Analog resonances in isospin-forbidden deuteron channels

I

2*c

I

Nuclear Physics

A184 (1972) 303-320;

@

North-Holland

PubIishing

Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

ANALOG RESONANCES IN ISOSPIN-FORBIDDEN DEUTERON CHANNELS A. F. R. DE TOLEDO PIZA t Laboratory Massachusetts

for Nuclear Institute

Science and Department

of Technology,

Cambridge,

of Physics, Massachusetts

and Institute

de Fisica,

Universidade

de S&o Pa&o, SLZOPaula, Brasil

Received 25 May 1971 (Revised 13 December 1971) Abstract: The isospin-forbidden deuteron decay of isobaric analog resonances is examined and estimates of the contributions of a number of different deuteron escape mechanisms are given. The dominant contributions are found to be associated with the coupling of the deuteron to isospin-allowed proton channels. Non-statistical compound effects are small.

1. Introduction

This paper reports a study of the decay of isobaric analog resonances by deuteron emission. The presence of important analog resonant effects in deuteron channels was first revealed by the work of Hamburger ‘) in “‘Pb(d, p) cross sections near analog states in the 209Bi compound system. They were later confirmed by Stein “) who studied the inverse “‘Pb(p, d) reaction, and by Bernstein and Armstrong “) in other lead isotopes. The deuteron shares with the proton the property of being a charged object, and as such susceptible to the effects of the Coulomb forces. Notably, one must expect a negligible deuteron yield from compound nucleus de-excitation in view of the competition of barrierless neutron channels. The deuteronis, on the other hand, a composite object with a symmetric space-spin internal wave function, which implies a T = 0 character. As a result of this, the deuteron decay of the analog states is “isospin forbidden” to the same extent as neutron decays. In the deuteron case, however, transitions involving only the internal degrees of freedom of the projectile can be found that violate the isospin selection rule and that are likely to play an important role due to the rather loose internal structure of the deuteron. This opens the possibility of important isospin-violating processes which do not involve any rearrangement of the target nucleons, in contrast with the isospin-forbidden decays by emission of neutrons. The essential ingredient for this is the strong coupling-between the deuteron channel and some relevant stripping channels which are also strongly coupled to the analog state. Such coupling has been investigated in a series of papers by Tat This work is supported in part through funds provided by the Atomic Energy Commission under Contract AT(30-l)-2098. 303

304

A. F. R. DE TOLEDO

PIZA

mura and Coker 4*“)_ These papers have in common the idea of generating resonant behavior in the stripping amplitude by employing a resonance-distorted proton wave functionin an otherwise standard stripping DWBA calculation. The resonant distortion in the proton channel is then generated either by solving coupled Lane equations [ref. “)I or through specialization of a more comprehensive reaction formalism “). In neither case do deuteron widths of the analog states have to be explicitly introduced and the analog state effects reveal themselves directly in terms of the resulting energy dependence of the calculated stripping cross sections. The main purpose of this work is to explore in a more systema~c way to role and interplay of a range of different mechanisms, including the channel coupling mechanism of Tamura, in the analog resonant effects seen in deuteron channels. These mechanisms emerge in a rather direct way from a suitable speciahzation of a general formalism for analog state phenomena given earlier “). The relative importance of these mechanisms is most conveniently measured in terms of their respective contribution to the deuteron widths of the relevant analog states. A formulation along these lines has the intrinsic advantage of starting from an a priori symmetric treatment of all channels, so that special features relating, e.g., to isospin selection rules or to the composite character of the projectile become explicit in terms of the dynamic features which are fed into the formalism. The details of the procedure to be followed have been given before ‘). It consists essentially in seeking a consistent interpretation of the formalism in terms of available phenomeuologic~ material such as the optical model and the shell model. It turns out, in particular, that the actual values obtained for the deuteron widths of the analog states are found to be rather sensitive to the details of the phenomenological prescription used to evaluate nuclear amplitudes involving the deuteron channel (see sect. 3). This suggests then that one may use the observed analog resonant behavior of the stripping cross sections in a more demanding test of the phenomenological procedures. The contents of this paper are divided into three main sections. Sect. 2 deals with a general discussion of the structure of the analog state decay amplitudes, particularly in connection with the ~sosp~n-forbidde~ness of the deuteron channel. Sect. 3 presents resultsof n~erical estimates of the various con~butions to the deuteron width amplitudes, and finally the last section contains concluding remarks.

