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Calculation of the giant dipole resonance in a 1p-1h continuum shell-model approximation for 208Pb
1
2.c
1
(1971) 641-664;
Nuclear Physics Al73 Not to be reproduced
CALCULATION IN A lplh
by photoprint
@ North-Holland
Publishing
Co., Amsterdam
or microfilm without written permission from the publisher
OF THE GIANT DIPOLE RESONANCE
CONTINUUM
SHELL-MODEL
APPROXIMATION
FOR *08Pb R. F. BARRETT t Institut fiir Theoretische
Physik
der Universitiit
Frankfurt,
Frankfurt/Main,
Germany tt
and P. P. DELSANTO Department
of Physics,
University
of Puerto Rico, Mayaguez,
Puerto Rico, USA 77
Received 22 April 1971 Abstract: The results of a continuum shell-model calculation for ro8Pb are presented and discussed. A method of locating the bound states embedded in the continuum is described which enables the accurate location and identification of resonances in the total photo-absorption cross section. The final cross section is then calculated using the eigenchannel reaction theory. The partial photo cross sections and the angular distribution of the reaction products are also calculated. Results of the calculations are compared with those of experiment and of earlier bound-state calculations.
1. Introduction
**Many shell-model calculations in which the effect of the particle continuum is properly included have already been performed for the lighter magic nuclei i2C [refs. ‘9‘)I, 160 [refs. ‘, 3S4- ‘)I and 40Ca [refs. ‘, ‘)I. Such calculations have the advantage over bound-state shell-model calculations that the widths of resonances in the cross sections may be directly obtained. The doubly magic nucleus, “*Pb, has however not been previously treated in a shell-model calculation which takes proper account of the continuum. This fact is due to the large number of particle-hole configurations which need to be included in any meaningful calculation, and the consequent numerical difficulties and large amount of computing time involved. “*Pb is, however, an interesting nucleus to study in the framework of a lp-lh shellmodel calculation. Whereas previously it had been thought that all shell structure would be washed out by collective effects in such a heavy nucleus, experiments have revealed that this is not the case “). It has been shown for example that the singleparticle and single-hole levels in 20gPb, *07Pb, 2o ‘Bi and ‘07Tl are of a remarkable purity, superior to that of the isotopes contiguous to lighter magic nuclei s). t Postdoctoral fellow of the Alexander v. Humboldt-Stiftung. tt This work has been supported by the Bundesministerium fur Bildung und Wissenschaft Deutsche Forschungsgemeinschaft and by the University of Puerto Rico, Mayaguez. 641
by the
642
R. F. BARRETT
AND P. P. DELSANTO
Previous bound-state calculations 9*1“) for *08Pb have shown that the dipole strength of the unperturbed configurations is largely concentrated by the effect of the residual interactions into one coherent giant resonance state, shifted by several MeV from the others, as predicted by the Brown-Bolster% schematic model ‘l). This effect is more pronounced in lead than in the lighter nuclei because of the existence of a large number of configurations with nearly degenerate unperturbed energies lying between 6 and 9 MeV. The assumptions made in the schematic model are thus more nearly fulfilled. As this coherent state lies in the continuum, an accurate determination of its strength and width can only be accomplished in a calculation which correctly treats the particle continuum. Recent experimental results for *“Pb [refs. ‘*, “)I have revealed structure in the photonuclear cross sections. The purity of the single-particle and single-hole levels in the nuclei *“Pb, *“Pb, *“Bi and *“Tl lead to the speculation that this structure may be described by the shell model. The nucleus *“Pb thus provides a good opportunity for a comparison of the results of a shell-model continuum calculation with those of experiments in the upper region of the periodic table. Such a calculation represents the necessary first step towards a detailed understanding of the structure of the 208Pb photonuclear cross section. The most recent bound-state calculation by Perez 14)obtains a better agreement of the position of the giant dipole resonance with experiment than was achieved in the earlier calculations of Kuo et al. ’ “) and Gillet et al. lo). Perez concludes, however, that the correct treatment of the particle continuum may have an effect on the position of the giant resonance state. The necessity of a continuum shell-model calculation for *‘*Pb is thus indicated. In this paper we present details and the results of a Ip-lh continuum calculation carried out for *‘sPb in the framework of the eigenchannel reaction theory. Preliminary results have been outlined in an earlier letter 16). In sect. 2 of the paper, we describe some features of the eigenchannel technique. Sect. 3 contains details of the configurations considered and potential parameters used in our calculation. The results are presented and discussed in sect. 4. In sect. 5 we summarize our conclusions.
