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Doorway structures in the radiative capture of neutrons by 28Si and 32S
ANNALS
OF PHYSICS
Doorway
114,452--466 (1978)
Structures
in the Radiative Capture by 2%i and s2S*
of Neutrons
M. MICKLINGHOFF Institut fir Experimentalphysik,
Cyclotron Laboratory,
Universitiit Hamburg, Germany
AND B. CASTELI The Niels Bohr Institute, University of Copenhagen, Denmark Received February 9, 1978
A method allowing the inclusion of single-particle resonances in the microscopic treatment of the nuclear reaction process is presented. The method based on the K matrix notation has for result to dampen the scattering wave function in the internal region and to remove all single-particle resonances from the continuum. As an application the quantitative analysis of the %i(n, y) and %(n, y) reaction is presented. In both reactions the trend and magnitude of the cross sections are reproduced very satisfactorily. The calculation also predicts the existence of doorway states embedded on the giant dipole and contributing significantly to the cross section. These doorway states are indeed consistent with the existence of rapid fluctuations observed in recent (n, yo) and (n, yJ measurements.
1. INTRODUCTION The reaction mechanism underlying the radiative capture of neutrons has attracted much attention recently and has been generally described by invoking macroscopic structure models mainly becauseof the numerical difficulties inherent to a shell model description for the excitation process. The main problem concerns the occurrence of single-particle resonancesarising from potential scattering and the resultant difficulty in discretizing the continuum. In the region of resonances the continuum is varying rapidly bringing conventional calculations to require a large mesh to carry out the numerical integration satisfactorily. This leads to the inversion of extremely large matrices which is some casesmake the numerical solution unpractical. In the method presented below this difficulty is avoided by separating the singleparticle resonance spectrum from the continuum. In the region of interaction, the nuclear wavefunction is expanded in terms of a finite number of harmonic oscillator * Supported by the Bundesministerium fiir Forschung und Technologie (Federal Republic Germany) and the National Research Council (Canada). + Permanent address: Physics Department, Queen’s University, Kingston, Canada.
452 OOO3-4916/78/1142-0452$05.00/0 Copyright All rights
0 1978 by Academic Press, Inc. of reproduction in any form reserved.
of
DOORWAY
STRUCTURES
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AND
32S
453
functions, whereas the external region is represented by modified scattering wavefunctions, orthogonal to the h.o. representation. This procedure then dampens the scattering functions in the internal region and removes all the single-particle resonances from the continuum. The bound-state-continuum coupling is represented by a nonlocal one-bo’dy interaction describing the decay of the h.o. states into the continuum. This formalkm is developed in conjunction with a particle-core model adapted to the nuclear structure aspect of the reaction. The formalism is presented in detail in the next section, whereas Section 3 deals with the application to the (n, y) cross section calculation. Two specific examples involving 28Si and % targets are then presented since both reactions have been extensively studied experimentally. The rapid fluctuations in cross section seen experimentally and previously unexplained are seen to be consistent with our prediction of a large number of resonances embedded in the continuum.
2. THE FORMALISM 2.1. The Model In the particle-core coupling model the core can be described by collective (either vibrational or rotational) coordinates. Expanding the deformed potential in a Taylor series around the spherical single-particle potential u(r) = -V, yields a particle-core
[l + exp !-
r BQRo )I-’
interaction of the form
v = ---k(r)CA A CLLQoYA,(e, #)
(rotation),
(2.2)
(vibration).
(2.3)
If we denote in both cases the collective core coordinates by Q,,, , we can simply write v = --k(r) c A
(Qh* YA
(2.4)
with
In order to avoid the distinction between the rotational and vibrational core operator QAPcan be related to the electrical multipole operator [l]
M&J‘vz
WQA, .
picture, the
(2.6)
454
MICKLINGHOFF
The particle-core
Hamiltonian
AND CASTEL
can therefore be written as
H = Ho + V = H, + h - ((314~) ZeR:)-l
k(r) 1 (MC&) A
- YJ.
(2.7)
Here H, denotes the core Hamiltonian, which yields the core experimental spectrum. The core transition rates and moments are related to the matrix elements of M,(Eh). The single-particle Hamiltonian h is just given by h=
732 82
2n1 ar2
I
h2Z(z+ij
2m
~
+ u(r) -
r2
V&o * rj 1%
$1.
cw
The antisymmetrization of the valence particle with the particle components of the core excitations can be treated in a perturbative manner [2]. The most important exchange terms are displayed in Fig. 1 and can be calculated as follows i
li.h>l”
Ii ~YA
I(0 II PA II Al2 1 (2h + 1) E,, - EplL’ (2.9)
l(.i,ll~Y~I~,AX
lKllQ211~)12~.
The second term occurs only if the quadrupole operator contributes diagonal matrix elements (rotational model, anharmonic vibrator). Further diagrams which involve higher core states (e.g., two phonon states, rotational 4+ state, etc.) can also be included in a similar fashion [2]. 2.2. The Model Space
In standard weak coupling calculations as in shell model calculations, the basis is defined by requiring the Hamiltonian without interaction Ho = H, + h to be diagonal A
B
jh
jh
FIG. 1. Exchange diagrams due to the antisymmetrization of the valence particle with the particle in the first excited core state. The second diagram contributes only if the quadrupole moment of the core state differs from zero.
DOORWAY
STRUCTURES
IN
%
AND
455
32S
in it. This leads, however, to major problems in continuum calculations spectrum of h possesses a discrete as well as a continuous part (En -
since the
h) 1n} = 0,
(e - h) 1 E) = 0,
(2.10)
c I n>(~ I de = 1. n The occurrence of single-particle resonances from the potential scattering creates difficulties in discretizing the continuum. In the region of the resonances the continuum is rapidly varying and requires a dense mesh. This leads to the inversion of large matrices which in some cases makes the numerical solution unpractical [3,4J. We therefore separate the single-particle resonances from the continuum by a method described earlier [5]. In this method based on a procedure suggested by Wang and Shakin [6, 711the single-particle resonances are described by bound states (e.g. harmonic oscillator states) orthogonal to the genuine bound states. Denoting then all bound single-particle states by 1 p) we define the following operator, which projects onto the bound states part of our model space:
e = cuxI PXXPXI.
