleus scattering. Such an interpretation follows since the V(bc, bb) matrix elements can scatter a particle only via an intermediate state where all particles are bound, i.e., via the compound-nucleus phase. Since we assume a high density
234
AGASI
of compound states in the energy region considered and of “random” character, such trapping-leaking processes will be numerous and each one equally important. The philosophy of the OP [I] is that it describes, in a simple fashion, the auerage effect of this host of processes. Hence, the OP is defined to reproduce the average of its T-operator [3]. Using the Schrodinger equation (2.11) for (3.2) and the model hamiltonian, we get (/r(‘)(b) defined in Eq. 2.5) ~‘“‘WU4 ~ovW
or, introducing
+ J,‘tb, +j&j
= &WI, (3.3)
+ W, O/W4 = W@L
the boundary conditions, M4 = E - $,jtbJUb~ 4iW~
wo - WM
= 0, (3.4)
4w = $0) + E - i.
+ $c>
W@J-
We solve first (3.3) for the exact T-operator, then take its energy average, and finally define the OP in such a way that it reproduces the average T. The exact T is obtained by solving (3.4) first for $(b) and then obtaining 1,9(c).By comparing the obtained $(c) with (2.7) we readily recover the exact T-operator: E - h(‘)(b) - V(b, c)
11
’ V(c, b) E - K. + iv
V(b, c).
(3.5)
The T-operator in (3.5) represents the full T-matrix. Its energy dependence is strong if the shift term in the denominator of (3.5) is small compared to the spacing between the eigenvalues of h(‘)(b). The smooth average T-operator, (T), is E - h(‘)(b) + iI - V(b, c)E - ;
+ iI J’k 4 0
11
W, 4.
(3.6)
There are two kinds of OP that could be defined: The first, referred to as OPl, reproduces the jiilj T-operator (3.6) whereas the second, OP2, reproduces (3.6) only for the elastic scattering of a particular channel co. The OP2 differs from OPl in that it takes into account the effect of all the other open channels except co and, in fact, corresponds to the phenomenological OP. The OPI, however, is the one to be used in a coupled-channel approach [5,6]. Both OP are expected to resemble one another provided the open channels are not too numerous, and that the couplings between the particular channel co (usually the ground state) and the other open channels is “weak.” In that case, the coupledchannels system yields the DWBA in first order. These conditions are probably
DWBA
FOR
235
STRIPPING
satisfied if projectile energy is low (say 10 MeV) and the target is not deformed. The condition of weakly coupled channels is basic for the rest of the discussion ; otherwise a DWBA form is somewhat artificial. However, the restriction that the open channels be not numerous can be relaxed at the cost of slightly complicating the expressions. We shall proceed with OP2, hereafter denoted by OP. This generalizes slightly the original derivation [5, 61 which has used the OPl. The formal steps, nevertheless, are similar. From (3.3) we note that (cJ$(c), the cO-component in e(c), satisfies E -
K.
-
1
V(co,b)
E - h”)(b) - V(b, c)[Q/(E - &)]V(c,
b)
Vh
co)
I
x (co~l+b(c)= 0.
(3.7)
In (3.7), V(co, b) is the co-projection of the V(c, b) interaction, whereas Q projects into all open channels except co. Following Feshbach [3], the OP is defined V$!(E) = V(co, b)
E - h(‘)(b) - V(b, c) E - i
11
+ in V(c, b) + iZ 0
so the OP wavefunction
W,
co),
(3.8)
(co]$(c) satisfies [E - K. - V~;,fE)](co~$(c) = 0.
(3.9)
In order to compute T, the T-matrix associated with (co]$(c), we note (3.9) is consistent with (3.4) but
11
+ in V(c, b)
E - hco)@l + d - V(b, c) E - i 0
V(b, c)(col$(c).
(3.10)
Hence, solving for the OP T-operator ?=as before we obtain
Comparing (3.11) and (3.6) we note that the two expressions are not identical. However, they become equal if we use the observation that the shift operator in the denominator of (3.1 l), which contributes the width and shift of the resonances associated with h(‘)(b), is smoothly energy dependent [5,6] :
236
AGAW
Hence, definition (3.8) of the OP satisfies the physical criterion, and will be adopted throughout with slight generalizations. The smoothness property (3.12) is important. It is guaranteed if an eigenfunction of K0 does not vary rapidly as a function of its eigenvalue in that region of space where V(c, b) is nonzero. Physically, it amounts to the assumption, already made, that there are no resonances or threshold effect of K0 in the energy region considered. B. The D WBA for Nucleon-Nucleus
Scattering
The complete hamiltonian (3.1) differs from (3.2) by the I’@, bc) matrix elements, which give rise to direct scattering. The philosophy underlying the DWBA is that these direct contributions are weak compared to the compoundnucleus scattering, and therefore could be treated to first order. This is indeed the case for the present model as we shall see below. In common use, the optical potential fits the elastic scattering. Hence, we should include the elastic portion of V@c, bc) in the OP so that, in first order, the DWBA matrix element for elastic scattering vanishes. We are thus motivated to decompose V’(bc, bc) as in (2.10): + &I’].
(3.13)
to appear in the two-potential
formula (1.3)
V(bc, bc) = (V(bc, bc)) + A[V(bc, bc)] = (V)
The “unperturbed” will be chosen to be
hamiltonian
(3.14)
H’ = K. + V(bc, bb) + V(bb, bc) + (V(bc, bc)).
The Schrodinger equation corresponding ~‘“VM4 ~o$b-d
+
W,
to (3.14) takes the form 4,W
+ VG WU4 +
Parallel to the treatment conditions
=
Waft,
(3.15)
= Wd
of (3.3) we have after introducing
the boundary
(3.16)
DWBA
Using the manipulations 7- = (E - &)
FOR
231
STRIPPING
as before, we readily obtain 1
E - K(J - (V)
+ iv
1 + w - Ko). E - K. - (V) + ii
(3.17) The propagator l/[E - K. - (V)] by its first Born term [5,6]
is diagonal in our basis. It is approximated
1
(3.18)
i - K. - (V)
Approximation (3.18) holds under the same conditions that (3.12) hold. The energy average of (3.17) is evaluated using (1.1). The OP (3.8) is only slightly modified in the present case. One easily obtains ,Y -
Ko
- Wo, W
,!T- hco’V4 - V@, c)E - ;
+ in V(c, b) 0
11
V(b, c,,) - (v)
(3.19)
. ~%lW~ = 0, so that the OP is defined by V&Y?@)= V(co, b)
E - h”)(b) + iZ - V(b, c) E - i
Vfh q,) +(v) 11
+ in V(c, b) 0
(3.20)
and (co]$(c), the OP wavefunction, satisfies (3.9) with (3.20). The OP T-matrix ?’ is obtained using (3.10) and (3.19) To = (E - Ko)-
1 E - K. - (V)
+ iq
(V)
. W,, , bl E - h”‘(b) + iZ - V(b, c)
+ (E - Ko)
Q E - K. + il
1 E - K. - (V)
+ iv
VCc,bl
(3.21)
238
AGASSI
Comparing T with the average of (3.17) we note that both expressions are not identical, as they should be. However, they become identical using (3.18) to zero order for the first term in (3.21), which amounts to using approximation t(c) of (3.14) in the first order term, and approximation (3.18) everywhere else. One needs also a variant of (3.12), namely,
We have just justified the choice (3.20) ofthe OP associated with the hamiltonian H’ (Eq. 3.14). Let us now turn to the two-potential formula (1.4) and choose II of (3.14) as the common unperturbed hamiltonian of the entrance and exit channels. This choice, as indicated above, is reasonable from the physical point of view, since it absorbs the multitude of the compound processes and the direct elastic scattering. Hence, we are led to the decomposition of (3.1): H = H’ + A[V]. Consequently, the DWBA form for the uueruge transition complete hamiltonian (3.1) will hold provided that :
(3.la) operator
T of the
1. The A[V](l/E - kI + iZ)A[V] term in (1.4) can be neglected. 2. The eigenfunction $(,5 -I- i1) of H’ can be replaced by q(E), the eigenfunction of the OP, for that portion of space where A[V] is nonzero (the internal region). Because of the special choice of H’, and the explicit model we assume, the answer to the first condition is affirmative. The norm of A[V] can be roughly estimated, and for Z large enough (about I MeV) this correction term is small [5,6]. In this estimate the number of open channels is assumed to be not exceedingly large. The validity of the second condition is almost obvious. Since (2.7) holds we have
The equality of T and ‘f@ + iZ) follows from the very choice of the OP. The smoothness property (3.12), on the other hand, implies that the distorted basis as well as the propagator are smooth in energy for small values of the coordinate. We note,however, that the OP involved in the DWBA is nor the one corresponding
DWBA FOR STRIPPING
239
to thefill hamiltonian (3.1) but rather to the hamiltonian (3.14), which is, in turn, close to the OP of (3.2). This does not correspond strictly to the phenomenological procedure, which uses the OP of the complete hamiltonian. However, the error made by this is of the order inherent to the DWBA. Secondly, the rough estimate of IId k’\l implies that the present analysis cannot justify the DWBA for highresolution experiments, say, of several keV.