2. Eigenchannel techniques
In this section we describe some of the techniques used in the *‘*Pb eigenchannel calculation. The main features of the eigenchannel method are outlined briefly in subsect. 2.1. In recent years a considerable amount of study has been done on the origin of resonances in the continuum shell model “). It is believed that the origin of resonances lies in the existence of bound states embedded in the continuum (BSEC). A method of locating the BSEC, and thus the positions of resonances in the cross section is discussed in subsect. 2.2.
GIANT DIPOLE RESONANCE
643
In subsect. 2.3 a method of calculating the relative contribution of the various configurations to the cross section in an eigenchannel calculation is explained. The results are used to infer the BSEC origin of the resonances. 2.1. THE EIGENCHANNEL
METHOD
We have used in this calculation the eigenchannel reaction theory developed by Danos and Greiner I’). This method has recently been discussed in detail by Biedenharn et al. ‘) in a review article, in which the formalism of eigenchannel theory is presented, together with the results of the lp-1 h continuum calculations for “C, ’ 6O and 40Ca. In this paper, we present only an outline of the theory. The eigenchannel method is a technique for the calculation of the cross sections of nuclear photoprocesses and particle-particle reactions in which the continuum particle states are properly taken into account. Essentially the method consists of a search for the eigenvalues E, and eigenvectors I’(‘) (the so-called eigenchannels) of the S-matrix. One has SI;‘“’ = s, I/C”), (2.1.1) where E, = exp (2i6,),
(2.1.2)
with eigenphases 6, which are real owing to the unitarity of the S-matrix. Eigenchannels and eigenphases can be obtained directly from a shell-model calculation by using the R-matrix method of separating the configuration space into inner and outer regions at some finite radius RM from the nucleus. All of the nuclear interaction is assumed to take place in the inner region. For a given excitation energy, “natural” boundary conditions are introduced at the matching radius RM in terms of the kinetic energy of the nucleon and the common phase shift 6, for all experimental channels. These boundary conditions result in a discretization of the continuum in the inner region and a normal lp-lh shell-model calculation may then be performed in this region. The eigenvalues EA of the shell-model Hamiltonian, diagonalized on the basis provided by this discretization, are then compared with the original excitation energy. An iteration procedure is used to vary phase 6, until one of the eigenvalues E,, after diagonalization of the shell-model Hamiltonian, coincides with the excitation energy. There exist N different values of 6, where N is the number of open channels at a given excitation energy. Once the eigenchannels and eigenphases are known, it is very simple to calculate the cross sections both for y-nucleon and nucleon-nucleon reactions since the S-matrix is given by S,,, = C VP’ exp (2i6,)VJY. II
(2.1.3)
If it were possible to have a fore-knowledge of the likely position of resonances in the calculated cross sections, the task of scanning the cross section in arbitrary steps to obtain the detailed shape would be greatly simplified.
R. F. BARRETT
644
2.2. BOUND
STATES EMBEDDED
AND P. P. DELSANTO
IN THE CONTINUUM
It is known that sharp resonances in the continuum may be brought about by the coupling of single-particle resonances or bound states to the continuum states 19). As the various holes have different threshold energies, a nuclear excitation E, which is sufficient to lift a particle from a level near the Fermi surface into the continuum may not be sufficient to lift a particle from a deeper lying level into the continuum. At certain sharply defined energies, however, E, may be sufficient to excite the particle from this low-lying level into a higher bound state. Subsequent particle emission may then occur by the coupling of this bound channel to an open continuum channel by means of the residual interaction. The effect of the residual interaction is to shift the energy of the resonance, broaden it and change its character by the admixture of other configurations.