2. General discussion of escape ampritudes

The general expression for the channel width amplitudes of an (isolated) analog state, $$I = (AIc9P]#c”>, (2.1) can

be analysed in a variety of formally equivalent ways to exhibit explicitly the effects associated with the many classes of modes of excitation of the compound system in which it occurs. It is easy to show a~gebraic~ly [see ref. “)] that the channel state

ANALOG

RESONANCES

305

vector ]$‘)> in eq. (2.1) satisfies the equation

(where, as usual, HPP E PHP)

with the effective Hamiltonian

2+P= H+Hq

1

E+iI-H,,

qH>

(2.3)

occurring both in (2.1) and in (2.2); His the full Hamiltonian of the compound system, q is a projection operator selecting the modes not explicitly included in the channel state vector and I is essentially the width of an energy-averaging interval introduced in order to smooth out the finer structure associated with the spectrum of H4,; IA) represents the analog state and P is defined such that the identity operator is split into the sum of the three commuting projectors 1 = P+q+lA)
(2.4)

The use of eqs. (2.1)-(2.4) to evaluate the yAnow hinges on their interpretation within the framework of workable models of the relevant structura1 and scattering phenomena. In particular, eq. (2.2) can be immediately interpreted in terms of the current phenomenology of elastic and inelastic scattering processes (including particle-transfer reactions) +. If we let the open channel projector P include the several most important individual open channels PC, i.e., p = CL C Ix’“‘> = 5 &IX’“‘> = F I#>, eq. (2.2) can be written as a set of coupled equations 5 [E&r

-P,

c%Pc,]@)

= 0,

(24

which can be identified with the familiar many-channel optical-model coupled equations:

(2.6) The channel superscript in (2.5) (and preceding equations) indicates the c-channel incident wave boundary condition to be imposed on the channel state vector. In this way eq. (2.1) becomes ~2’ = 5 <~=@Ix$‘>(2.7) The terms with c’ # c in this equation correspond to so-called channel coupling contributions to the escape amplitude. They describe the escape amplitude, from the t For an extensive discussion of the following treatment see ref. ‘).

306

A. F. R. DE TOLEDO PIZA

analog state to channel c, proceeding virtually through channels c’ coupled to c by the non-diagonal terms in eq. (2.6). It is worth noting that each matrix element in the sum (2.7) splits further into two different contributions, namely one from H and one from the q-space part of the effective Hamiltonian j’t’ [see eq. (2.3)]. These have previously been called the direct and the compound contributions to the escape amplitude. While compound effects in the channel state vectors can be included eventually through the optical potentials introduced in eq. (2.6), they have to be dealt with explicitly, at least where escape transitions are concerned, in eq. (2.7). The evaluation of compound contributions to eq. (2.7) involves directly the more difficult problem of the spectral distribution of those compound nucleus modes which couple both to the analog state and to the channel state vectors. The contributions of a number of such modes have been considered in connection with isospin-allowed decays ‘). They have been found to be smaller by as much as two orders of magnitude as compared to direct contributions to the escape amplitude. 2.1. THE ANALOG STATE AND THE CHARGE RAISING INTERACTION

In order to make eq. (2.7) more explicit we now introduce the ansatz (2.10)

]A> = ,

where In) is the parent analog state and T, are the total charge lowering and raising operators respectively. We get y,)

=


F
fm) N

I

>

(2.11)

where both the direct and the compound width amplitudes have been rewritten in terms of the parent state and of the charge raising coupling interaction [H, T-1. This has been studied in detail in ref. 7), and the results given there can be summarized by writing (2.12) [H, T--j = [I+“, T_]f[T/, T_]+[:l/, T-1, where

Here the a and b are second-quantized operators, for protons and neutrons respectively. The first term of eq. (2.11) thus involves a one-body interaction which turns

ANALOG

RESONANCES

307

a parent state neutron into a proton. It includes, in general, dynamic effects of the proton-neutron mass difference and electromagnetic spin-orbit effects ?. The two-body terms involving VP and V,, account for the two-body Coulomb interaction and for a charge-dependent part of the nuclear interaction. The former is contained in VP, which makes proton-proton pairs out of proton-neutron pairs in the parent nucleus. As shown in refs. ‘, ‘), the effects of a charge-symmetric charge-dependent nuclear interaction can be included through an appropriate combination of terms like VP and V,. The latter gives rise to proton-neutron pairs when acting between neutronneutron pairs of the parent state. The analog state ansatz (2.10) and the resulting form (2.12) for the coupling interaction have some immediate consequences for the coupling of the analog states to deuteron channels. To see this we must only note that, since the interaction (2.12) must be symmetric in all particle labels, neither of the two-body terms can connect states of the interacting pair having different space-spin symmetry. Since either the initial or the final states for these interactions contain two nucleons in the same charge state, it follows that they can act between T = 1 pairs only ti. This implies that the third term of eq. (2.12), which alone gives rise to proton-neutron pairs when acting on the parent state [see eq. (2.1 I)], cannot cause a transition to a state having a deuteron (T = 0) and a residual nucleus. Thus except for contributions involving the small overlap between different open channels configuration to be discussed later, there is no direct coupling of the analog state to the deuteron channel itself. 3. The deuteron width amplitudes We proceed now to a more detailed discussion of eq. (2.11) in the special case in which the decay channel (c) is a deuteron channel. It is convenient for this purpose to work in configuration space and to introduce explicitly a set of open-channel projectors PC written in canonical form as