IV.THE
(d,p) REACTION
The derivation of the DWBA in the stripping case follows a parallel route and technique as described in the previous section. We construct first the OP of nucleons and deuterons and eventually use again the two-potential formula to establish the DWBA form. Since we include now two continua (deuterons) in an Hilbert space, all terms of the two-body interation (2.4) will intervene and consequently the expressions obtained are more complex. The discussion is presented in order of increasing sophistication of the (d, p) scattering model. A. Pure Compound-Nucleus
(d, p)
The simplest model for the (d, p) reaction is assuming a pure compoundnucleus mechanism. In this model the incoming deuteron is at first trapped in a compound-nucleus state and later one particle leaks back to the continuum via a bound continuum coupling. Thus we consider first the model hamiltonian (in the notations of Eqs. 2.8, 2.9) H’ = I& + V(c, b) + V’(b, c) + V(d, b) + V(b, d).
(4.1)
The hamiltonian (4.1) can also lead to nucleon and deuteron scattering; however, only via the compound-nucleus phase. Adopting expansion (2.6) for the exact solution $, the Schrodinger equation (2.11) takes the form
or, introducing
the boundary conditions, we have from (4.2):
240
(4.3)
The exact (d,p) amplitude associated with (4.1) can be obtained using the previous method; we express all components in terms of $(b), solve for t+h(b), and then insert back into (4.3). The result is V(b, d). (4.4)
Comparing the denomination of (4.4) with its counterpart (3.5) we note that there are now two sources for shifts and widths of the resonances: the nucleon and deuteron couplings to the continuum. The energy average of (4.4) is correspondingly
Although the energy dependence of (4.4) may be strong, the average (4.5) is smooth. The counterpart of the smoothness assumption (3.12) is Ub, 4
1
1
V(d, b) z V(b, d)VM W. E - K. + itj E - K. + il
PW
In order for (4.6) to hold, we must exclude “deuteron resonances” of K. or threshold effects in the energy region considered. As in Section III.A, the smoothness property (4.6) is essential for the construction of the OP. The average T-operator (4.5) can be further examined by introducing statistical assumptions. The smoothness properties (4.6) and (3.12) imply that the widths and shifts of the compound-nucleus resonances are smooth in the energy interval considered. Now, since I contains a large number of these resonances, and assuming their shift is small compared to E, (4.5) can be approximated by (d is the average spacing between ho(b) eigenvalues Ei and ( ) denotes average
DWBA
FOR
241
STRIPPING
over states ii))
(4.7) We now assume the matrix elements in (4.7) are random variables without any correlation, i.e., that M.8)
C’J&Eb’J&‘I>izz 0.
Assumption (4.8) is reasonable since deuteron and nucleon capture are different processes, and there are no common doorway states of nucleon and deuteron scattering. Recently [9], however, there has been detected a case where such a common doorway state exists, so that (4.8) is not valid. We exclude such situations. Assumption (4.8) is consistent with the statistical assumptions required for the construction of the OPl [5, 61. Using (4.8) and the approximation (4.7), the amplitude averages out to zero. We consider now the deuteron and nucleon scattering associated with (4.1). These are purely compound-nucleus reactions and will serve to define the OP. From (4.2) and (4.3) we obtain the exact amplitude for deuteron and nucleon scattering, namely : T&d, = w,
!4 E - h(‘)(b) - V(b, c)
The (do/$(d) component
1 1 E - K. + iv
V@, 4,
(4.9)
V(b, c).
(4.10)
V(c, bl
for deuteron scattering and the (col+(c) component for
242
AGASSI
nucleon scattering satisfy, from (4.2), 1
E - K. - V(d, b) E - h(‘)(b) - V(b, c)
’ UC, @ E - K. + irj
1
E - K. - V(c,
Equations (4.11) and (4.12) generalize (3.7). Using the method of Section III, it is suggestive to define the OP as J,‘:$Wl = Wo 74
1 E - h(‘)(b) + il - V(b c) ’ E-K0 + V(b, d)
V@/(c) = V(co, b)
V(b, do), (4.13)
1 + iv
J’k bl
’ W, W E - K. + iv
1 1
E - h(‘)(b) + il - V(b d) ’ E-Ko+iv + V(b, c)
W, co), (4.14) W, 4
’ Vk, W E - K. + iv
where Viz/(d) is the deuteron optical potentia1 and V$‘(c) is the nucleon proton potential. We note that due to the smoothness properties (3.12) and (4.6), one can omit the iZ term in the shift operators of the denominators in (4.13) and (4.14). In order to justify definitions (4.13) and (4.14) we have to prove that the T-operator T associated with the OP deuteron wavefunction (doJt,h(d),
DWBA
FOR
243
STRIPPING
equals the energy average of the exact amplitude (4.9). A similar statement may be made for nucleons. We show it for deuterons, but it is equally easy to prove for nucleons. Equation (4.15) is consistent with the first equation in (4.3) and
(4.16)
Hence solving for $(d), using the method in Section III, we obtain 1
c,d, = V(d, b) E - h(‘)(b) + iZ - V(b, c)
1 E - K. + iv
V(b, d). (4.17) UG 4
We see, by comparing T&d, to the average of (4.9), that both are indeed equal within the standard smoothness property (4.6). B. Model for (d, p) with Compound-Nucleus Scattering
and Direct Nucleon and Deuteron
The next stage in the sophistication of the model is to switch on the V(bc, bc) interactions. In such a model the (d, p) reaction can still proceed only via the compound-nucleus phase, and will therefore vanish again on the average as we shall see. However, the optical potentials for nucleon and deuteron scattering will be modified. Since this model already includes inelastic nucleon and deuteron scattering, it is the stage where the DWBA form for these processes should be considered. The following discussion resembles the one given in Section 1V.A and therefore is reviewed briefly. We consider first the hamiltonian (notations of Eq. 2.10): H” = K. + V(c, b) + V(b, c) + V(d, b) + V(b, d) + (V(c, c)) + (V(d, d)).