1
0Den continuum
---------unfilled / levels \
A Fermi surface
Fig. 1. Potential well (schematic) showing possible transitions from a bound level to (a) a higher bound level; (b) a single particle resonance level; (c) the open continuum.
Accepting the origin of sharp resonances to be that described above, we would expect some correspondence between the states found in a bound-state calculation and the peaks obtained in a continuum calculation, although we would not expect them to be at exactly the same energies, even if the same strength and type of residual interaction were used in both cases ‘O). In the eigenchannel method it is known that the basis vectors of the Hilbert space are a function of the eigenphase 6,. This is a result of the dependence of the boundary conditions applied to the wave function at the matching radius on the phase 6 (see subsect. 2.1). At a given excitation energy above the particle emission threshold, some channels may be closed, others may be open but with particle emission restricted by the presence of a Coulomb or centrifugal barrier, while others are completely open. These three situations are illustrated in fig. 1. To obtain the single-particle energies
GIANT DIPOLE RESONANCE
645
and wave functions of the basis states, the Schrodinger equation is solved with the boundary conditions implied by 6 and E, applied at the matching radius. We select a value of the excitation energy which is in the open continuum region for all channels and examine the variation of the eigenvalues of the Schrddinger equation solved in the inner region as a function of the phase 6. The energies of the bound single-particle states lying in the bound region of fig. 1 are independent of any boundary conditions applied at the comparatively large matching radius; for the single-particle resonance states in fig. 1, this is approximately true; while in the open continuum region, the energies of the single-particle states depend directly on the boundary conditions. A typical plot (ideal) of the eigenvalues of the Schrddinger equation as a function of the phase 6 is shown in fig. 2 for given values of the angular quantum numbers I andj.
.;i rll .Z
s
d
3
u”
0
180
6 (degrees)
Fig. 2. A plot (schematic) of the eigenvalues of the Schrbdinger equation as a function of the phase 6 (excitation energy fixed) for arbitrary values of the angular quantum numbers I and j.
It is now instructive to consider the diagonal energies of the shell-model Hamiltonian in the internal region at a given excitation energy as a function of phase 6 when the residual interaction is switched off. The diagonal energies E:“‘(6) for a given configuration L are then simply given by
E:“‘(6)=
P?)(d)-Q,
,
(2.2.1)
where P:“‘(6) are the single-particle energies obtained from fig. 2, and Q, is the threshold energy of the configuration. The threshold energies for the different configurations may be quite different. The dependence of the diagonal energies Ep’ upon phase 6 for the case of “*Pb is shown in fig. 3 for two different values of the excitation energy. The positions of the bound-state and single-particle resonance configurations may be clearly seen. As the excitation energy enters the internal shell-model calculation only through the boundary conditions applied at the matching radius, and as the eigenvalues corresponding to bound states are independent of these boundary conditions, the positions of the bound states in fig. 3 are also independent of the choice of the excitation energy E,. This may be seen from a comparison of figs. 3a and 3b.
R. F. BARRETT AND P. P. DELSANTO
646
PHASE (DEGREES)
(a)
PHASE (DEGREES)
Ib)
Fig. 3. ‘The unperturbed configuration energies as a function of the phase 6 for ‘OsPb with the excitation energy E, equal to (a) 16 MeV; (b) 20 MeV.
We now consider the effect of switching on the residual interaction between the configurations. As the shell-model Hamiltonian cannot have two equal eigenvalues when the off-diagonal terms are non-zero, the channel lines in fig. 3 now do not cross. The residual interaction also shifts the energies of the states (sometimes substantially as in the case of the giant dipole resonance) and changes their character by the ad-
647
GIANT DIPOLE RESONANCE
mixture of different configurations, with the effect, for instance, of concentrating most of the dipole strength in the giant resonance region. The unperturbed diagonal energies of the bound states embedded in the continuum are independent of the phase 6. This tendency persists even after diagonalization in the sense that the corresponding eigenvalues are relatively independent of the phase 6 over a wide range of values of 6. Plots of eigenvalues En(“) after diagonalization versus phase 6 were shown for the case
15
74
13
> f -12 lJ.7
11
10
9
PHASE(DEGREES) (4
PHASE
(DEGREES)
(b)
Fig. 4. The eigenvalues of the shell-model Hamiltonian solved in the inner region as a function of the phase S for zoaPb with the excitation energy E, equal to (a) 16 MeV; (b) 20 MeV.