where the channel position vectors Y, include spin and isospin coordinates and the integration implies also a sum over these coordinates. The symbols 6, stand for the internal states of the two fragments in channel c ‘. In general, in constructing a set of operators (3.1) one is confronted with ambiguities (or with some freedom) relating to the lack of orthogonality between different channel configurations for small sept It is usefulto define o(l) so as to include the average Coulombfield of the target as well as an eventual charge-dependent nuclear field. These must then be duly subtracted from the full two-body interaction VP and J/,. tt This restriction has been explicitly obtained in ref. 9, for a nuclear charge-dependent chargesymmetric force. 5 For a general discussion of the technicalities relating to the construction of the projectors PC see ref. 7).

308

A. F. R. DE TOLEDO PIZA

arations between the fra~ents. We indicate for de~~it~ness in the appendix, an explicit formal realization of eq. (3.1) in which such freedom is used to minimize the distortion of the deuteron channel coming from the orthogonalization process. This, in particular, is consistent with the ~va~ua~o~ of the direct deuteron channel contribution discussed below. As was mentioned in sect. 2, however, it will be seen that effects of channel-overlapping amplitudes are negligible in the present context. With eq. (3.1) we can write the direct deuteron width amplitude in the standard ‘) form

where

is the so-called analog state form factor for channel c and

is the wave function in channel c when the system is fed by an incident wave in the deuteron channel. In particular q (dd)describes the elastic scattering of deuterons. The problem of obtaining y$i is thus generally separated in two parts which consist, respectively, in obtaining the analog state form factors (this being essentially a structure problem) and in obtaining the wave functions (3.3). The latter effort involves a coupled-channels problem that can either be solved numerically within the context of a phe~omenolo~ica~ mode1 [see eq. (2.6)], or simplism further by means of some approximation. Since in the present context the essential coupling occurs between the deuteron and nucleon channels, for which the bulk of the available phenomenological material is based on the use of the distorted-wave Born a~pro~mation, we shall choose the latter course and evaluate all channel coupling effects in this approximation. The compound amplitude, on the other hand, has the form

with

being justdeuteron escape amplitudes of the compound states q, En order to estimate the contribution (2.4) we use the fact that the relative simplicity both of the parent state and of the channel configurations will considerably restrict the class of compound foliations which can contribute to this attitude (doo~ay co~g~ations with respect to the parent and open-channel configurations) ‘). The more complicated excitations manifest themselves only by helping to regulate the actual spectral ~stribution of the doorway con~~ra~o~s. In simple cases one may thus consider in the summation in eq. (3.4) only a small number of doorway states E)i with

ANALOG

RESONANCES

309

average energies Ei and damping widths r/ :

(3.6)

Y

The yid’ are now expressions similar to (3.5) involving the doorways. The spectral distribution of the compound widths is described in an average way by the energy denominators of (3.6). The problem of picking the relevant doorway states out of q-space, on the other hand, has been repeatedly studied in the context of the damping of analog states ‘,lc). It is believed that the most important effects are due to the so called anti-analog states. Recently, isovector monopole excitations of the target have also been considered 79“). We shall here restrict ourselves to the former. 3.1. DIRECT CONTRIBUTION

OF THE DEUTERON

CHANNEL

The simplest contribution to the direct deuteron width amplitude (3.2) is that which involves the deuteron elastic scattering wave function and the deuteron channel form factor v:(R) = <711T+ T-/n)-‘. (3.7) Here R refers to the deuteron c.m. coordinates and spin and 0, stands for the internal states of the (undistorted) deuteron and of the target nucleus. The construction of these channel state vectors is sketched in the appendix, following the procedure used in refs. 6, ‘). Using eq. (A.2), (3.7) can be rewritten as @@df#,