(4.18)
In (4.18) we include only the diagonal portions of the nucleon and deuteron scattering to meet the common meaning of the optical potentials. The Schrodinger
244
AGASSI
equation associated with (4.18) takes the form
We examine just the optical potentials for nucleon and deuteron scattering that are induced by (4.19). For deuteron scattering one obtains from (4.19)
whereas, for nucleon scattering, introducing one obtains
the boundary conditions
into (4.19)
Using the manipulation employed previously and (4.20) (4.21) we obtain for deuteron scattering, for instance, the exact T-amplitude
‘E - K. - (V(d,d))
CE -
Ko).
(4.22)
DWBA
FOR
245
STRIPPING
Equation (4.22) has a similar form to (3.17). A similar formula holds for r,c,, the nucleon scattering T-matrix, where (I’(& d)) in (4.22) is replaced by (k’(c, c)). The exact (do\+(d) and (cJ+(c) satisfy
D2 = E - h”)(b) - V(b, c)
Q E - K. + iv
V(G bl
1 E - K. - (V(d, d)) + iv
W
&
(4.23)
1
E - K. - V(c, b)+V(b,
c) - (V(c, c)) (colt
2
=
0,
Consequently the OP is taken to be of the form:
(4.24)
We shall omit the proof that the OP (4.24) indeed reproduces the average of the exact T-operators, like (4.22). The proof runs along the same lines that were presented in Section 1V.A. Here, however, we need in addition to approximate the deuteron propagator to the first Born term, i,e., as in (3.22): 1 E-Ko-(V(d,d))=m+E-K.
1
L<
(4.25)
Wt dL&. 0
We should point out here that t(d) of Eq. (4.20) satisfies an ordinary Lippman-Schwinger equation, just as for nucleon scattering. This follows because the hamiltonian of t(d) was chosen to scatter on/y deuterons elastically. Hence, the usual difficulties of the three-body problems are absent, and we could infer (4.25) under the same conditions that (3.21) holds, i.e., their lack of resonances of K. or threshold effects.
246
AGASSI
For the (d, p) reaction associated with the hamiltonian the exact amplitude T(d,p)
(4.18) we obtain for
1
=
@
-
E - K. - (V(c, c)) + iv UC,W
&)
1 E - h(‘)(b) - V(b, c)
11 E - K. - (V(c, c)) +
iqW, 4
1
. V, 4 E - K. - (V(d, d)) + iv (E - Koh
(4.26)
The amplitude (4.26), when we use for simplicity (4.25) and the smoothness properties will average out to zero under assumption (4.8). We are now in the position to consider inelustic nucleon and deuteron scattering for the model Ii”
= H” + A(V[c, c]) + A(V[d, d]).
(4.27)
fl”’ cannot induce rearrangements as we saw. If we use the two-potential formula (1.4) for H”‘, with the decomposition (4.27), we recover the DWBA form for the average T-matrix amplitude provided the interaction in (4.27) can be taken to the first order only. The other necessary condition for the DWBA to hold is similar to that discussed in Section 1II.B and is automatically fulfilled because the present discussion is completely parallel to the previous case. The new features in (4.27) are the matrix elements A( V[d, d]) which have to be estimated, or rather 11 A[V(d, q]i]. Suppose the V(c, c) part of the interaction (2.3) is modified such that the deuteron is approximated by a point object. Then, the motion of the point deuteron will be essentially governed by Ho(c). Therefore, it is effectively again a nucleon-nucleus situation. The previous estimates [5, 61 will hold for 11V(d, d)l[, so the DWBA form holds. The finite range of the deuteron @(d) is represented, effectively, by a point deuteron wave packet over a finite range of energy. Hence I]A V(d, d)li will increase somewhat, but definitely not in the order of magnitude. Hence, the averaging interval Z valid for nucleon scattering is certainly large enough to assure the convergence of the Born series for the interaction in (4.27). C. The D WBA for the Stripping Process
The final part of the discussion is to switch on the V(c, d) matrix elements in (2.4). This part of the force induces, as remarked in Section II, a (d, p) reaction
DWBA
FOR
STRIPPING
247
via a direct process, i.e., one nucleon scatters directly and the other is trapped. This direct mechanism is the common stripping process and we shall indeed recover it for the average T-operator. Thus we consider the full hamiltonian: H = H” + A[V(c, c)] + A[ V’(d, d)] + V(d, c) + V(c, d) = H”’ -I- W.
(4.28)
The hamiltonian (4.28) can induce (d, p), nucleon, and deuteron scattering. Consequently there are three DWBA formulas to consider if the decomposition (4.28) is used in the two-potential formula (1.4). All these DWBA formulas hold provided 1. Only the first order W-term in (1.4) can be retained. 2. The appropriate eigenfunctions $(,!Z -!- i1) of H”’ could be replaced by q(E), the OP wavefunctions in the internal region. The second assertion concerning the wavefunction holds according to the arguments given in Sections 1II.B and 1V.B. Here, again, the analogy in the construction of the OP and smoothness properties render automatically the conclusions deduced for the simple nucleon-nucleus case. Next, in order to prove the first assertion, we need an estimate of 11WI1 (Eq. 4.28). This will be achieved using the following qualitative argument. Suppose the interaction I’ is of very short range. Then, a V(d, c) matrix element will involve an integration over a jinite portion of space of a product of four orbitals, three of them scattering orbitals. A I’(& k) matrix element involves, however, integration over the same portion of space of two scattering orbitals and two bound ones. Since scattering orbitals oscillate more strongly than bound orbitals, and their amplitude is of the order unity in the internal region (off resonance) it follows that, if the number of open channels is not exceedingly large,
IIJ’h W 2 IIW 411.
(4.29)
The finite range of the two-body force cannot change (4.29). Hence, since we gave in Section 1V.B arguments that 11v(k, bc)il is small compared to Z, we conclude that
IIWII <
(4.30)
Equation (4.30) is the affirmative answer to the first assertion [5,6]. Returning now to the (d, p) reaction, and recalling that H” gives a vanishing contribution to the rearrangement process, we presume that :
=
Gripping
g
<$‘-kF’k 4l$‘+W
(4.31)
since v(c, d) is the only term of the interaction (2.4) that survives on the average. We note further that there are two sources for V(d, c) matrix elements. The first
248
AGASSI
comes from V(& !K) elements (Eq. 2.8) and the second from V(cc, !IC) elements. The first is probably small if theinert-core assumption [1] is made and, furthermore, does not correspond to a stripping picture. The second contribution does correspond to a stripping picture and is probably the dominant term. Furthermore one should remark that the OP that intervene do not correspond to the fill hamiltonian but rather to H”. This is in variance with the phenomenological procedures that use OP referring to the full hamiltonian; however the error committed is within the error of the DWBA. The rough estimates of the neglected term 11wii do not allow one to justify good resolution experiments of a few tens of keV. It is instructive to compare (4.31) with the standard result [lo] Gripping
g
(6,
k$D)
(4.32)
where C& and &. are OP wavefunctions of the deuteron and proton, respectively. The Pauli Principle is only partly taken into account, and the choice of the OP as the generating potentials of c&., 6,, is physically reasonable. We have shown that, within an explicit model, that DWBA has one term is a consequence of the statistical assumption (4.8) and VP” is replaced by V(c, d) which is a portion of the interaction localized in the internal region. This portion of the interaction mainly scatters directly one nucleon and traps the other nucleon of the deuteron. Antisymmetrization is included throughout the discussion, and $,, $, SIUIUU be the OP wavefunctions. We are unable however to relate the V&,,(d) to V&(c) of (4.24) without a numerical examination of the expressions, neither to establish a connection with phenomenological OP. From the discussion, the analogy between the simple nucleon-nucleus and the (d, p) scatterings is exhibited.