648
R. F. BARRETT AND P. P. DELSANTO
of “*Pb in fig. 4 for two different values of the excitation energy E,. The positions of the BSEC may be seen. In the region of any BSEC there exists an eigenvalue E$“’ of the shell-model Hamiltonian which is approximately independent of the phase 6 and thus of the boundary conditions at the matching radius. It is also independent of E, for the reasons given above, as may be seen from a comparison of figs. 4a and 4b. The expected positions of the resonances in the calculated cross sections may therefore be obtained by a series of diagonalizations at only one excitation energy. This eliminates the likelihood of narrow resonances, such as those occurring in “‘Pb in the region below 10 MeV, being missed, as could otherwise occur in a search procedure carried out in finite energy steps. The strength and shape of the resonances are found by a detailed determination of the eigenphases at different excitation energies and described briefly in subusing the procedure outlined by Wahsweiler et al. sect. 2.1. As the positions of the resonances are known, excitation energies around these positions may be chosen and the computation of the cross section is greatly facilitated. The origin of a specific resonance can be inferred from the relative contribution of the different configurations to its dipole matrix element. If at the resonance energy, but nowhere else in the immediate neighbourhood, the dipole matrix element contains a large contribution from a bound-state configuration, it means that this bound state is strongly coupled to the continuum by the residual interaction at this excitation energy. The BSEC origin of the resonance is then obvious. A procedure enabling the calculation of the configuration mixing occurring in resonances calculated from the eigenchannel theory is now described. 2.3. CONFIGURATION
MIXING
In a bound-state catculation, a diagonalization of the shell-model ~amiltonian is performed on a basis of particle-hole configurations. After diagonalization, an eigenstate Iv} of the Hamiltonian is represented by a mixture of the basis configurations fen), i.e., (2.3.1) Iv> = c X&n>, c,n where X,, is the amplitude of the basis state Jcn), c represents all quantum numbers except the radial quantum number, and thus characterizes a configuration; n is the radial quantum number. The dipole strength of the state Iv) is proportional to the square of the dipole matrix element between the ground state Ii> and the state Iv): i.e., the dipole strength of Iv> cc ~~’= ] C Xzm.The contribution of a configuration to different states can be meaningfully compared because the basis wave functions ICR>are the same for all states iv>.
GIANT
DIPOLE
RESONANCE
649
In the eigenchannel theory, the basis states are a function of the boundary conditions applied at the matching radius, and are thus different for different eigenphases. Consider the lp-lh basis state wave function I&ln,;j,j+4;
JM) = C(-l)j”-“(ja
j,Mfm-mlJM)
X N,I,SJje M-l- m)*i(lA SA)jA ?n>ri i Ul,,l,i.(Y,)$~lUnAIAjA(rA)IT~ WXrA m:‘>, (2.3.3) see e.g., ref. “I). Uppercase subscripts or superscripts refer to particles; lower-case subscripts or superscripts refer to holes. The kets containing r characterize the charge of the nucleon; II is the radial quantum number. A variation of the boundary conditions at the matching radius results in a different radial wave function for the particle, i.e., ril umAIAjA(rA) is different for different eigenchannels. The angular part of the particle wave function, represented by l(l,s,)j,m) in eq. (2.3.3) is, however, unchanged. The basis state wave functions are of course orthogonal. The orthogonality between different branches of the same configuration, i.e., between states ln,n~;j,j,; JM) which differ only by the quantum number n, arises from the fact that the radial wave function unAIAjA(rA) is an eigenfunction of the radial Schrodinger equation solved in the inner region with the boundary conditions of the vth eigenchannel applied at the matching radius. The orthogonality between configurations i.e., where any quantum number other than n, differs, arises from the inherent orthogonality of the single-particle and single-hole angular wave functions l(i, 5’,)j,m), l(Z=~~)~=~+~>* and the single-hole radial wave functions. This orthogonality is independent of the boundary conditions applied at the matching radius. If we simplify the nomenclature as before by letting c represent all quantum numbers except n, and letting n represent n,, we have the lp-1 h basis wave function in the form lcn’). The v indicates the dependence of lcn’) upon the boundary conditions of the vth eigenchannel. From the arguments of the last paragraph, we have (cW’]cnV) = &cc’(cn’V’]c?zV>.