T-lb>

=I

dR’F;(R, R’)(R’@,I[ff, T-]/n),

where

Here @,” and @,” are proton and neutron creation operators in configuration space and the kernel F,(R, R’) differs from a &function only to the extent that the twonucleon density,matrix in the parent state overlaps with the internal state of the deuteron. The kernel thus represents a distortion of the deuteron amplitude due to the exclusion principle near the target and will be ignored for the purposes of our discussion. We consider thus the form factor as written in eq. (3.8). This is just a twonucleon decay amplitude of the analog state projected onto the internal state of the deuteron Z+(Y). In terms of the analog state coupling interaction, we see that the possible contributions to this term have the typical structure shown in fig. la. They involve a one-particle transition to a state having non-zero overlap with the internal state of the deuteron and therefore give rise to a non-vanishing amplitude in the deuteron channel as defined here. More complicated contributions are of course also

310

A. F. R. DE TOLEDO PEA

easily extracted from this term. As an example, fig. 1b shows a contribution involving the anni~lation of parent-state ~o~elations in a process involving the two-body parts of the coupling interaction.

(a)

(bf

Fig. 1. Simple deuteron channel contr~bntions. Shading between parallel proton and neutron lines indicates projection onto the internal state of the dcuteron.

1

-.*4

I

0

IO

5

15

R (fm) Fig. 2. Radial deuteron channel form factor for the &+ analog state in 2ogBi.

We have evaluated the contribution of fig. la in the simplest case of a parent state consisting of two neutrons added into appropriate orbits to a purely closed-shell target. Aside from angular momentum coupling, the resulting form factor takes the form

where u1 and u2 are the two neutron orbits involved and r = r1 -P,; T/,(Y~) is the field causing the transition. Since the deuteron wave function is symmetric in rl and r2 and the field VC(rI) is of long range, this form factor is strongly reduced by the antisymmetry of the product of the two neutron orbits. It can be easily evaluated by expanding the integrand in harmonic oscillator wave functions and using the Moshinski

ANALOG

RESONANCES

311

transformation. The result obtained in the case of the J” = 3’ analog resonance in ‘*‘Bi is shown in fig. 2, where the radial part of the form factor is plotted. The Coulomb field of the target has been taken for V,(rl). A comparison of this form factor with the corresponding function which appears in connection with typical channel coupling contributions [see fig. 4) shows at once that the former is negligibly small. In fact, the escape amplitude obtained from it corresponds to widths in the eV range. This has, for our purposes, an important practical significance, since it guarantees a near orthogonality of the proton channels to the deuteron channel. When, in fact, unlike the formal prescription given in the appendix, all deuteron-target components are not removed from the proton channels, there will be some spurious overcounting of these components when the channel coupling amplitudes are obtained. Because their contribution is very small, however, the effects resulting from such an overcounting will be numerically unimportant. 3.2. DIRECT

CHANNEL

COUPLING

CONTRIBUTIONS

To first order in the channel coupling interaction [see eq. (2.5)] the channel wave functions tiy) (with c # d) can be written as

&)(r)

NN

ss

dv’ dRg,(v, u’)(p~)(R),

(3.9)

where gc(r, r’) is the Green function for channel c; the matrix element involves the channel coupling interaction and $‘,“’ is the deuteron elastic scattering wave function. Inserting (3.9) into (3.2) one gets channel coupling contribution to the direct width amplitude as J&p’ M

IS

dv

dR 5~*(r)
(3.10)

where the appropriate analog state form factor has been combined with the channel Green function to form the amplitude E?(r)

=I dY’Y~(~‘)g&‘, r).

(3.11)

With this grouping of factors, eq. (3.10) acquires the typical structure of standard DWBA transition amplitudes, which is in a form of a matrix element of the transition operator between two distorted waves. In the spirit of reinterpreting eq. (2.2) in terms of optical potentials [see eq. (2.6)], it is easy to see that, while @id’is an ordinary optical-model scattering wave function, the wave (3.11) is obtained as a solution to the inhomogeneous Schrddinger equation b-

6

- E,(F)-iw,,(r)]

t%*(r) = v:*(Y),

(3.12)

c with appropriate asymptotic boundary conditions. It can thus be simply interpreted as the virtual amplitude appearing in channel c as a result of its coupling to the ana-