REFERENCES 1. I. E. MCCARTHY, “Introduction to Nuclear Theory,” refs. in Chap. 13. Wiley, New York, 1969. 2. A. JAFFE AND A. RINAT (REINER), Phys. Rev. 161 (1967). 935, 3. W. F. JUNKIN AND F. VILLAR, Ann. P/zys. N. Y. 45 ( 1967), 93 ; ibid, 51 ( l969), 68 : H. FESHBACH. Ann. Phys. N. Y. 19 (1962). 287. 4. D. AGASSI. Am. P/IKV. 65 (1971). 212. 5. C. MAHAIJ~ ANII H. WEIDENMULLER. ‘Shell Model Approach to Nuclear Reactions.” NorthHolland* Amsterdam, 1969. 6. J. HUFNER, C. MAHAUX, AND H. WEIDENMULLER, NucI. Phys. Al05 (1967), 489. 7. G. E. BROWN, “Unified Theory of Nuclear Models.” North-Holland, Amsterdam, 1967, 8. R. NEWTON, “Theory of Scattering of Waves and Particles.” McGraw-Hill, New York, 1969. 9. A. M. LANE, preprint (1970). IO. L. R. DODD AND K. R. GREIDER. Phys. Rev. 146 (1966), 67.
The
OF PHYSICS
DWBA
65. 227-248 (I9711
for Stripping
in the Shell Model
Approach
to Reactions
DAN AGASSI Department
of Nuclear
Ph.vsics,
Weizmann
Institute
of Science,
Rehovot,
Israel
Received July 21, 1970
The DWBA form for the energy-averaged stripping amplitude is derived in the framework of the shell-model approach to reactions. This derivation generalizes for a rearrangement process, a previous work of Mahaux and Weidenmuller which concerns nucleon-nucleus scattering. The present derivation follows a similar line of argumentation. The required statistical assumptions and optical potentials are stated. Antisymmetry is included in the discussion.
I. INTRODUCTION
Stripping is a direct-reaction mechanism for (d, p) scattering, and is therefore analyzed using the DWBA approach [l]. Although much used, this approximation has received little theoretical attention. It has been discussed for a three-body model [2] and to some extent for the general many-body case [3]. It is clear that three-body difficulties form an additional source of complications in any attempt to analyze the (d, p) reaction. A merit of the so-called shell-model approach to reactions is that the many-body complications and the three-body difficulties separate out in a simple way. In fact, once we can describe a deuteron scattering off the average well generated by the target particles, the deuteron-nucleus scattering reduces, effectively, to a nucleon-nucleus problem [4]. Since the DWBA has been already analyzed for nucleon-nucleus scattering [5,6], we are thus motivated to study the DWBA for the stripping within the framework of the shell-model approach. The (d, p) process provides the simplest example of rearrangement collision. In the usual shell-model treatments of reaction [5], rearrangement processes are excluded. In this sense, the present work generalizes the previous ones. Because the underlying extension of the model for a twocontinuum has a similar structure to the one-continuum model [4], the analysis of nucleon-nucleus and deuteron-nucleus direct scattering turn out to be similar. Hence, two related points are demonstrated below, i.e., the similarity between a nonrearrangement and rearrangement direct reaction, and that a stripping picture does emerge from a microscopic model for a direct-rearrangement collision. 227
228
AGASSl
The discussion below may also be viewed as an application of Feshbach’s [3] philosophy to the shell-model approach to reactions in the case of rearrangements. The DWBA is fabricated to apply when a direct-reaction mechanism is dominating the scattering. The cross section resulting from such a mechanism is smoothly energy dependent since it describes a short time-delay process. We adopt [6] the attitude of associating the energy-uueruged T-matrix element with this smoothly energy-dependent process. Hence, “derivation” of the DWBA in a dynamic model amounts to proving that the energy average exact T-matrix element can be recast in the usual DWBA form, on the assumption that the “theoretical” optical potential is properly defined. This kind of analysis cannot justify the common phenomenological means of fitting first elastic scattering data and then computing inelastic transitions. Rather, it has been shown [6] that the DWBA procedure and the optical potential (OP) associated with it, is a consistent approximation wihin the shell-model approach. The relevance of such an analysis is to give some confidence to the legitimacy of the phenomenological procedures. It is this philosophy which underlines the present work. We show that using approximations and qualitative arguments of the type needed for the nucleonscattering case [6], the energy average T-matrix element for the (d, p) process can be expressed in the DWBA form and corresponds to a stripping picture. We are unable, however, to connect the OP defined below with the phenomenological ones, nor can we establish the wellknown relation [3] between the nucleon and deuteron optical potentials. We shali exclude also any complicating factors like doorway states and strongly-coupled channels. In the following, energy averages of the T-matrix are obtained by the common device (T(E))
= T(E + iI),
(I.11
where Z is the averaging interval. The averaging interval is assumed to include a host of compound nucleus resonances. The Z-dependence of the results can usually be neglected if the direct reaction mechanism is correct and there are, for example, no doorway states in the energy region considered. The order of magnitude of Z is 1 MeV for nucleon-nucleus scattering [5]. The DWBA form will be derived using the two-potential scattering formula which holds also for rearrangement collisions [8]. If we consider two decompositions of the complete hamiltonian H, H = Ha + HA = Hb + H;
(1.2)
where the “u” decomposition is appropriate for the incoming channel and the “b” decomposition for the outgoing channel (Ha, Hb include distortion) H&J;*) = EC);*)
H&i*’
= E&*‘,
(1.3)
DWBA FOR STRIPPING
then the T-matrix element for the transition
229
is given by the familiar formula
In order to make the presentation self-contained, we review briefly in Section II the shell-model approach using, however, the second quantization language [4]. We feel it is instructive to consider first the case of nucleon-nucleus scattering [5,6] in order to examine in the simpler situation the line of argumentation for the stripping DWBA. This is carried out in Section III, while in Section IV we proceed to analyze the (d, p) scattering. We make use of the Feshbach projection operator techniques in the formal manipulations.
II. THE SHELL-MODEL APPROACH OF NUCLEAR REACTIONS The basic idea underlying this approach is to make use of the distinction between the bound and the continuous single-particle orbital of the unperturbed, finite well, shell-model hamiltonian. We then truncate the complete Hilbert space of many-body states to include at most two particles in the continuum, approximate the bound states of the system to have no continuum components, and solve the Schrodinger equation in the model space just as in the usual shellmodel. In the present case, however, the final product is a system of coupled integral equations instead of an eigenvalue problem. Analogous to the situation for bound-state problems, the second quantization language seems to be convenient here. Consider the complete hamiltonian H in the second quantization form
where (c@~k’[$) is the nonantisymmetrized matrix element and the state indices run over bound and continuous distorted states of Ho. For the continuum states of Ho it is convenient to choose the outgoing boundary conditions [4]. Since we use second quantization throughout all derivations, antisymmetry is included automatically in all expressions. Using the distinction between the bound and continuum orbitals, the hamiltonian (2.1) can be decomposed in the following way H = Ho(b) + Ho(c) + V(b, b) + V(c, c) + W(b, c),
cv
230
AGASSI
where
(2.3)
Hqb, c) = v - vyb, b) - V(c, c).
We note from (2.3) that IV(b,c) is that portion of the interaction where at least one orbital is bound. If we denote by “b” and “c” bound and continuum orbitals, W(b, c) can be further decomposed : W(b, c) = V(bc, bb) + V(bb, bc) + V(cc, bb) + V(bb, cc) + V(bc, bc) + V(bc, cc) -t V(cc, bc),
(2.4)
where, for instance, (2.4a) The other terms in (2.4) have obvious meanings. The many-particle basis wavefunctions are chosen to be eigenfunctions of K. = Ho(b) + Ho(c) + V(b, b) + V(c, c) = h(‘)(b) + Ho(c) + V(c, c).