(2.3.4)
The dipole cross section at a specific excitation energy is given in the eigenchannel theory by (2.3.5) where M, = (+)*Ct~fi T
(G)‘l-
Z:-C,
(2.3.6)
[see e.g. ref. 21)] where Ii) is the ground state’if the nucleus and Iv> is the vth eigenchannel = ccR X&n”>. The normali~tion of the basis wave functions to unity inside the matching radius is arbitrary. Hence a renormalization must be introduced to normalize the eigenvector of the S-matrix to unit flux. This is represented by the factor cc, (Cc.)” in eq. (2.3.6) [see ref. ““)I.
R. F. BARRETT
650
AND P. P. DELSANTO
Let us define a vector
f-K> = M, c XcnlCfiV)n
(2.3.7)
Comparing eq. (2.3.7) with eq. (2.3.1) we could choose z,, X,‘, to be the percentage contribution of the configuration c to the eigenchannel iv) and 6,
=
L@
h:fl@c 1412~ 0_e_
(2.3.8)
Y
the total contribution of a configuration c to the cross section cr at a given energy. Then P = ~,cF,, For this to be a meaningful definition, we must show that ~nlXcn12 is independent of the change of basis produced by the changing boundary conditions at the matching radius. We have (2.39) We change the basis I&‘) to another complete set Ic’n”“) describing the same space by changing the boundary conditions to those of the vth eigenchannel: (2.3.10)
i.e., = ; [ ;
X~n(C’n’~‘~CnP)](Crn’v’>, (2.3.11)
from eq. (2.3.4), where A’&, the equivalent of A’,,,on the new basis, is thus given by XLn = ; x,(cn’“‘[cI2”>.
(2.3.12)
Similarly (2.3.13)
X,, X~~(cmY~c~~‘v’)(c~‘Y’~cnV)(c~z’Y’~cn’Y’> (2.3.14) =
5 [ ; Xcn(cn'Y'lcnY)][
= ; IXd12~ &,JX,,,Iz is thus independent of any change boundary conditions at the matching radius, to calculate the configuration mixing in the Results of the calculation of contributions are presented and discussed in subsect. 4.1.
C X~~(cmYIcdv’)“j
(2.3.15)
m
(2.3.16)
in the basis produced by the changing and we are justified in using eq. (23.8) eigenchannel theory. from different configurations for *‘*Rb
GIANT DIPOLE RESONANCE
651
3. Choice of parameters iu the *“Pb calculation In this section, we discuss the choice of parameters
in the “*Pb
calculation.