312

A. F. R. DE TOLEDO

PIZA

log state. This coupling, in fact, is represented by the source term of the right-hand side of eq. (3.12). There are several comments to be made on the structure of the direct channel coupling contributions as given by eqs. (3.9)-(11) ?. We may note first that the simplest, and at the same time the most important contributions of the form (3.10) are clearly those involving nucleon (stripping) channels in the intermediate state (see fig. 3a). This, on the other hand, is just a statement of fact that the stripping channels are the most important non-elastic channels for deuteron-nucleus scattering. But this

k n

P

P ___x

n

n

(a)

(b)

Fig. 3. Direct channel coupling contribution involving proton channels. In (a) the virtual proton is represented by the double particle line and the two-body interaction is a stripping interaction. In (b) different configurations corresponding to important proton channels for the *“Pb+d reaction are shown explicitly.

result is also related, on the other hand, to the importance of the coupling of the analog state to such channels. Moreover, there are important differences between contributions involving proton and neutron stripping channels in the intermediate state. These differences come, on the one hand, from the nature of the charge raising interaction, eq . (2.12), which behaves differently in each of the two cases; and, on the other hand, from different Coulomb distortions on their respective phase spaces +?. We believe that these facts provide for an adequate framework in which the mechanism for isospin violation in the decay of the analog state by deuteron emission can be understood. The work done on the width amplitudes of allowed proton decays of analog states ‘) indicates clearly the dominance of the Coulomb terms in eq. (2.12). These terms will couple the analog state to proton channel intermediate states only. The larger Q-value associated with proton channels will also enhance their contribution over that of neutron analog channels. We can then see that the most important t An exact formal expression for ye cd)having the same structure as eq. (3.9) can be written by suitably modifying the definitions of the Green function and of the channel coupling interaction. With these modifications, the following discussion can be extended beyond the context of the distorted waves Born approximation implied in eqs. (3.9) to (3.11). tt E.g., the neutron analog channel of a given open proton channel is energetically forbidden (i.e., closed) in a medium or heavy nucleus.

ANALOG

RESONANCES

313

direct channel coupling contributions must be associated with intermediate states involving proton channels with large spectroscopic factors relative to the target in the deuteron channel. It is also important to note that, even in the most schematic case where one describes the target states as purely closed-shell states of the maximum parentage, there will be in general several proton channels that must be included in an evaluation of the direct-channel coupling amplitude. This is due to the fact that the parent state has two neutrons more than the target state for the deuteron channel, either of which can be emitted as a proton by the charge raising interaction. Whenever they occupy different single-particle orbits (as in the case of the “‘Pb$ d reaction, see fig. 3b) these possibilities give rise to different amplitudes. In general, we expect the most important analog resonance effects in (d, p) excitation functions to show in proton channels that have large analog resonance partial widths. From what has been said, we see that these are also the important intermediate state channels in the directchannel coupling contributions to the deuteron widths of the analog states. In view of the structure of eq. (3.10) for the corresponding deuteron width amplitudes, we may in particular combine part of the analog resonant contributions with the usual direct DWBA transition amplitude for a given (d, p) process and write

The first term of this expression has again the form of a standard DWBA amplitude involving, in the square brackets, a proton wave function suitably distorted by the resonance. This term is essentially the one studied by Tamura and Coker 4, “) in twochannel calculations of the analog resonances in deuteron stripping. To this transition amplitude, however, one must still in general add a purely resonant contribution involving important proton channels other than the final channel in the stripping process. Results of some numerical evaluations of eq. (3.10) for a few analog states in “‘Bi are given in table 1. They have been obtained using the zero-range approximation in its usual form 11) for the stripping interaction, with the neutron form factor set equal to the appropriate Woods-Saxon single-particle wave function bound by the observed neutron separation energy for 2‘*Pb. The only point involving less standard techniques is the calculation of the virtual proton wave emitted by the analog state. This has been done by solving eq. (3.12) with the analog state form factor (3.13) Here z)(~““‘)is the Coulomb field of 208Pb and U,,j describes the appropriate (bound) neutron orbital in the parent nucleus; Unrjhas also been taken to be a Woods-Saxon wave function with the correct separation energy. The remaining factor of (3.13) is

314

A. F. R. DE TOLEDO PIZA

just the ap~ro~ria~~ spin-angul~ function. It should be noted that the virtual proton wave has to be properly orthogon~ized to the analog state ‘) ?. The values obtained in this way for the direct channel coupling width amplitudes are typically of the order of I keV*. This is consistent with the experimental widths extracted from the data by Hamburger ‘). A comparison of the last two columns of table 1 shows, however, that the results are rather sensitive to changes in the deuteron optical parameters. In particular, the values listed in the last column were obtained TABLE1 amplitude for the &+ and 4’ resonances

Channel coupling deuteronSwidth

8’