The model Hilbert space consists of three groups of antisymmetrized functions of K. :
(2.5)
eigen-
1. Those where all particles are in bound orbitals, denoted by [a(N + 2)) b space. 2. Those with one particle in a continuum orbital, denoted by jb(N + 1); E) -the c space, 3. Those with a deuteron in the continuum, denoted by [y(N); @g)(s))-the d space. -the
In the b and c subspaces, the V(c, c) interaction vanishes identically whereas in the d space it binds the two continuum particles to form the distorted deuteron. The three-body complications are confined in the construction of the @g)(s) of the d space, which satisfy [Ho(c) + v(cc)]@+q&) = E&q)(&)
(2Sa)
DWBA
FOR
STRIPPING
231
for the deuteron incoming wave. Since the hamiltonian (2Sa) cannot trap any particle in a bound state of the well, it corresponds to a problem which is easier than the usual three-body problem. in particular, if we are below the breakup threshold, (2Sa) is a one-channel scattering problem, and therefore has a Lippmann-Schwinger equivalent. According to the above subdivision of the Hilbert space, the exact solution of H within the model space can be decomposed to its projections on these orthogonal subspaces : ti = tw4 + $(c) + wb
WJ)
It should be realized that in e(b), I& c), and $(d) we lump together all channels with the same outgoing product, namely,
(2.6a)
For deuteron stripping, for instance, $(d) contains an incoming deuteron wave and outgoing scattered deuterons and $(c) contains only outgoing scattered nucleons. For nucleon scattering, on the other hand, we have an incoming nucleon wave in $(c) as well as scattered nucleons, and I&J) contains outgoing scattered deuterons. The T-operator corresponding to the various processes can be deduced easily by comparing (2.6) with the general definition [S]
where SC+) are the (distorted) unperturbed eigenstates of &, . In the following we shall manipulate with the T-operator of (2.7) defined in space of the distorted states. We should therefore bear in mind that in order to obtain the actual T-matrix element, there are additional phase factors [4] and contributions from the unperturbed hamiltonian. These, in turn, do not enter the discussion since they are assumed throughout to be smooth in energy. This assumption is equivalent to the hypotheses of the absence of narrow resonances of KO in the considered energy interval and no thresholds effects. These exclusions are assumed in the rest of the discussion.
232
AGASSI
In the context of the present work, the energy E of the system is assumed to be in the region where the spectrum of the compound nucleus hamiltonian /r(‘)(b) (Eq. 2.5) is very dense. We shall further assume that the “sea” of eigenstates of h(‘)(b) is “random,” i.e., there is no state or group of states that couple in a particular fashion with the continuum. In other words, we exclude doorway states or giant resonances in the energy interval considered [5,6]. The different components of the interaction (2.4) have a simple physical interpretation. The V(& M) term (and its conjugate) scatter a bound particle to the continuum or vice-versa. These matrix elements, therefore, generate processes where the incoming projectile is trapped in a bound orbital, so the whole system is temporarily bound and forms the compound system. Later, one particle leaks back to the continuum via the conjugate matrix elements. It is therefore expected, as we shall see, that they are important constituents of the OP [5] for nucleon scattering. In passing, we should note that the elements V(bc, M) couple also the d and c spaces, simply because one particle is added to the continuum. It will prove convenient to decompose these two functions of these matrix elements, Hence, we deime V’(bc, bb) = V(d, c) + V(c, b),
Gw
where, for instance, V(d, c) couples the d and c spaces. The decomposition (2.8) can be carried out by the use of projection operators of the d and c spaces. The V(cc, M) terms in (2.4) play for deuterons a similar role that the V(& M) play for nucleons. They are thus V(d, b) terms. They give rise to compoundnucleus scattering processes, and therefore will play a role in the construction of the deuteron OP. The V@c, bc) matrix elements scatter directly, elastically or inelastically, a one-continuum particle. Consequently these matrix elements are responsible for a direct reaction mechanism. We note that they can scatter directly either nucleons or deuterons and, as before, it will be convenient to disentangle these two functions. Hence we define J’(bc, bc) = V-(c,c) + V(d, d)
(2.9)
with obvious notations. We would like to further distinguish between the elastic and inelastic components in (2.9) using, again, the c and d projection operators. This separation will be needed since the OP usually fits the elastic scattering on the ground state, and elastic scattering, in turn, has some direct reaction portion in it. Denoting by (V(c, c)) the elastic scattering portion of V(c, c) and by A[V(c, c)] the inelastic part, we have
DWBA FOR STRIPPING
233
The last group of matrix elements, V(cc, k), in (2.4) scatter directly one particle in the continuum and one particle bound to the continuum and enter, therefore, the category V(d, c) without the need for projectors. They correspond, in particular, to a stripping process where one particle is believed to scatter directly while the other is trapped in the bound orbital. The standard equations in the shell-model approach are obtained when the $ of (2.6) is inserted in the Schrodinger equation (2.11)
E$ = E$.
In (2.11) we neglect those terms that couple with portions outside the model Hilbert space, and finally we project (2.11) into the various channels in the b, c and d spaces. In the following we shall apply this procedure repeatedly.
III. NUCLEON-NUCLEUS
SCATTERING
We review in this section the results obtained previously [5,6], recast in the notation of Section II, and with some slight modification-this in order to outline the generalization in the next section. For the one-continuum problem, the model space is restricted to the b and c spaces. Consequently the complete hamiltonian to be considered, in the notation of Eq. (2.4), is I-l = KO + V(bc, bb) + V(bb, bc) + V(bc, bc), = KO + V(c, b) + V(b, c) + V(bc, bc),
(3.1)
with the exact solution decomposition $ = WI
+ $64.
(3.la)
In the present case, there is no need for projection operators since only one particle is in the continuum. A. Modelfor
Pure Compound-Nucleus
Suppose for the moment deleted, i.e., we consider
Scattering Mechanism
that the V(bc, bc) matrix elements in (3.1) are
H’ = KO + V(c, b) + V(b, c).
(3.2)
The resulting hamiltonian will correspond to a model of purely compoundnucleus scattering. Such an interpretation follows since the V(bc, bb) matrix elements can scatter a particle only via an intermediate state where all particles are bound, i.e., via the compound-nucleus phase. Since we assume a high density
234
AGASI
of compound states in the energy region considered and of “random” character, such trapping-leaking processes will be numerous and each one equally important. The philosophy of the OP [I] is that it describes, in a simple fashion, the auerage effect of this host of processes. Hence, the OP is defined to reproduce the average of its T-operator [3]. Using the Schrodinger equation (2.11) for (3.2) and the model hamiltonian, we get (/r(‘)(b) defined in Eq. 2.5) ~‘“‘WU4 ~ovW
or, introducing
+ J,‘tb, +j&j
= &WI, (3.3)
+ W, O/W4 = W@L
the boundary conditions, M4 = E - $,jtbJUb~ 4iW~
wo - WM
= 0, (3.4)
4w = $0) + E - i.
+ $c>
W@J-
We solve first (3.3) for the exact T-operator, then take its energy average, and finally define the OP in such a way that it reproduces the average T. The exact T is obtained by solving (3.4) first for $(b) and then obtaining 1,9(c).By comparing the obtained $(c) with (2.7) we readily recover the exact T-operator: E - h(‘)(b) - V(b, c)
11
’ V(c, b) E - K. + iv
V(b, c).