The
lp-1 h configurations considered are listed in subsect. 3.1. Details of the central singleparticle potential for lead are given in subsect. 3.2, and the residual interaction between the lp-lh configurations is detailed in subsect. 3.3. TABLE 1 The l- configurations Neutron configuration
included in the zosPb calculation, Unperturbed energies (MeV)
Proton configuration
11.50 (estimated)
i* -‘_i+ iy-l
3.1. CHOICE
and their unperturbed
energies
Unperturbed energies (MeV) 15.45 (estimated)
14.65 (estimated
7.21
j,
6.51
h+-i ig
7.70
hs-’
ga
9.41
8.46
h+-’ gf ft-’ da
6.92
10.57
6.56
7.44
6-i
dt
5.58
7.73
f*-’ g*
6.51
8.25
P*-i s+
5.49
7.90
p_t-id+
5.99
7.38
P*- ’ s*
6.39
9.05
P+-’ ds
6.89
8.76
P+- i di f;-’ d+
5.91
6.83
f*- i gj;
8.29
f$- i gp
5.80
hq-’ gq
13.40 (estimated) 7.47
7.36
OF CONFIGURATIONS
In the calculation described in this paper, we have included all of the lowest order l- configurations emanating from the last filled shell in “‘Pb. The configurations considered are listed in table 1, and the unperturbed energies shown. In order to diagonalize the shell-model Hamiltonian in the internal region, only a limited number of radial quantum numbers can be considered for each configuration, i.e., the infinite set of basis vectors of the Hilbert space must be arbitrarily truncated. In this calculation, we have considered four branches for each configuration. The
652
R. F. BARRETT AND P. P. DELSANTO
effect of truncation of the Hilbert space in an eigenchannel calculation has been investigated by Delsanto et al. 22). 3.2. SINGLE-PARTICLE
POTENTIAL
is customary in the particle-hole model to represent the average single-particle potential by some form which, although somewhat arbitrarily, reproduces the basic features of the nucleus studied (e.g., the nuclear radius and the single-particle energy level scheme as observed experimentally). In table 2, the single-particle energy level scheme is listed for “‘Pb. The particle and hole energies are known from experiments performed on the nuclei “‘Pb, 2ogBi, “‘Tl and 207Pb. References to such experiments are given in table 2. It
TABLE
The single-particle Ref. ’ 3, neutron particles
energy (MeV)
2h+ 3d3. 2gg 4%
energy (MeV)
proton particles
Ref. *6)
Ref. t5)
Ref. 24)
li+
lj*
2
and single-hole energy level scheme for “*Pb
neutron holes
proton holes
(MN
energy WeV) ______-
34
-
7.38
3%
-
8.05
2%
-
7.95
2d3
-
8.40
lhJ+
-
9.39
3di, lgr;
-
9.72
-1.39
2ge 3pt.
-0.15
3P+
-
8.28
-1.44
3p*
-0.67
lig
-
9.02
-1.89
2f+
-0.96
2f;
-
9.73
3dt
-2.37
lip
-2.18
lh%
- 10.85
lj,
-2.51
24
-2.89
Ii+
-3.15
*he
-3.79
2g*
-3.93
-11.53
In this calculation, we have assumed the single-particle potential Y(r) to be of the Woods-Saxon form: V(r) = K [~(~)-r
(~)2(z
- 4 ‘, !$I
+ fLl”i1
(3.2.1)
where V_u*=gJ-i,il-
Ze2 r
9
for
r 5 Rc
for
r > R,
(3.2.2)
and p(r) = [l-i-exp ((r-l-Ro)lt)]-i.
(3.2.3)
GIANT DIPOLE RESONANCE
653
Hzre R. denotes the nuclear radius, Z is the charge of the residual nucleus, M is the particle mass, t is the surface thickness parameter of the Fermi distribution, V, is the depth of the Woods-Saxon well, y is the spin-orbit coupling constant and Rc is the Coulomb radius. Values of R, = 7.52 fm and t = 0.7 fm were chosen in agreement with the parameters used by Blomqvist and Wahlborn “) for a Woods-Saxon potential in the lead region. A value of Rc = 7.0 fm was used in agreement with the equivalent uniform radius obtained from electron scattering data “‘). The parameters Vc and y were then optimized for each level (or doublet) in order to fit the calculated energies of the singleparticle and single-hole states accurately to the experimentally measured values. In this way the symmetry potential (i.e. the part of Vc due to the neutron excess) was automatically included. To discretize the continuum, boundary conditions are applied at an arbitrary “matching radius”, which must be sufficiently large that all nuclear interaction can be considered to take place within it. An eigenchannel matching radius of 17.0 fm was used in this calculation. This value was chosen by comparison of the magnitude of the single-hole wave functions at various radii for “*Pb with those obtained in an earlier eigenchannel calculation for I60 at the I60 matching radius ‘). In order that the nuclear interaction between configurations is negligible at the matching radius, it is necessary that the hole wave functions are vanishingly small there. In the case of 160, it has been shown by trial that the chosen matching radius (12 fm) is adequately large, so we infer that 17.0 fm is a large enough value in the case of lead.