(2.11)
The index x describes the core states with energy E, . It is also useful to include in x all angular momentum quantum numbers of the single-particle state (x ? rj x 1, : JM, denotes then the channel index). The remaining space P=l-Q
(2.12)
is continuous and can then be represented by P = c j de / ~~x>(~~ I.
x
(2.13)
The modified continuum states I +) are solutions of the unperturbed Hamiltonian Ho = h + H, in a space orthogonal to the bound state basis (including the singleparticle resonances) (c + E, - If;,)
[ Ex> = 0.
(2.14)
The expression for the calculation of the modified continuum states 1 C) is given in Ref. [5]. As ito the closure relation in the single-particle space (cf. Eq. (2.10)) it can now be written as
; I pL)
(2.15) states ] 2) no longer
456
MICKLINGHOFF
AND
CASTEL
exhibit resonance behavior, since the single-particle resonances are already included in some of the states 1 CL).It is important however to note that in the basis { 1 px), I Cx)} the unperturbed Hamiltonian Ho is no longer diagonal
H!o f 0,
(2.16)
We first diagonalize the Hamiltonian
in Q space
G - Ho,) I w> = 0.
(2.17)
For E, < 0, the states 1w) are bound, while the states with positive energy lie in the continuum and can decay via the one-body field fi,J~) and the particle-core interaction V. This gives rise to resonances in the scattering process. The wavefunction can now be expanded in the following form: I #,(O) = 1 0, I u> + c s de a,(~) I qy). w x
(2.18)
Since the main part of the calculation will remain real we can then use the principal value integral and the reaction operator K. The K operator formalism has already been described in detail in Ref. [5]. The “free” principal value propagator and the “K propagator” are given by 1 (2.19) Go0 = E -
Ho0 - Hpp ’
(2.20)
Gk = Go0 + GoovGk with P = V + Hi,
+ H&, . The K operator is defined by K = P + ~%,OK
(2.21)
= v + vGkv.
Because of the separation of the single-particle resonances from the continuum, the coupling within the continuum is reduced drastically. We can therefore neglect the continuum-continuum coupling and approximate the eigenstates of Hpp by the modified continuum states 1CX). With this assumption, the solution to the full Hamiltonian now reads I ~Ex,Y”’
= I coxo) + Gkv I zoxo) (2.22) = I Eoxo) + G;ovo,
I 4x0)
+ G:ovop I cgyo>
DOORWAY
STRUCTURES
IN
28SI AND
457
32S
with ~~ = E - E,, .
The K propa.gator in Q space can easily be represented as [5] GsQ =
1 E -
-
HQQ
wQQ
= c [ 5) (E - Ii’,) (CT,1. w
(2.23)
The shift operator is given by W OQ =
POP@
- HPP)-~
and the states j 6) are obtained by diagonalization Eq. (2.17) @w(E) -
-
HQQ
wQQ(E))
(2.24)
vPQ
of (HQQ + WQQ) instead of using (2.25)
1 @ = O.
FinaJly using the relations G~Q~QP
(2.26)
= GooKpp,
and KPP
=
(2.27)
~PQG;QPQP,
the wavefunction (2.22) can then be written in the new basis as
(2.28)
with (ZX I K,, I ioxo) = C (q I V I ~3) (E - E,)-‘) w
(c;, I
V i E~x~).
(2.29)
One can also transform this wavefunction with standing wave boundary into a wavefunction which satisfies physical boundary conditions. This can be done with the on-shell matrix operator & I +K,Y+)
= 6 I Wx,Y”) = 1 I Cx,,; I $4;)
(2.30)
x0
with C x,x; = P,,~ + i~K,,~l~~; . The matrix elements K,,t are the K matrix elements on the energy shell which can be used for the calculation of the elastic and inelastic cross section .K,,* =
E, = E - E, ,
E./ = E - E./ .
(2.31)
458
MICKLINGHOFF
AND
CASTEL
They can be calculated in a similar manner as the half off-shell matrix elements seen in Eq. (2.29). 2.4. Coupling to the Giant Dipole Resonance
In this work we also include the dipole vibration of the core. In a previous study, Clement, Lane, and Rook [13] introduced an isovector dipole-dipole interaction term of the form V’ = (-$)
2 ($$)
6&r -
where the strength of the isovector potential “finite width 6 function” a,(r -
&) = 4
R,) 3
TV ,
(2.32)
was denoted by V, and 6, stood for a
?$ E (R,V,,-l
(2.33)
k(r).
The collective core coordinate n was then related to the dipole radiation operaior M,(El)
= e (&]“’
F
(2.34)
q
Using Eqs. (2.33) and (2.34) the isovector particle-core written in analogy to (2.4)
interaction
(2.32) can be (2.35)
v’ = Mr)(Ql * Yd 73 with Qlgu
= [G
R,,]-l 5
M,(El,
p),
for
N = 2.
One problem arises from the fact that the GDR state decays by particle emission, which gives it a width r. Brown [14] took this into account in his first-order perturbative approach by assuming a complex energy for that state. This is simply not possible for a full continuum calculation where an eigenstate of an Hermitian Hamiltonian is sought. In an earlier calculation [4] we therefore included several GDR core states with a Breit-Wigner strength distribution. This made the calculation very tedious especially as we still retained the single-particle resonances in the continuum. With the modified continuum the formalism described in Section 2.3 is in some respects similar to the perturbative approach. We therefore included in first order
as an additional term in the Hamiltonian energies E, will become complex
after diagonalizing
i?, = E, + iG,
Ho, . By doing this the (2.38)
DOORWAY
STRUCTURES
IN
%I
AND
32S
459
with (co - HQ, - WQQ> I 6) = 0 and G,,, = (W I GQQ / 6). These energies can then be used in the evaluation of the wavefunction using Eqs. (2.28) and (2.29). 2.5. The Cqoture Cross Section
The capture cross section can be calculated as related to a dipole transition from the continuum to a bound state 1f). The final state is to be calculated by diagonalizing H QQ > while the continuum state is given by Eq. (2.30). With a normalization of basis states like (2.39) (PX I P’X’) = fLs%,~ and (Q 1 2x’) = 6,,,6(E - 6’) (2.40) the scattering states 1 &?&)(+) become also orthonormal is achieved by the matrix operator e (cf. Eq. (2.30)) (+)(&x6 1 $,$(+)
in contrast to 1 $ExO)(0). This
= 6,,,6(E - E’).