(3.5)
The T-operator in (3.5) represents the full T-matrix. Its energy dependence is strong if the shift term in the denominator of (3.5) is small compared to the spacing between the eigenvalues of h(‘)(b). The smooth average T-operator, (T), is E - h(‘)(b) + iI - V(b, c)E - ;
+ iI J’k 4 0
11
W, 4.
(3.6)
There are two kinds of OP that could be defined: The first, referred to as OPl, reproduces the jiilj T-operator (3.6) whereas the second, OP2, reproduces (3.6) only for the elastic scattering of a particular channel co. The OP2 differs from OPl in that it takes into account the effect of all the other open channels except co and, in fact, corresponds to the phenomenological OP. The OPI, however, is the one to be used in a coupled-channel approach [5,6]. Both OP are expected to resemble one another provided the open channels are not too numerous, and that the couplings between the particular channel co (usually the ground state) and the other open channels is “weak.” In that case, the coupledchannels system yields the DWBA in first order. These conditions are probably
DWBA
FOR
235
STRIPPING
satisfied if projectile energy is low (say 10 MeV) and the target is not deformed. The condition of weakly coupled channels is basic for the rest of the discussion ; otherwise a DWBA form is somewhat artificial. However, the restriction that the open channels be not numerous can be relaxed at the cost of slightly complicating the expressions. We shall proceed with OP2, hereafter denoted by OP. This generalizes slightly the original derivation [5, 61 which has used the OPl. The formal steps, nevertheless, are similar. From (3.3) we note that (cJ$(c), the cO-component in e(c), satisfies E -
K.
-
1
V(co,b)
E - h”)(b) - V(b, c)[Q/(E - &)]V(c,
b)
Vh
co)
I
x (co~l+b(c)= 0.
(3.7)
In (3.7), V(co, b) is the co-projection of the V(c, b) interaction, whereas Q projects into all open channels except co. Following Feshbach [3], the OP is defined V$!(E) = V(co, b)
E - h(‘)(b) - V(b, c) E - i
11
+ in V(c, b) + iZ 0
so the OP wavefunction
W,
co),
(3.8)
(co]$(c) satisfies [E - K. - V~;,fE)](co~$(c) = 0.
(3.9)
In order to compute T, the T-matrix associated with (co]$(c), we note (3.9) is consistent with (3.4) but
11
+ in V(c, b)
E - hco)@l + d - V(b, c) E - i 0
V(b, c)(col$(c).
(3.10)
Hence, solving for the OP T-operator ?=as before we obtain
Comparing (3.11) and (3.6) we note that the two expressions are not identical. However, they become equal if we use the observation that the shift operator in the denominator of (3.1 l), which contributes the width and shift of the resonances associated with h(‘)(b), is smoothly energy dependent [5,6] :
236
AGAW
Hence, definition (3.8) of the OP satisfies the physical criterion, and will be adopted throughout with slight generalizations. The smoothness property (3.12) is important. It is guaranteed if an eigenfunction of K0 does not vary rapidly as a function of its eigenvalue in that region of space where V(c, b) is nonzero. Physically, it amounts to the assumption, already made, that there are no resonances or threshold effect of K0 in the energy region considered. B. The D WBA for Nucleon-Nucleus
Scattering
The complete hamiltonian (3.1) differs from (3.2) by the I’@, bc) matrix elements, which give rise to direct scattering. The philosophy underlying the DWBA is that these direct contributions are weak compared to the compoundnucleus scattering, and therefore could be treated to first order. This is indeed the case for the present model as we shall see below. In common use, the optical potential fits the elastic scattering. Hence, we should include the elastic portion of V@c, bc) in the OP so that, in first order, the DWBA matrix element for elastic scattering vanishes. We are thus motivated to decompose V’(bc, bc) as in (2.10): + &I’].
(3.13)
to appear in the two-potential
formula (1.3)
V(bc, bc) = (V(bc, bc)) + A[V(bc, bc)] = (V)
The “unperturbed” will be chosen to be
hamiltonian
(3.14)
H’ = K. + V(bc, bb) + V(bb, bc) + (V(bc, bc)).
The Schrodinger equation corresponding ~‘“VM4 ~o$b-d
+
W,
to (3.14) takes the form 4,W
+ VG WU4 +
Parallel to the treatment conditions
=
Waft,
(3.15)
= Wd
of (3.3) we have after introducing
the boundary
(3.16)
DWBA
Using the manipulations 7- = (E - &)
FOR
231
STRIPPING
as before, we readily obtain 1
E - K(J - (V)
+ iv
1 + w - Ko). E - K. - (V) + ii
(3.17) The propagator l/[E - K. - (V)] by its first Born term [5,6]
is diagonal in our basis. It is approximated
1
(3.18)
i - K. - (V)
Approximation (3.18) holds under the same conditions that (3.12) hold. The energy average of (3.17) is evaluated using (1.1). The OP (3.8) is only slightly modified in the present case. One easily obtains ,Y -
Ko
- Wo, W
,!T- hco’V4 - V@, c)E - ;
+ in V(c, b) 0
11
V(b, c,,) - (v)
(3.19)
. ~%lW~ = 0, so that the OP is defined by V&Y?@)= V(co, b)
E - h”)(b) + iZ - V(b, c) E - i
Vfh q,) +(v) 11
+ in V(c, b) 0
(3.20)
and (co]$(c), the OP wavefunction, satisfies (3.9) with (3.20). The OP T-matrix ?’ is obtained using (3.10) and (3.19) To = (E - Ko)-
1 E - K. - (V)
+ iq
(V)
. W,, , bl E - h”‘(b) + iZ - V(b, c)
+ (E - Ko)
Q E - K. + il
1 E - K. - (V)
+ iv
VCc,bl
(3.21)
238
AGASSI
Comparing T with the average of (3.17) we note that both expressions are not identical, as they should be. However, they become identical using (3.18) to zero order for the first term in (3.21), which amounts to using approximation t(c) of (3.14) in the first order term, and approximation (3.18) everywhere else. One needs also a variant of (3.12), namely,
We have just justified the choice (3.20) ofthe OP associated with the hamiltonian H’ (Eq. 3.14). Let us now turn to the two-potential formula (1.4) and choose II of (3.14) as the common unperturbed hamiltonian of the entrance and exit channels. This choice, as indicated above, is reasonable from the physical point of view, since it absorbs the multitude of the compound processes and the direct elastic scattering. Hence, we are led to the decomposition of (3.1): H = H’ + A[V]. Consequently, the DWBA form for the uueruge transition complete hamiltonian (3.1) will hold provided that :
(3.la) operator
T of the
1. The A[V](l/E - kI + iZ)A[V] term in (1.4) can be neglected. 2. The eigenfunction $(,5 -I- i1) of H’ can be replaced by q(E), the eigenfunction of the OP, for that portion of space where A[V] is nonzero (the internal region). Because of the special choice of H’, and the explicit model we assume, the answer to the first condition is affirmative. The norm of A[V] can be roughly estimated, and for Z large enough (about I MeV) this correction term is small [5,6]. In this estimate the number of open channels is assumed to be not exceedingly large. The validity of the second condition is almost obvious. Since (2.7) holds we have
The equality of T and ‘f@ + iZ) follows from the very choice of the OP. The smoothness property (3.12), on the other hand, implies that the distorted basis as well as the propagator are smooth in energy for small values of the coordinate. We note,however, that the OP involved in the DWBA is nor the one corresponding
DWBA FOR STRIPPING
239
to thefill hamiltonian (3.1) but rather to the hamiltonian (3.14), which is, in turn, close to the OP of (3.2). This does not correspond strictly to the phenomenological procedure, which uses the OP of the complete hamiltonian. However, the error made by this is of the order inherent to the DWBA. Secondly, the rough estimate of IId k’\l implies that the present analysis cannot justify the DWBA for highresolution experiments, say, of several keV.