I,2 3, 2 3, 3

IC H

190

1, 1

5+

4-

1Y B’

%’ 2I-

-0.532+0.252i -0.826+0.8931 O.lOl--0.05% 0.166-0.1883

%+ 1z

.&-

-0.453+0.2473

t+

z

1H

5-t E

1+ 3 $Q"

!,

-0.882f0.999i 0.512-0.192i 0.579-0.55Oi

;-

-0.724$0.2721'

g-

B+

-1.76

+1.68i

in z0gBi

-0.309 -0.513

+0.259i +0.88!%

0.0570-0.05001

0.102 -0.255 -0.543 0.254 0.371 -0.359 -1.13

-0.17% 3-0.2251‘ +0.944i -0.229i

-0.5751’ 3-0.3233 +1.7.%

The second column identifies the deuteron channel. The third and fourth columns identify the virtual proton channel and the captured neutron respectively. I and II refer to two different sets of deuteron optical potentials given by Jeans Is). The potentials of Becchetti i4) have been used for the Proton channels.

by using the stripping optical potential of Jeans et ai. 12) which is somewhat deeper and considerably more absorptive than the ones used by the same authors to fit the elastic scattering of deuterons. The width amplitudes obtained by using the latter potential are given in table 1 in the column preceding the last. The deepening and increased absorptivity of the deuteron optical potential results in the damping of the scattering wave function inside the target nucleus and thus of the width amplitudes. This effect can be seen clearly in figs. 4-6. Fig. 4 shows the radial distribution of the zero-range approximation to the effective deuteron channel form factor [see eq. (3.1O)l

for a typical contribution to the channel coupling amplitude. In figs. 5 and 6 this distribution is multiplied by the deuteron radial wave functions obtained by using each of the two optical potentials mentioned above. It may be worth pointing out that the t The orthogonalization can be easily carried out by adding to the source (3.13) a term proportional to the nucleon parentage function of the analog state (&\p(r)jA>. This technique has been described in ref. 13). A Fortran program (TABOO) to solve eq. (3.12) with such an orthogonality constraint is available.

ANALOG

RESONANCES

315

contributions to the direct stripping amplitude (i.e., in which the virtual proton wave is replaced by a proton scattering wave) in corresponding partial waves show the same

Channel Coupiing Form Factor (zero range) Re Im ___-

I

I

I 5

0

I 95

IO

;

R(fm)i

Fig. 4. Channel coupling form factor in the deuteron channel for the g+ analog state in *OgBi. The intermediate channel is the “‘Pb ground state proton channel,

xi(R)

V0 = 103.5 MeV

,,“-“\,

(keV1/2/fm)

:

Ws = 19.44 MeV :

:

:

:

:

Channei Coupling Width Ampi. Re Im ----

90 R (fm)

Fig. 5. Radial distribution

95

3

-

of the width contribution from fig. 4 with the Jeans elastic scattering deuteron potential.

qualitative behavior of figs. 4-6. We may thus conclude that, even though the opticalmodel wave functions (and the neutron form factor) may be inadequate in the nuclear

316

A. F. R. DE TOLEDO

HZ4

interior, the resulting error in the width amplitude is no worse than that in the corresponding partial wave contribution to the direct stripping amplitude. 3.3. COMPOUND

CONTRIBUTIONS

To obtain an estimate of the compound contribution (3.4) to the deuteron width amplitude we consider the antianalog states as the relevant group of doorway states in the sense of eq. (3.6). The structure of these states is closely related to that of the corresponding analog state. The anti-analogs involve the same configurations and 0.11 V. = 120.0 MeV

t y,‘(R)(keV”yfm)

Ws = 40.0

MeV

0.05 -

-0.05

-

0

I 5

Re Im ____ I 15

I IO R (fm) -

2(

Fig. 6. Same as fig. 5, for the Jeans stripping deuteron potential.

spin coupling as the analog but have a different overall symmetry characterized by the normal isospin of the low-lying states of the compound system. As a result of this, their average excitation energy can be roughly related to the analog state energy by EA-jjl$

-+ E,-

AE;?-

&iI$,

with AEz M AE,,,

N-Z w -

2A

x 100 MeV;

that is, the displacements AE;;i are dominated by the symmetry energy AEsym associated with the change of isospin by one unit in the compound system ‘). The damping widths I$, on the other hand, have been estimated to be small with respect to the value of AEsym in heavy nuclei. The anti-analog states can be thus regarded as degenerate in the sense that their contribution to the escape amplitudes does not depend, in any important way on the particular representation chosen in the anti-analog space. This