(2.41)
With this convention the capture cross section is given by
dn, 14 = iv c Kfll WEl) IIhJ(+) 1’3 x01 lj x o;.,.
(2.42)
lj
with
and (2.43)
It is instructive to separate the transition
amplitude
into four parts (2.44)
The first term represents the direct capture and is given by D,j = G-l I MW)I
I ~00x0).
(2.45)
The Clj and Slj stand for the collective and single particle decay of the resonance state 1 W), respectively,
S$j
595/114/I/2-30
= C
873) e-Y, II 6) (En -
-&J-l
(2.47)
460
MICKLINGHOFF
AND CASTEL
Then the “final state capture” or capture from the continuum, process is denoted by F,j = (fll M(El)(Q
which is a nondirect
- 1) II EOXO)
+ c J A G-II WE11 II Cx) c& - E - E)-l (Ex 1K,,e 1Q). x
(2.48)
f
Dlj =
---
\
r’j
El
k b------_, f
f
---El
I-
1-
+
s.p. r.
- El
-7
j
j
f
f
\
h ‘lj
---El
= $
j
+ .. .
+
FI - -.
I
+ ..
s.p.r. -_ G
j
1 f
---
-- G s.p. r. -- i;
EI
+ .. .
j
2. The diagrams contributing to the capture cross section are classified as corresponding to direct (&), collective (C,), single particle (S,,), and final state capture (FLj). Some graphs are shown which involve the bound “single particle resonance” (SPR). In the conventional continuum (i.e., still containing the SPR) the second term of & would be counted as direct capture. In the calculation diagrams to all orders within the model space are included. FIG.
DOORWAY STRUCTURES IN %
AND 32S
461
This term may be interpreted as follows: the particle first enters the resonance state 1 W); after its reemission into the continuum (on- or off-shell) it is finally captured by radiative emission. The different terms can be easily visualized by the graphs described in Fig. 2.
3. CALCULATION
AND RESULTS
The solution of the Schrodinger equation in Q space leads to a matrix diagonalization as usual in bound state structure calculations. For the 2gSi and ?S bound state calculation, we repeated our earlier weak-coupling calculation [8,9] departing from them in using a Saxon-Woods potential to generate the single-particle states. The potential parameters (V, = 53 MeV, V,, = 8.5 MeV, a, = 0.6 fm, r0 = 1.25 fm) were then determined so as to obtain single-particles energies identical to our previous results. As (expected, the results of the diagonalization in the low energy region were therefore si!milar to bound state calculations, most of the differences occurring above threshold since the Saxon-Woods potential does not bind the 2pllZ and lf5,2 singleparticle states. In Fig. 3 we plot the calculated phase shifts 6 for the 28Si + PZreaction where we note the existence of single-particle resonances. These resonances were removed from the continuum as described above and included as bound states in the Q space. This modifies considerably the phase shift results which now exhibit a smooth variation with energy (see s”in Fig. 3). The effect is also obvious from Fig. 4: here some singleparticle matrices of the type contributing to the interaction are displayed. Figure 4a shows the matrix elements obtained when the single-particle resonances are still incorporateld within the continuum. As these functions occur as integrands in the
FIG. 3. ThLe pl,z and & phase shift calculated within the conventional tinuum (8 and ii, respectively).
and the modified con-
462
MICKLINGHOFF
AND
CASTEL
b 0
E lMeV1 FIG. 4. (a) Some single-particle matrix elements which contribute to the interaction are shown. They are calculated within the conventional continuum, and therefore exhibit a resonance structure. (b) In the modified continuum the single-particle resonances are removed. The size of the matrix elements is also substantially reduced.
Lippmann-Schwinger equation, the discretization of the integral would require a large number of mesh points. With 40 mesh points and 10 channel indices, for instance, the solution of the Lippman-Schwinger equation demands the inversion of 400 x 400 matrices for each channel spin and energy [3,4]. In Fig. 4b the value of the matrix elements calculated between the modified continuum states is displayed. The effect of the modification is twofold. First, the functions behave very smoothly, so that the integration can be performed with only very few mesh points. This reduces the numerical expenditure drastically [3]. Second, the size of the continuum-continuum coupling becomes very small now (compare the scale). Both effects are compensated by the fact that the (2 space is now larger and that the bound-continuum coupling contains the term C,(E). From the numerical point of view, however, it is desirable to shift as much as possible from the continuum to the discrete space. In this work the smallness of the continuum-continuum coupling allows us to treat the modified states j Cx) as eigenstates of the Hamiltonian in the continuum space Hpp . This approxi-
DOORWAY
STRUCTURES
IN
‘%I
AND
463
%
mation has also been adopted by other authors [7, lo] who have, however, neglected the one-body bound-continuum coupling term E,,(,$). Compared to the earlier bound state calculations for 2sSi and 33S the present one includes a much higher energy region and therefore includes the coupling to the GDR core states. Following Lindholm et al. [ll] we derived the GDR parameters from experiment (I!?- = 19.5 MeV, r = 5.5 MeV). The core-dipole matrix element was related to the absorption cross section by I(0 j 1 M,(El)I
1 l-)1” E 0.249o-,e2 fm2/mbar.