IV.THE
(d,p) REACTION
The derivation of the DWBA in the stripping case follows a parallel route and technique as described in the previous section. We construct first the OP of nucleons and deuterons and eventually use again the two-potential formula to establish the DWBA form. Since we include now two continua (deuterons) in an Hilbert space, all terms of the two-body interation (2.4) will intervene and consequently the expressions obtained are more complex. The discussion is presented in order of increasing sophistication of the (d, p) scattering model. A. Pure Compound-Nucleus
(d, p)
The simplest model for the (d, p) reaction is assuming a pure compoundnucleus mechanism. In this model the incoming deuteron is at first trapped in a compound-nucleus state and later one particle leaks back to the continuum via a bound continuum coupling. Thus we consider first the model hamiltonian (in the notations of Eqs. 2.8, 2.9) H’ = I& + V(c, b) + V’(b, c) + V(d, b) + V(b, d).
(4.1)
The hamiltonian (4.1) can also lead to nucleon and deuteron scattering; however, only via the compound-nucleus phase. Adopting expansion (2.6) for the exact solution $, the Schrodinger equation (2.11) takes the form
or, introducing
the boundary conditions, we have from (4.2):
240
(4.3)
The exact (d,p) amplitude associated with (4.1) can be obtained using the previous method; we express all components in terms of $(b), solve for t+h(b), and then insert back into (4.3). The result is V(b, d). (4.4)
Comparing the denomination of (4.4) with its counterpart (3.5) we note that there are now two sources for shifts and widths of the resonances: the nucleon and deuteron couplings to the continuum. The energy average of (4.4) is correspondingly
Although the energy dependence of (4.4) may be strong, the average (4.5) is smooth. The counterpart of the smoothness assumption (3.12) is Ub, 4
1
1
V(d, b) z V(b, d)VM W. E - K. + itj E - K. + il
PW
In order for (4.6) to hold, we must exclude “deuteron resonances” of K. or threshold effects in the energy region considered. As in Section III.A, the smoothness property (4.6) is essential for the construction of the OP. The average T-operator (4.5) can be further examined by introducing statistical assumptions. The smoothness properties (4.6) and (3.12) imply that the widths and shifts of the compound-nucleus resonances are smooth in the energy interval considered. Now, since I contains a large number of these resonances, and assuming their shift is small compared to E, (4.5) can be approximated by (d is the average spacing between ho(b) eigenvalues Ei and ( ) denotes average
DWBA
FOR
241
STRIPPING
over states ii))
(4.7) We now assume the matrix elements in (4.7) are random variables without any correlation, i.e., that M.8)
C’J&Eb’J&‘I>izz 0.
Assumption (4.8) is reasonable since deuteron and nucleon capture are different processes, and there are no common doorway states of nucleon and deuteron scattering. Recently [9], however, there has been detected a case where such a common doorway state exists, so that (4.8) is not valid. We exclude such situations. Assumption (4.8) is consistent with the statistical assumptions required for the construction of the OPl [5, 61. Using (4.8) and the approximation (4.7), the amplitude averages out to zero. We consider now the deuteron and nucleon scattering associated with (4.1). These are purely compound-nucleus reactions and will serve to define the OP. From (4.2) and (4.3) we obtain the exact amplitude for deuteron and nucleon scattering, namely : T&d, = w,
!4 E - h(‘)(b) - V(b, c)
The (do/$(d) component
1 1 E - K. + iv
V@, 4,
(4.9)
V(b, c).
(4.10)
V(c, bl
for deuteron scattering and the (col+(c) component for
242
AGASSI
nucleon scattering satisfy, from (4.2), 1
E - K. - V(d, b) E - h(‘)(b) - V(b, c)
’ UC, @ E - K. + irj
1
E - K. - V(c,
Equations (4.11) and (4.12) generalize (3.7). Using the method of Section III, it is suggestive to define the OP as J,‘:$Wl = Wo 74
1 E - h(‘)(b) + il - V(b c) ’ E-K0 + V(b, d)
V@/(c) = V(co, b)
V(b, do), (4.13)
1 + iv
J’k bl
’ W, W E - K. + iv
1 1
E - h(‘)(b) + il - V(b d) ’ E-Ko+iv + V(b, c)
W, co), (4.14) W, 4
’ Vk, W E - K. + iv
where Viz/(d) is the deuteron optical potentia1 and V$‘(c) is the nucleon proton potential. We note that due to the smoothness properties (3.12) and (4.6), one can omit the iZ term in the shift operators of the denominators in (4.13) and (4.14). In order to justify definitions (4.13) and (4.14) we have to prove that the T-operator T associated with the OP deuteron wavefunction (doJt,h(d),
DWBA
FOR
243
STRIPPING
equals the energy average of the exact amplitude (4.9). A similar statement may be made for nucleons. We show it for deuterons, but it is equally easy to prove for nucleons. Equation (4.15) is consistent with the first equation in (4.3) and
(4.16)
Hence solving for $(d), using the method in Section III, we obtain 1
c,d, = V(d, b) E - h(‘)(b) + iZ - V(b, c)
1 E - K. + iv
V(b, d). (4.17) UG 4
We see, by comparing T&d, to the average of (4.9), that both are indeed equal within the standard smoothness property (4.6). B. Model for (d, p) with Compound-Nucleus Scattering
and Direct Nucleon and Deuteron
The next stage in the sophistication of the model is to switch on the V(bc, bc) interactions. In such a model the (d, p) reaction can still proceed only via the compound-nucleus phase, and will therefore vanish again on the average as we shall see. However, the optical potentials for nucleon and deuteron scattering will be modified. Since this model already includes inelastic nucleon and deuteron scattering, it is the stage where the DWBA form for these processes should be considered. The following discussion resembles the one given in Section 1V.A and therefore is reviewed briefly. We consider first the hamiltonian (notations of Eq. 2.10): H” = K. + V(c, b) + V(b, c) + V(d, b) + V(b, d) + (V(c, c)) + (V(d, d)).
(4.18)
In (4.18) we include only the diagonal portions of the nucleon and deuteron scattering to meet the common meaning of the optical potentials. The Schrodinger
244
AGASSI
equation associated with (4.18) takes the form
We examine just the optical potentials for nucleon and deuteron scattering that are induced by (4.19). For deuteron scattering one obtains from (4.19)
whereas, for nucleon scattering, introducing one obtains
the boundary conditions
into (4.19)
Using the manipulation employed previously and (4.20) (4.21) we obtain for deuteron scattering, for instance, the exact T-amplitude
‘E - K. - (V(d,d))
CE -
Ko).
(4.22)
DWBA
FOR
245
STRIPPING
Equation (4.22) has a similar form to (3.17). A similar formula holds for r,c,, the nucleon scattering T-matrix, where (I’(& d)) in (4.22) is replaced by (k’(c, c)). The exact (do\+(d) and (cJ+(c) satisfy
D2 = E - h”)(b) - V(b, c)
Q E - K. + iv
V(G bl
1 E - K. - (V(d, d)) + iv
W
&
(4.23)
1
E - K. - V(c, b)+V(b,
c) - (V(c, c)) (colt
2
=
0,
Consequently the OP is taken to be of the form:
(4.24)
We shall omit the proof that the OP (4.24) indeed reproduces the average of the exact T-operators, like (4.22). The proof runs along the same lines that were presented in Section 1V.A. Here, however, we need in addition to approximate the deuteron propagator to the first Born term, i,e., as in (3.22): 1 E-Ko-(V(d,d))=m+E-K.