ANALOG

317

RESONANCES

be now used to reduce the number of terms contributing to the sum in (3.6). An important part of the amplitude (3.6) is the anti-analog state deuteron width amplitudes. The simplest ones involve just a two-particle transition leading directly from the anti-analog to a final state having a deuteron in the continuum. In shellmodel terms, the residual state is just a neutron hole (3p&‘, in the “‘Pb core for the ground state deuteron channel and the “‘Bi analog resonances. In this case there is a single relevant anti-analog state 15) which can be written as the product state fact will

IA,) = W), where I&> = ~~l(n~j),l(3P~),“>--~l(n~j)~~(3p~)~(3pl):1,=0>, is the T = + state built with two particles in the 3p, orbit and an odd nucleon in an orbitj, while (b) is the shell-model ground state of ” 6Pb. The corresponding escape amplitude is then

+

(E1V,nb,P'(3p3:>9

where V,, is the proton-neutron force, and this is easily evaluated in the zero-range approximation. Also needed to evaluate (3.6) is the analog-antianalog coupling matrix element

+ &J30[l(nZjlvC”“‘lnEj>

-

(3p,~vc”“‘~3p+>],

and the analog-antianalog splitting, which we take to be just the symmetry energy A&y,. The results for the 3’ and for the f’ resonances are given in table 2 below. Anti-analog

compound contribution

7 (4

Ld, Ja

z5+

1, 2

**

3, 2 3, 3 1, 0 1,1

TABLE2 to the analog state deuteron width amplitudes

yi(l)

(keM)

0.0376 -0.02583 -0.00734+0.003693 0.0328 -0.0165i -0.0466 $0.008941 -0.0887 -0.0127i

~~(11) (keM) 0.00566+0.05293 -0.00359-0.009~1i 0.0161+O.O425i -0.0111 -0.08343 0.0156+O.llSi

The first two columns label the resonance quantum numbers and the deuteron channels respectively. I and II refer to two sets of deuteron potentials given by Jeans i2).

The mainconclusionto be drawn from these numbers is thattheyaresmallcompared to the channel coupling contributions of subsect. 3.2. They were however relatively more important than in the case of allowed proton decays, where the proton amplitude of the anti-analog state is reduced by a partial cancellation between Coulomb and symmetry potentials ‘). We find here that the deuteron escape amplitudes of the anti-analog states tend to be larger than those of the analog, rather than smaller as in the case of proton decays. Channel coupling corrections to the deuteron escape amplitudes of the anti-analogs, on the other hand, have been found to be smaller than the

31s

A. F. R. DE TOLEDO PJZA

direct two-body escape by one of order magnitude. This happens again as a result of the reduction of the single-nucleon widths of the anti-analogs due to Coulombsymmetry destructive interference. 4. Discussion The estimates described in the preceding sections account for the known fact ‘) that deuteron widths of analog states are of the order of 1 keV. They show them to be dominated by amplitudes involving the coupling of the deuteron to proton (deuteron stripping) channels. There are in general several such channels that can contribute significantly to the deuteron emission process. Compound (non statistical) contributions to the escape amplitude appear to be smaller by one order of magnitude, while the direct charge-dependent coupling of the analog state to the deuteron channel is negligible. A detailed comparison with measured (d, p) cross sections near analog resonances, not attempted here, will also be sensitive to various phase relationships between nonresonant and resonant parts of the transition amplitude 1), and can be used as an additional test for the approximations and models used in the calculations. It seems worth mentioning in this connection that the angular dependence of the interference pattern observed in the stripping excitation ftmctions near resonances will reflect the angular dependence of the calculated non-resonant stripping amplitude. This fact can eventually be used to provide for an additional test for the current approximations (such as the DWBA) used to calculate direct amplitudes. A more direct check on the overall phase of the resonant term (which includes the phases of the amplitudes given in tables 1 and 2) would be provided, on the other hand, by fitting total stripping cross sections in order to eliminate interference effects involving several partial waves. The data available thus far are unfortunately barely enough to indicate the qualitative behaviour of the interference effects in the “‘Bi analog resonances r6), and more quantitative data would be extremely useful to study direct-resonant plase relationships in individual partial waves. On the basis of the present estimates it is apparent that uncertainties and ambiguities in optical-potential parameters, as determined from elastic scattering and stripping fits, give rise to appreciable variation in both phase and magnitude of the deuteron with amplitudes, as shown by a comparison of the last two columns of tables 1 and 2. While this certainly discourages any intent to use these particular isospinforbidden processes as an analog-state spectroscopic device, it probably stresses their utility for a detailed study of reaction calculations ‘). I must thank E. W. Hamburger for several challenging and illuminating discussions and for correspondence concerning his data and A. K. Kerman for numerous discussions, suggestions and remarks which played a major role in the rethinking, reshaping and final formulation of the contents of this paper during a stay at MIT.