(3.1)
The experi:mental value for the first moment of the absorption cross section in 28Si is about 17.5 mbar [12] which corresponds to about seven single-particle units. The isoscalar potential strength was taken to be V, = 70 MeV. In calculating the antisymmetrization diagrams we discovered that they play no important role above threshold. Finally, instead of diagonalizing H,, we diagonalized H,, + W,,,(E) for each energy value. For that purpose, we calculated the shift matrix W,, via the energy representation
where yi denote the integration
weights for singular integrands [3, 51. TABLE
Results of the Diagonalization PllZ Pslz x o+>
J?r
CM%
-___A B C D E F G H I K
*f-Hf:I-i::;-
1.70 2.36 2.64 4.42 5.05 5.80
5.93 7.44 7.59 9.45
I
of Ho0 + Woo above Threshold in 2QSi”
I P3/2 x 2+>
I PllZ
x 2+>
Ifs/z x 2+>
%
f&/z 0.08
0.27 -0.51 0.83 0.85 -0.12 0.12 0.48 0.06 0.17
0.83 0.82 0.76 0.49 -0.01 -0.19 0.10
-0.08 -0.04 -0.07
-0.04 0.21 0.23 0.91 -0.75 0.33 0.20
0.02 0.05 -0.05 0.22 -0.38 0.15 -0.48 0.76 0.98 0.86
70 79 84 98 92 90 82 92 98 77
a Many of the negative parity states have large components which involve 2pl,z, 2paiz, or lf5,2 configurations. Wave functions of J = s-, s-, or t- states with more than 50 % in the given configurations are given. All J = i-, g-, and -!- states between 0 and 10 MeV are also plotted in Fig. 5.
464
MICKLINGHOFF
AND
CASTEL
In Table I some of the wavefunctions resulting from the diagonalization are given. The eigenvalues E, + (w 1 W,, 1w) are displayed in Fig. 5. Because of the role of G@(E),the energy shift was particularly large for the states containing important components of the bound single-particle resonances. The states above threshold are mainly doorway states. These 2p-lh states are, however, split due to the coupling to more complicated configurations (particle coupled to higher core states). As we included only a finite number of core states, the equilibrium compound states were not reached in our model space. We therefore included an additional spreading width of 0.2 MeV to each state 10). The results were not essentially modified by this procedure which allowed us to perform the calculation with a reasonable step size in energy of 0.1 MeV without overlooking a resonance. Having performed the Q space calculation we can now evaluate the wavefunction according to Eqs. (2.28) and (2.29) and the capture cross section with the help of Eq. (2.42). In order to see which term contributes most in a particular energy region, we calculated the following quantities, displayed also in Figs. 5 and 6
C = iv c I G 12, Ti
(3.3)
I &j 12,
The singular integral in the final state term (cf. Eq. (2.48)) was solved by the same technique and using the same weights as described by Eq. (3.2). The results of the calculation indicated that this term was always more than two orders of magnitude smaller than all the other contributions. Also in order to account for the fact that at the energies of interest other channels were open, we reduced the isovector neutron .charge from -0.5e to -0.3~~ (cf. Eq. (2.43)). Two main conclusions can be drawn from our study of the radiative capture of neutrons over a wide energy range. The first one concerns the feasibility of the calculation of single-particle resonances linked to our modification of the shell model continuum. The method has been shown to be numerically simple and allows to treat bound states and continuum exactly on the same level with a great economy of computational effort. Our second conclusion should be related to our quantitative analysis of the results of the 2sSi(n, y) and 32S(n, r) reactions. In both reactions, the general trend and magnitude of the cross sections are reproduced very satisfactorily. The calculation also predict the existence of doorway states embedded on the giant dipole and contributing significantly to the cross section. These doorway state resonances are indeed consistent with the existence of rapid fluctuations observed
DOORWAY
STRUCTURES
IN “SI
AND
=S
465
in recent (n, y,,) and (n, yI) measurements. Additional experiments with increased energy resolution and resonance spin assignments would be of great value in testing the model Ipredictions presented here. NEUTRON 5.
10
ENERGY
IMeVl
1:
16'
~(n,x,l P""""""'r
*‘Si
FIG. 5. Iladiative capture cross section for the %i(n, y)Wi reaction. The thin full line represents direct capture D, the dashed line denotes the single-particle decay of the resonance states S, whereas the dot-dashed line represents the collective decay contribution C. For the bound states see Table I.
NEUTRON
ENERGY
[ MeVl
FIG. 6. Radiative capture cross section for the YS(n, y)YG reaction. Because the experiment cannot resolve yO and y1 we also calculated u(n, y,, + n) (right-hand side). The same notations as in Fig. 5 were used. The relative smallness of the (n, rO) cross section is due to the quasiparticle character of the final state.
466
MICKLINGHOFF
AND CASTEL
ACKNOWLEDGMENT One of the authors (B.C.) is grateful to Professor Aage Bohr for kind hospitality Bohr Institute.
at the Nielr
REFERENCES AND G. R. SATCHLER, Nucl. Phys. 51 (1964), 155; A. M. BERNSTEIN, in “Advances in Nuclear Physics” (M. Baranger and E. Vogt, Ed.), Vol. 3, Plenum, New York, 1970. B. R. MOTTELSON, “Proceedings of the International Conference on Nuclear Structure,” Tokyo, 1967; I. HAMOMOTO, Nucl. Phys. A 205 (1973), 205; H. R. MEDER AND J. E. PURCELL, Phys. Rev. C 12 (1975), 2056. J. RAYNAL, M. A. MELKANOV, AND T. SAWADA, Nucl. Phys. A 101; M. MICKLINGHOFF, Ph. D. Thesis, Hamburg, 1977, and to be published. M. MICKLINGHOFF AND B. CASTEL, 2. Physik A 282 (1977), 117. M. MICKLINGHOFF, Nucl. Phys. (in print). W. L. WANG AND C. M. SHAKIN, Phys. Lett. B32 (1970), 421. W. L. WANG AND C. M. SHAKIN, Phys. Rev. C5 (1972), 1898. I. P. JOHNSTONE AND B. CASTEL, Nucl. Phys. A 213 (1973), 341; B. CASTEL and I. P. JOHNSTONE, Cunud. J. Phys. 51 (1973), 988. B. CASTEL, K. W. C. STEWART, AND M. HARVEY, Nucl. Phys. A 162 (1971), 273. K. W. SCHMID AND G. Do DANG, Phys. Rev. Cl5 (1977), 1515. A. LINDHOLM, L. NILSSON, I. BERQUIST, AND B. PALSSON, Nucl. Phys. A 279 (1977), 445. N. BEZIC, D. JAMNIK, G. KERNEL, J. KAJNIK, AND J. SNAJDER, Nucl. Phys. A 117 (1968), 124. C. F. CLEMENT, A. M. LANE, AND J. R. ROOK, Nucl. Phys. 26 (1965), 66. G. E. BROWN, Nucl. Phys. 57 (1964), 339.