1
L<
(4.25)
Wt dL&. 0
We should point out here that t(d) of Eq. (4.20) satisfies an ordinary Lippman-Schwinger equation, just as for nucleon scattering. This follows because the hamiltonian of t(d) was chosen to scatter on/y deuterons elastically. Hence, the usual difficulties of the three-body problems are absent, and we could infer (4.25) under the same conditions that (3.21) holds, i.e., their lack of resonances of K. or threshold effects.
246
AGASSI
For the (d, p) reaction associated with the hamiltonian the exact amplitude T(d,p)
(4.18) we obtain for
1
=
@
-
E - K. - (V(c, c)) + iv UC,W
&)
1 E - h(‘)(b) - V(b, c)
11 E - K. - (V(c, c)) +
iqW, 4
1
. V, 4 E - K. - (V(d, d)) + iv (E - Koh
(4.26)
The amplitude (4.26), when we use for simplicity (4.25) and the smoothness properties will average out to zero under assumption (4.8). We are now in the position to consider inelustic nucleon and deuteron scattering for the model Ii”
= H” + A(V[c, c]) + A(V[d, d]).
(4.27)
fl”’ cannot induce rearrangements as we saw. If we use the two-potential formula (1.4) for H”‘, with the decomposition (4.27), we recover the DWBA form for the average T-matrix amplitude provided the interaction in (4.27) can be taken to the first order only. The other necessary condition for the DWBA to hold is similar to that discussed in Section 1II.B and is automatically fulfilled because the present discussion is completely parallel to the previous case. The new features in (4.27) are the matrix elements A( V[d, d]) which have to be estimated, or rather 11 A[V(d, q]i]. Suppose the V(c, c) part of the interaction (2.3) is modified such that the deuteron is approximated by a point object. Then, the motion of the point deuteron will be essentially governed by Ho(c). Therefore, it is effectively again a nucleon-nucleus situation. The previous estimates [5, 61 will hold for 11V(d, d)l[, so the DWBA form holds. The finite range of the deuteron @(d) is represented, effectively, by a point deuteron wave packet over a finite range of energy. Hence I]A V(d, d)li will increase somewhat, but definitely not in the order of magnitude. Hence, the averaging interval Z valid for nucleon scattering is certainly large enough to assure the convergence of the Born series for the interaction in (4.27). C. The D WBA for the Stripping Process
The final part of the discussion is to switch on the V(c, d) matrix elements in (2.4). This part of the force induces, as remarked in Section II, a (d, p) reaction
DWBA
FOR
STRIPPING
247
via a direct process, i.e., one nucleon scatters directly and the other is trapped. This direct mechanism is the common stripping process and we shall indeed recover it for the average T-operator. Thus we consider the full hamiltonian: H = H” + A[V(c, c)] + A[ V’(d, d)] + V(d, c) + V(c, d) = H”’ -I- W.
(4.28)
The hamiltonian (4.28) can induce (d, p), nucleon, and deuteron scattering. Consequently there are three DWBA formulas to consider if the decomposition (4.28) is used in the two-potential formula (1.4). All these DWBA formulas hold provided 1. Only the first order W-term in (1.4) can be retained. 2. The appropriate eigenfunctions $(,!Z -!- i1) of H”’ could be replaced by q(E), the OP wavefunctions in the internal region. The second assertion concerning the wavefunction holds according to the arguments given in Sections 1II.B and 1V.B. Here, again, the analogy in the construction of the OP and smoothness properties render automatically the conclusions deduced for the simple nucleon-nucleus case. Next, in order to prove the first assertion, we need an estimate of 11WI1 (Eq. 4.28). This will be achieved using the following qualitative argument. Suppose the interaction I’ is of very short range. Then, a V(d, c) matrix element will involve an integration over a jinite portion of space of a product of four orbitals, three of them scattering orbitals. A I’(& k) matrix element involves, however, integration over the same portion of space of two scattering orbitals and two bound ones. Since scattering orbitals oscillate more strongly than bound orbitals, and their amplitude is of the order unity in the internal region (off resonance) it follows that, if the number of open channels is not exceedingly large,
IIJ’h W 2 IIW 411.
(4.29)
The finite range of the two-body force cannot change (4.29). Hence, since we gave in Section 1V.B arguments that 11v(k, bc)il is small compared to Z, we conclude that
IIWII <
(4.30)
Equation (4.30) is the affirmative answer to the first assertion [5,6]. Returning now to the (d, p) reaction, and recalling that H” gives a vanishing contribution to the rearrangement process, we presume that :
=
Gripping
g
<$‘-kF’k 4l$‘+W
(4.31)
since v(c, d) is the only term of the interaction (2.4) that survives on the average. We note further that there are two sources for V(d, c) matrix elements. The first
248
AGASSI
comes from V(& !K) elements (Eq. 2.8) and the second from V(cc, !IC) elements. The first is probably small if theinert-core assumption [1] is made and, furthermore, does not correspond to a stripping picture. The second contribution does correspond to a stripping picture and is probably the dominant term. Furthermore one should remark that the OP that intervene do not correspond to the fill hamiltonian but rather to H”. This is in variance with the phenomenological procedures that use OP referring to the full hamiltonian; however the error committed is within the error of the DWBA. The rough estimates of the neglected term 11wii do not allow one to justify good resolution experiments of a few tens of keV. It is instructive to compare (4.31) with the standard result [lo] Gripping
g
(6,
k$D)
(4.32)
where C& and &. are OP wavefunctions of the deuteron and proton, respectively. The Pauli Principle is only partly taken into account, and the choice of the OP as the generating potentials of c&., 6,, is physically reasonable. We have shown that, within an explicit model, that DWBA has one term is a consequence of the statistical assumption (4.8) and VP” is replaced by V(c, d) which is a portion of the interaction localized in the internal region. This portion of the interaction mainly scatters directly one nucleon and traps the other nucleon of the deuteron. Antisymmetrization is included throughout the discussion, and $,, $, SIUIUU be the OP wavefunctions. We are unable however to relate the V&,,(d) to V&(c) of (4.24) without a numerical examination of the expressions, neither to establish a connection with phenomenological OP. From the discussion, the analogy between the simple nucleon-nucleus and the (d, p) scatterings is exhibited.
REFERENCES 1. I. E. MCCARTHY, “Introduction to Nuclear Theory,” refs. in Chap. 13. Wiley, New York, 1969. 2. A. JAFFE AND A. RINAT (REINER), Phys. Rev. 161 (1967). 935, 3. W. F. JUNKIN AND F. VILLAR, Ann. P/zys. N. Y. 45 ( 1967), 93 ; ibid, 51 ( l969), 68 : H. FESHBACH. Ann. Phys. N. Y. 19 (1962). 287. 4. D. AGASSI. Am. P/IKV. 65 (1971). 212. 5. C. MAHAIJ~ ANII H. WEIDENMULLER. ‘Shell Model Approach to Nuclear Reactions.” NorthHolland* Amsterdam, 1969. 6. J. HUFNER, C. MAHAUX, AND H. WEIDENMULLER, NucI. Phys. Al05 (1967), 489. 7. G. E. BROWN, “Unified Theory of Nuclear Models.” North-Holland, Amsterdam, 1967, 8. R. NEWTON, “Theory of Scattering of Waves and Particles.” McGraw-Hill, New York, 1969. 9. A. M. LANE, preprint (1970). IO. L. R. DODD AND K. R. GREIDER. Phys. Rev. 146 (1966), 67.