ANALOG

RESONANCES

319

I must also thank him and Professor II. Feshbach for making this stay possible and such a rich experience. Finally, I. M. Cohenca provided me with several computer sub-routines and the benefits of programming know-how in the crucial. initial states of the numerical work done at the University of %o Paulo.

OPEN CHANNELS,

ANALOGS,

ANTIANALOGS

In this appendix we just indicate how a pair of projection operators Pd and P, which are approp~aie for a stripping process in the presence of an analog state can be defined. For more channels and more analogs the generalization is straigh~o~ard, and can be found in ref. 7), along with some intermediate steps in the construction. Our main concern here is to indicate how one may handle the lack of orthogonality between deuteron-target and nucleon-target configurations minimizing the distortion of the deuteron-target space while preserving the integrity of the analog state. A convenient starting point is the construction of the deuteron-target projector Pd. This can be written as

where dR’F,(R, R’)[R%,),

I&)

=

(A.4

~1-I~~<~il~d*u,(r)~~(~-tEr)Jrt(~-4r (A-3)

the $* are nucleon creation operators, Us is the wave function describing the internal state of the deuteron and Fd is defined in terms of the two-nucleon density matrix of the target projected onto the internal state of the deuteron and of a deuteron parentage function of the analog state. It differs from a &function essentially on account of effects due to the exclusion principle. One can then proceed to construct P,. It is again obtained in the form

P, = with I&>

s

=f

W,>dr
(A4

dr’F,,(~, @4&J,

(A.9

where now the round kets

are constructed orthogonal to the deuteron-target space. Besides the target density matrix and analog state parentage function, F,, involves now a term related to the overlap of deuteron-target and nucleon-target states.

included in eqs. (A-3) and (A.6). This has the effect of adding suit&k antiana~og Components to the P-spaces. These can however be treated together with the gspace by ~e~ovi~~ the analog state projector from these equations and o~hogo~al~~i~g to the analog space using the technique indicated in ref. ‘). Ia the special case when the malog state /A} has pure isospin TO+ I and the target state ict,> has pure isospin TO the state vector represented by the integral on the right-hand side of eq. (A.3) will be trivially orthogonal to the analog state jif). In that case the projector I- jA){AI can be dropped and the discussioa of autian~ogs witl be restricted to eq. (A.6) only.

1) ELW. Hamburger, Phys. Rev. Let% 19 (1967) 36 2) N. Stein, J. P. C&In, C. A. Whitten, Jr. and D. A. Bromley, Phys. Rev. Lett. 21 (196s) 3456 3) E. M. Bernstein and D. D. Armstrong, Phys. Lett. 268 (1968) 365: Phys. Rev. Lett. 20 (1968) 936 4) T. .Tamura, Proc. Int. Conf. on nuclear structwe (Tokyo, X967) p. 288; R. Coker and T. Taraura, Pbys. Rev. 182 (1967) 1277 5) ‘I’. Tamura and R. Coker, Phys. Lett. 3OB (1969) 581 6) A. F. R. de Toledo Pia and A. K. Kerman, Ann. of Phys. 43 (1967) 363 7) N. Auerbach, J. Htiner, A. K. Kerrnan and C. M. Shakin, Rev. Mod. Pbys., to be published 8) A. Z. Mekjian, Phyys. Rev. Lett. 25 (1970) 888 9) ha. di Tore, P. Numberg aud E. 3. Rihimaki, Phys. Rev. C2 (3970) 13 10) A. F. R. de Toledo Piza, A. I(. Karman, S. Failieros and R. M. Venter, Nucl. Phys. 89 (1966) 369 11) G. R. Satchler, Nucl. Phys. 55 (lQ64) 1 12) A. F. Jeans, W. Darcey, W. G. Davies, K, N. &ones and P. K. Smith, Nucf. Phys. MB? (1969) 224 13) W. L. Wang and 6. Shakin, Phys. Lett. 323 (19’70) 421 14) F. D. Becchetti, Sr. and G. W. Greenlees, Phys. Rev. 182(1969) 1190 15’) I. F. Perez, MSc. thesis, ~~~versidade do S%o Pauto, 1969 16) E. W. Hamburger, private commuuication