1. L. W. OWEN 2.
3. 4. 5. 6. 7. 8.
9. 10. 11.
12. 13.
14.
OF PHYSICS
Doorway
114,452--466 (1978)
Structures
in the Radiative Capture by 2%i and s2S*
of Neutrons
M. MICKLINGHOFF Institut fir Experimentalphysik,
Cyclotron Laboratory,
Universitiit Hamburg, Germany
AND B. CASTELI The Niels Bohr Institute, University of Copenhagen, Denmark Received February 9, 1978
A method allowing the inclusion of single-particle resonances in the microscopic treatment of the nuclear reaction process is presented. The method based on the K matrix notation has for result to dampen the scattering wave function in the internal region and to remove all single-particle resonances from the continuum. As an application the quantitative analysis of the %i(n, y) and %(n, y) reaction is presented. In both reactions the trend and magnitude of the cross sections are reproduced very satisfactorily. The calculation also predicts the existence of doorway states embedded on the giant dipole and contributing significantly to the cross section. These doorway states are indeed consistent with the existence of rapid fluctuations observed in recent (n, yo) and (n, yJ measurements.
1. INTRODUCTION The reaction mechanism underlying the radiative capture of neutrons has attracted much attention recently and has been generally described by invoking macroscopic structure models mainly becauseof the numerical difficulties inherent to a shell model description for the excitation process. The main problem concerns the occurrence of single-particle resonancesarising from potential scattering and the resultant difficulty in discretizing the continuum. In the region of resonances the continuum is varying rapidly bringing conventional calculations to require a large mesh to carry out the numerical integration satisfactorily. This leads to the inversion of extremely large matrices which is some casesmake the numerical solution unpractical. In the method presented below this difficulty is avoided by separating the singleparticle resonance spectrum from the continuum. In the region of interaction, the nuclear wavefunction is expanded in terms of a finite number of harmonic oscillator * Supported by the Bundesministerium fiir Forschung und Technologie (Federal Republic Germany) and the National Research Council (Canada). + Permanent address: Physics Department, Queen’s University, Kingston, Canada.
452 OOO3-4916/78/1142-0452$05.00/0 Copyright All rights
0 1978 by Academic Press, Inc. of reproduction in any form reserved.
of
DOORWAY
STRUCTURES
IN
28SI
AND
32S
453
functions, whereas the external region is represented by modified scattering wavefunctions, orthogonal to the h.o. representation. This procedure then dampens the scattering functions in the internal region and removes all the single-particle resonances from the continuum. The bound-state-continuum coupling is represented by a nonlocal one-bo’dy interaction describing the decay of the h.o. states into the continuum. This formalkm is developed in conjunction with a particle-core model adapted to the nuclear structure aspect of the reaction. The formalism is presented in detail in the next section, whereas Section 3 deals with the application to the (n, y) cross section calculation. Two specific examples involving 28Si and % targets are then presented since both reactions have been extensively studied experimentally. The rapid fluctuations in cross section seen experimentally and previously unexplained are seen to be consistent with our prediction of a large number of resonances embedded in the continuum.
2. THE FORMALISM 2.1. The Model In the particle-core coupling model the core can be described by collective (either vibrational or rotational) coordinates. Expanding the deformed potential in a Taylor series around the spherical single-particle potential u(r) = -V, yields a particle-core
[l + exp !-
r BQRo )I-’
interaction of the form
v = ---k(r)CA A CLLQoYA,(e, #)
(rotation),
(2.2)
(vibration).
(2.3)
If we denote in both cases the collective core coordinates by Q,,, , we can simply write v = --k(r) c A
(Qh* YA
(2.4)
with
In order to avoid the distinction between the rotational and vibrational core operator QAPcan be related to the electrical multipole operator [l]
M&J‘vz
WQA, .
picture, the
(2.6)
454
MICKLINGHOFF
The particle-core
Hamiltonian
AND CASTEL
can therefore be written as
H = Ho + V = H, + h - ((314~) ZeR:)-l
k(r) 1 (MC&) A
- YJ.
(2.7)
Here H, denotes the core Hamiltonian, which yields the core experimental spectrum. The core transition rates and moments are related to the matrix elements of M,(Eh). The single-particle Hamiltonian h is just given by h=
732 82
2n1 ar2
I
h2Z(z+ij
2m
~
+ u(r) -
r2
V&o * rj 1%
$1.
cw
The antisymmetrization of the valence particle with the particle components of the core excitations can be treated in a perturbative manner [2]. The most important exchange terms are displayed in Fig. 1 and can be calculated as follows i
li.h>l”
Ii ~YA
I(0 II PA II Al2 1 (2h + 1) E,, - EplL’ (2.9)
l(.i,ll~Y~I~,AX
lKllQ211~)12~.
The second term occurs only if the quadrupole operator contributes diagonal matrix elements (rotational model, anharmonic vibrator). Further diagrams which involve higher core states (e.g., two phonon states, rotational 4+ state, etc.) can also be included in a similar fashion [2]. 2.2. The Model Space
In standard weak coupling calculations as in shell model calculations, the basis is defined by requiring the Hamiltonian without interaction Ho = H, + h to be diagonal A
B
jh
jh
FIG. 1. Exchange diagrams due to the antisymmetrization of the valence particle with the particle in the first excited core state. The second diagram contributes only if the quadrupole moment of the core state differs from zero.
DOORWAY
STRUCTURES
IN
%
AND
455
32S
in it. This leads, however, to major problems in continuum calculations spectrum of h possesses a discrete as well as a continuous part (En -
since the
h) 1n} = 0,
(e - h) 1 E) = 0,
(2.10)
c I n>
e = cuxI PXXPXI.