3.3. RESIDUAL
INTERACTION
The effective two-body force which is responsible for the residual interaction is assumed to be of the form T/(1,2) = v0J(r,,)[a0+a,a(1)~0(2)+a,z(1)~~(2)+a,,a(1)~a(2)r(1)~2(2)].
(3.3.1)
In this calculation, we have made similar assumptions to those made in earlier eigenchannel calculations ‘); i.e., we have employed a contact force Jh2)
=
6(rl -r2)
(3.3.2)
and we have taken the exchange mixture to be of the Soper type. Due to the large size of the matrix to be diagonalized in the case of lead, it was not possible to vary the strength of the zero-range force I/, in order to optimize the calculated position of the giant resonance. However, in an existing bound-state calculation, Balashov “) employs a similar zero-range Soper force with a strength I’, of - 1220 MeV * fm3. He obtains the position of the giant resonance at 13.8 MeV in reasonable agreement with experiment. We have used the same value, V, = - 1220 MeV * fm3 in the continuum calculation described here.
654
R. F. BARRETT
4. Presentation
and discussion
AND
P. P. DELSANTO
of the results of the eigenchannel
calculation
In this section, we present and discuss the results of the eigenchannel continuum calculation for “‘Pb. In subsect. 3.1, the results of the continuum calculation for the total cross section are compared with the predictions of bound-state calculations, and 2 5
3) Perez
5,"
I-
I 2
d)
z
2m
Balashov et a1. (1962)
-f
5
;
lcoO-2 a, g d 7
9
10 Excitation
Fig.
5. The results
I
I, 8
of bound-state
11
12
13
li
15
lk
Energy(MeV)
shell-model
calculations
for
“@Pb.
GIANT
DIPOLE
655
RESONANCE
in subsect. 3.2, a comparison with experiment is made. In subsect. 3.3, the calculated partial cross sections are presented and the angular distributions given. 4.1. COMPARISON BOUND-STATE
OF RESULTS OF EIGENCHANNEL CALCULATIONS
CALCULATION
WITH THOSE
OF
Previous bound-state calculations for “‘Pb have been performed by Balashov Gillet et al. lo), Kuo et al. ’ “) and Perez ’ “). The results of their calculations are summarized in fig. 5. The energy of the giant dipole resonance obtained by Balashov et al. is higher than that obtained by Gillet et al. and Kuo et al. This is possibly due to the somewhat higher values of the unperturbed configuration energies employed by him. Kuo et al. comment that the energy of the giant dipole resonance obtained from their calculation is too low by several MeV. They indicate that the use of appropriate Woods-Saxon wave functions instead of oscillator wave functions for the single-particle states could shift the dipole strength to higher energies. This has been verified in the most recent “‘Pb bound-state calculation of Perez. He employs Woods-Saxon wave functions, and obtains the position of the giant dipole resonance in much better agreement with experiment. His results are shown in fig. 5a. et al. 9),
Pb
8 Fig. 6. The total photonuclear
10 Excitation
12 Energy
1L
208
16
(MeV)
cross section of zosPb from the eigenchannel
calculation.