(2.11)
The index x describes the core states with energy E, . It is also useful to include in x all angular momentum quantum numbers of the single-particle state (x ? rj x 1, : JM, denotes then the channel index). The remaining space P=l-Q
(2.12)
is continuous and can then be represented by P = c j de / ~~x>(~~ I.
x
(2.13)
The modified continuum states I +) are solutions of the unperturbed Hamiltonian Ho = h + H, in a space orthogonal to the bound state basis (including the singleparticle resonances) (c + E, - If;,)
[ Ex> = 0.
(2.14)
The expression for the calculation of the modified continuum states 1 C) is given in Ref. [5]. As ito the closure relation in the single-particle space (cf. Eq. (2.10)) it can now be written as
; I pL)
(2.15) states ] 2) no longer
456
MICKLINGHOFF
AND
CASTEL
exhibit resonance behavior, since the single-particle resonances are already included in some of the states 1 CL).It is important however to note that in the basis { 1 px), I Cx)} the unperturbed Hamiltonian Ho is no longer diagonal
H!o f 0,
(2.16)
We first diagonalize the Hamiltonian
in Q space
G - Ho,) I w> = 0.
(2.17)
For E, < 0, the states 1w) are bound, while the states with positive energy lie in the continuum and can decay via the one-body field fi,J~) and the particle-core interaction V. This gives rise to resonances in the scattering process. The wavefunction can now be expanded in the following form: I #,(O) = 1 0, I u> + c s de a,(~) I qy). w x
(2.18)
Since the main part of the calculation will remain real we can then use the principal value integral and the reaction operator K. The K operator formalism has already been described in detail in Ref. [5]. The “free” principal value propagator and the “K propagator” are given by 1 (2.19) Go0 = E -
Ho0 - Hpp ’
(2.20)
Gk = Go0 + GoovGk with P = V + Hi,
+ H&, . The K operator is defined by K = P + ~%,OK
(2.21)
= v + vGkv.
Because of the separation of the single-particle resonances from the continuum, the coupling within the continuum is reduced drastically. We can therefore neglect the continuum-continuum coupling and approximate the eigenstates of Hpp by the modified continuum states 1CX). With this assumption, the solution to the full Hamiltonian now reads I ~Ex,Y”’
= I coxo) + Gkv I zoxo) (2.22) = I Eoxo) + G;ovo,
I 4x0)
+ G:ovop I cgyo>
DOORWAY
STRUCTURES
IN
28SI AND
457
32S
with ~~ = E - E,, .
The K propa.gator in Q space can easily be represented as [5] GsQ =
1 E -
-
HQQ
wQQ
= c [ 5) (E - Ii’,) (CT,1. w
(2.23)
The shift operator is given by W OQ =
POP@
- HPP)-~
and the states j 6) are obtained by diagonalization Eq. (2.17) @w(E) -
-
HQQ
wQQ(E))
(2.24)
vPQ
of (HQQ + WQQ) instead of using (2.25)
1 @ = O.
FinaJly using the relations G~Q~QP
(2.26)
= GooKpp,
and KPP
=
(2.27)
~PQG;QPQP,
the wavefunction (2.22) can then be written in the new basis as
(2.28)
with (ZX I K,, I ioxo) = C (q I V I ~3) (E - E,)-‘) w
(c;, I
V i E~x~).
(2.29)
One can also transform this wavefunction with standing wave boundary into a wavefunction which satisfies physical boundary conditions. This can be done with the on-shell matrix operator & I +K,Y+)
= 6 I Wx,Y”) = 1 I Cx,,; I $4;)
(2.30)
x0
with C x,x; = P,,~ + i~K,,~l~~; . The matrix elements K,,t are the K matrix elements on the energy shell which can be used for the calculation of the elastic and inelastic cross section .K,,* =
E, = E - E, ,
E./ = E - E./ .
(2.31)
458
MICKLINGHOFF
AND
CASTEL
They can be calculated in a similar manner as the half off-shell matrix elements seen in Eq. (2.29). 2.4. Coupling to the Giant Dipole Resonance
In this work we also include the dipole vibration of the core. In a previous study, Clement, Lane, and Rook [13] introduced an isovector dipole-dipole interaction term of the form V’ = (-$)
2 ($$)
6&r -
where the strength of the isovector potential “finite width 6 function” a,(r -
&) = 4
R,) 3
TV ,
(2.32)
was denoted by V, and 6, stood for a
?$ E (R,V,,-l
(2.33)
k(r).
The collective core coordinate n was then related to the dipole radiation operaior M,(El)
= e (&]“’
F
(2.34)
q
Using Eqs. (2.33) and (2.34) the isovector particle-core written in analogy to (2.4)
interaction
(2.32) can be (2.35)
v’ = Mr)(Ql * Yd 73 with Qlgu
= [G
R,,]-l 5
M,(El,
p),
for
N = 2.
One problem arises from the fact that the GDR state decays by particle emission, which gives it a width r. Brown [14] took this into account in his first-order perturbative approach by assuming a complex energy for that state. This is simply not possible for a full continuum calculation where an eigenstate of an Hermitian Hamiltonian is sought. In an earlier calculation [4] we therefore included several GDR core states with a Breit-Wigner strength distribution. This made the calculation very tedious especially as we still retained the single-particle resonances in the continuum. With the modified continuum the formalism described in Section 2.3 is in some respects similar to the perturbative approach. We therefore included in first order
as an additional term in the Hamiltonian energies E, will become complex
after diagonalizing
i?, = E, + iG,
Ho, . By doing this the (2.38)
DOORWAY
STRUCTURES
IN
%I
AND
32S
459
with (co - HQ, - WQQ> I 6) = 0 and G,,, = (W I GQQ / 6). These energies can then be used in the evaluation of the wavefunction using Eqs. (2.28) and (2.29). 2.5. The Cqoture Cross Section
The capture cross section can be calculated as related to a dipole transition from the continuum to a bound state 1f). The final state is to be calculated by diagonalizing H QQ > while the continuum state is given by Eq. (2.30). With a normalization of basis states like (2.39) (PX I P’X’) = fLs%,~ and (Q 1 2x’) = 6,,,6(E - 6’) (2.40) the scattering states 1 &?&)(+) become also orthonormal is achieved by the matrix operator e (cf. Eq. (2.30)) (+)(&x6 1 $,$(+)
in contrast to 1 $ExO)(0). This
= 6,,,6(E - E’).
(2.41)
With this convention the capture cross section is given by
dn, 14 = iv c Kfll WEl) IIhJ(+) 1’3 x01 lj x o;.,.
(2.42)
lj
with
and (2.43)
It is instructive to separate the transition
amplitude
into four parts (2.44)
The first term represents the direct capture and is given by D,j = G-l I MW)I
I ~00x0).
(2.45)
The Clj and Slj stand for the collective and single particle decay of the resonance state 1 W), respectively,
S$j
595/114/I/2-30
= C
873) e-Y, II 6) (En -
-&J-l
(2.47)
460
MICKLINGHOFF
AND CASTEL
Then the “final state capture” or capture from the continuum, process is denoted by F,j = (fll M(El)(Q
which is a nondirect
- 1) II EOXO)
+ c J A G-II WE11 II Cx) c& - E - E)-l (Ex 1K,,e 1Q). x
(2.48)
f
Dlj =
---
\
r’j
El
k b------_, f
f
---El
I-
1-
+
s.p. r.
- El
-7
j
j
f
f
\
h ‘lj
---El
= $
j
+ .. .
+
FI - -.
I
+ ..
s.p.r. -_ G
j
1 f
---
-- G s.p. r. -- i;
EI
+ .. .
j
2. The diagrams contributing to the capture cross section are classified as corresponding to direct (&), collective (C,), single particle (S,,), and final state capture (FLj). Some graphs are shown which involve the bound “single particle resonance” (SPR). In the conventional continuum (i.e., still containing the SPR) the second term of & would be counted as direct capture. In the calculation diagrams to all orders within the model space are included. FIG.
DOORWAY STRUCTURES IN %
AND 32S
461
This term may be interpreted as follows: the particle first enters the resonance state 1 W); after its reemission into the continuum (on- or off-shell) it is finally captured by radiative emission. The different terms can be easily visualized by the graphs described in Fig. 2.
3. CALCULATION
AND RESULTS
The solution of the Schrodinger equation in Q space leads to a matrix diagonalization as usual in bound state structure calculations. For the 2gSi and ?S bound state calculation, we repeated our earlier weak-coupling calculation [8,9] departing from them in using a Saxon-Woods potential to generate the single-particle states. The potential parameters (V, = 53 MeV, V,, = 8.5 MeV, a, = 0.6 fm, r0 = 1.25 fm) were then determined so as to obtain single-particles energies identical to our previous results. As (expected, the results of the diagonalization in the low energy region were therefore si!milar to bound state calculations, most of the differences occurring above threshold since the Saxon-Woods potential does not bind the 2pllZ and lf5,2 singleparticle states. In Fig. 3 we plot the calculated phase shifts 6 for the 28Si + PZreaction where we note the existence of single-particle resonances. These resonances were removed from the continuum as described above and included as bound states in the Q space. This modifies considerably the phase shift results which now exhibit a smooth variation with energy (see s”in Fig. 3). The effect is also obvious from Fig. 4: here some singleparticle matrices of the type contributing to the interaction are displayed. Figure 4a shows the matrix elements obtained when the single-particle resonances are still incorporateld within the continuum. As these functions occur as integrands in the
FIG. 3. ThLe pl,z and & phase shift calculated within the conventional tinuum (8 and ii, respectively).
and the modified con-
462
MICKLINGHOFF
AND
CASTEL
b 0
E lMeV1 FIG. 4. (a) Some single-particle matrix elements which contribute to the interaction are shown. They are calculated within the conventional continuum, and therefore exhibit a resonance structure. (b) In the modified continuum the single-particle resonances are removed. The size of the matrix elements is also substantially reduced.
Lippmann-Schwinger equation, the discretization of the integral would require a large number of mesh points. With 40 mesh points and 10 channel indices, for instance, the solution of the Lippman-Schwinger equation demands the inversion of 400 x 400 matrices for each channel spin and energy [3,4]. In Fig. 4b the value of the matrix elements calculated between the modified continuum states is displayed. The effect of the modification is twofold. First, the functions behave very smoothly, so that the integration can be performed with only very few mesh points. This reduces the numerical expenditure drastically [3]. Second, the size of the continuum-continuum coupling becomes very small now (compare the scale). Both effects are compensated by the fact that the (2 space is now larger and that the bound-continuum coupling contains the term C,(E). From the numerical point of view, however, it is desirable to shift as much as possible from the continuum to the discrete space. In this work the smallness of the continuum-continuum coupling allows us to treat the modified states j Cx) as eigenstates of the Hamiltonian in the continuum space Hpp . This approxi-
DOORWAY
STRUCTURES
IN
‘%I
AND
463
%
mation has also been adopted by other authors [7, lo] who have, however, neglected the one-body bound-continuum coupling term E,,(,$). Compared to the earlier bound state calculations for 2sSi and 33S the present one includes a much higher energy region and therefore includes the coupling to the GDR core states. Following Lindholm et al. [ll] we derived the GDR parameters from experiment (I!?- = 19.5 MeV, r = 5.5 MeV). The core-dipole matrix element was related to the absorption cross section by I(0 j 1 M,(El)I
1 l-)1” E 0.249o-,e2 fm2/mbar.
(3.1)
The experi:mental value for the first moment of the absorption cross section in 28Si is about 17.5 mbar [12] which corresponds to about seven single-particle units. The isoscalar potential strength was taken to be V, = 70 MeV. In calculating the antisymmetrization diagrams we discovered that they play no important role above threshold. Finally, instead of diagonalizing H,, we diagonalized H,, + W,,,(E) for each energy value. For that purpose, we calculated the shift matrix W,, via the energy representation
where yi denote the integration
weights for singular integrands [3, 51. TABLE
Results of the Diagonalization PllZ Pslz x o+>
J?r
CM%
-___A B C D E F G H I K
*f-Hf:I-i::;-
1.70 2.36 2.64 4.42 5.05 5.80
5.93 7.44 7.59 9.45
I
of Ho0 + Woo above Threshold in 2QSi”
I P3/2 x 2+>
I PllZ
x 2+>
Ifs/z x 2+>
%
f&/z 0.08
0.27 -0.51 0.83 0.85 -0.12 0.12 0.48 0.06 0.17
0.83 0.82 0.76 0.49 -0.01 -0.19 0.10
-0.08 -0.04 -0.07
-0.04 0.21 0.23 0.91 -0.75 0.33 0.20
0.02 0.05 -0.05 0.22 -0.38 0.15 -0.48 0.76 0.98 0.86
70 79 84 98 92 90 82 92 98 77
a Many of the negative parity states have large components which involve 2pl,z, 2paiz, or lf5,2 configurations. Wave functions of J = s-, s-, or t- states with more than 50 % in the given configurations are given. All J = i-, g-, and -!- states between 0 and 10 MeV are also plotted in Fig. 5.
464
MICKLINGHOFF
AND
CASTEL
In Table I some of the wavefunctions resulting from the diagonalization are given. The eigenvalues E, + (w 1 W,, 1w) are displayed in Fig. 5. Because of the role of G@(E),the energy shift was particularly large for the states containing important components of the bound single-particle resonances. The states above threshold are mainly doorway states. These 2p-lh states are, however, split due to the coupling to more complicated configurations (particle coupled to higher core states). As we included only a finite number of core states, the equilibrium compound states were not reached in our model space. We therefore included an additional spreading width of 0.2 MeV to each state 10). The results were not essentially modified by this procedure which allowed us to perform the calculation with a reasonable step size in energy of 0.1 MeV without overlooking a resonance. Having performed the Q space calculation we can now evaluate the wavefunction according to Eqs. (2.28) and (2.29) and the capture cross section with the help of Eq. (2.42). In order to see which term contributes most in a particular energy region, we calculated the following quantities, displayed also in Figs. 5 and 6
C = iv c I G 12, Ti
(3.3)
I &j 12,
The singular integral in the final state term (cf. Eq. (2.48)) was solved by the same technique and using the same weights as described by Eq. (3.2). The results of the calculation indicated that this term was always more than two orders of magnitude smaller than all the other contributions. Also in order to account for the fact that at the energies of interest other channels were open, we reduced the isovector neutron .charge from -0.5e to -0.3~~ (cf. Eq. (2.43)). Two main conclusions can be drawn from our study of the radiative capture of neutrons over a wide energy range. The first one concerns the feasibility of the calculation of single-particle resonances linked to our modification of the shell model continuum. The method has been shown to be numerically simple and allows to treat bound states and continuum exactly on the same level with a great economy of computational effort. Our second conclusion should be related to our quantitative analysis of the results of the 2sSi(n, y) and 32S(n, r) reactions. In both reactions, the general trend and magnitude of the cross sections are reproduced very satisfactorily. The calculation also predict the existence of doorway states embedded on the giant dipole and contributing significantly to the cross section. These doorway state resonances are indeed consistent with the existence of rapid fluctuations observed
DOORWAY
STRUCTURES
IN “SI
AND
=S
465
in recent (n, y,,) and (n, yI) measurements. Additional experiments with increased energy resolution and resonance spin assignments would be of great value in testing the model Ipredictions presented here. NEUTRON 5.
10
ENERGY
IMeVl
1:
16'
~(n,x,l P""""""'r
*‘Si
FIG. 5. Iladiative capture cross section for the %i(n, y)Wi reaction. The thin full line represents direct capture D, the dashed line denotes the single-particle decay of the resonance states S, whereas the dot-dashed line represents the collective decay contribution C. For the bound states see Table I.
NEUTRON
ENERGY
[ MeVl
FIG. 6. Radiative capture cross section for the YS(n, y)YG reaction. Because the experiment cannot resolve yO and y1 we also calculated u(n, y,, + n) (right-hand side). The same notations as in Fig. 5 were used. The relative smallness of the (n, rO) cross section is due to the quasiparticle character of the final state.
466
MICKLINGHOFF
AND CASTEL
ACKNOWLEDGMENT One of the authors (B.C.) is grateful to Professor Aage Bohr for kind hospitality Bohr Institute.
at the Nielr
REFERENCES AND G. R. SATCHLER, Nucl. Phys. 51 (1964), 155; A. M. BERNSTEIN, in “Advances in Nuclear Physics” (M. Baranger and E. Vogt, Ed.), Vol. 3, Plenum, New York, 1970. B. R. MOTTELSON, “Proceedings of the International Conference on Nuclear Structure,” Tokyo, 1967; I. HAMOMOTO, Nucl. Phys. A 205 (1973), 205; H. R. MEDER AND J. E. PURCELL, Phys. Rev. C 12 (1975), 2056. J. RAYNAL, M. A. MELKANOV, AND T. SAWADA, Nucl. Phys. A 101; M. MICKLINGHOFF, Ph. D. Thesis, Hamburg, 1977, and to be published. M. MICKLINGHOFF AND B. CASTEL, 2. Physik A 282 (1977), 117. M. MICKLINGHOFF, Nucl. Phys. (in print). W. L. WANG AND C. M. SHAKIN, Phys. Lett. B32 (1970), 421. W. L. WANG AND C. M. SHAKIN, Phys. Rev. C5 (1972), 1898. I. P. JOHNSTONE AND B. CASTEL, Nucl. Phys. A 213 (1973), 341; B. CASTEL and I. P. JOHNSTONE, Cunud. J. Phys. 51 (1973), 988. B. CASTEL, K. W. C. STEWART, AND M. HARVEY, Nucl. Phys. A 162 (1971), 273. K. W. SCHMID AND G. Do DANG, Phys. Rev. Cl5 (1977), 1515. A. LINDHOLM, L. NILSSON, I. BERQUIST, AND B. PALSSON, Nucl. Phys. A 279 (1977), 445. N. BEZIC, D. JAMNIK, G. KERNEL, J. KAJNIK, AND J. SNAJDER, Nucl. Phys. A 117 (1968), 124. C. F. CLEMENT, A. M. LANE, AND J. R. ROOK, Nucl. Phys. 26 (1965), 66. G. E. BROWN, Nucl. Phys. 57 (1964), 339.
1. L. W. OWEN 2.
3. 4. 5. 6. 7. 8.
9. 10. 11.
12. 13.
14.