In fig. 6, we present the total photonuclear cross section of “*Pb obtained from the eigenchannel calculation described here. As indicated in subsect. 2.2, a correspondence is expected between the resonances found in a continuum calculation and the states found in a bound-state calculation. From fig. 4, we have seen that BSEC are located at energies of 10.1, 10.85, 11.4, 12.875, 13.8 and 15.35 MeV, as well as at a number of low energies. These energies are therefore the expected positions of resonances in the photo cross sections. A complete determination of the photoabsorption cross section reveals that resonances are indeed located at these positions, with the
656
R. F. BARRETT
AND
P. P. DELSANTO
exception of the one expected at 10.1 MeV. This resonance is too weak to be resolved above the background. The relative strengths of the states in the bound-state calculation by Kuo et al. Is) are in good agreement with the results of the eigenchannel calculation. It must be remembered, however, that in the former calculation, the random-phase approximation has been used. The energies of the states obtained by Kuo et al. are lower than those of the corresponding resonances shown in fig. 6. This has been attributed to the use of harmonic oscillator wave functions by Kuo et al. At energies below IO MeV in the case of ‘08Pb, few channels are really open. Due to the high Coulomb and centrifugal barriers, most channels are of the single-particle resonance type shown in fig. 1. This results in a number of sharp resonances at these energies. We found, however, that these resonances do not carry much dipole strength, a result in agreement with the large number of weak states found in this region by Kuo et al. in the corresponding bound-state calculation. The technique described in subsect. 2.2 was used to accurately locate the positions of these resonances. The positions of the corresponding BSEC may be seen in fig. 4. Due to the extreme narrowness of these low-energy resonances, their location by means of a normal search procedure as used by Wahsweiler et al. ‘I) would have been extremely tedious, if not impossible. An analysis of the con~guration mixing occurring in the major resonances was carried out using the procedure detailed in subsect. 2.3. In fig. 7 are shown the contributions to the cross section from the most important configurations. The BSEC origin of the individual resonances may be deduced to be the fohowing: Tire 10.8 MeV resonance. Predominantly due to the i$‘h, neutron and h;‘g% neutron configurations which have unperturbed configuration energies of 11.5 MeV and 9.4 MeV, respectively. 7%e 11.4 MeV resonance. Due to the g; if, proton configuration turbed energy of 10.55 MeV.
with an unper-
The 12.875 MeV resonance. The giant dipole resonance, consisting of a mixture of many con~guratioils, the highest contribution from any one configuration being 11 %. The 13.75 MeV resonance. Due predominantly with an unperturbed energy of 13.4 MeV.
to the hi’&
proton configuration
The 15.35 MeV resonance. Due to the ii’j, neutron configuration with an unperturbed energy of 14.65 MeV. Two effects may be noted from these results. Firstly, the effect of the residual interaction is to shift the position of the BSEC by an amount of the order of 1 MeV from their unperturbed positions. Secondly, the giant dipole resonance state is a highly collective mixture of many configurations, as predicted by the Brown-Bolsterli schematic model. This result is also in agreement with the results of the bound-state calculations of Gillet et al. i”).
657
GIANT DIPOLE RESONANCE
In subsect. 4.2, we compare the results of the eigenchannel calculation for “‘Pb with those of experiment. -1
300
200 100
$2%,2
l---.-d h11,2 g%*
P
2cQtoo-
0--
0
n
200-
n
P
-1
92
=7/2
lOI0
loo0 loo 0
41,
ill,2
P
n 2005. &
1000. A too-f-’ 7/2 %A
n
b
10
Fig. 7. A plot of the contributions
-I
16
16
to the total photo cross section from the most important figurations.
4.2. COMPARISON OF THE RESULTS EXPERIMENTAL RESULTS
OF EIGENCHANNEL
CALCULATION
con-
WITH
Fig. 6 shows the total photonuclear cross sections obtained from the eigenchannel calculation. Because the extremely high Coulomb barrier for lead inhibits the emission of protons until excitation energies of about 17 MeV are reached, the predominant contribution to the total cross section comes from the photoneutron reaction.
R. F. BARRETT AND P. P. DELSANTO
658
The (y, n) cross section for lead has been measured by several authors using either quasi-monochromatic y-rays or bremsstrahlung radiation as a source of photons. These measurements are in general agreement as to the position, height and width of the giant resonance. The most recent measurement using the positon-annihilation-inflight technique
by Beil et al. 12) has detected
structure
.-. Excitation
_1_
on the increasing
slope of the
I--I
14 16 Energy(MeV)
18
b)
Excttation
Fig. 8. The experimental
Energy(MeV)
208Pb photoneutron cross section due to (a) Beil et al. I’); (b) Goryachev et al. 13); and (c) Harvey et a/. 29).
giant resonance which was not observed in earlier measurements by Harvey et al. 2 ‘) using the same technique. The bremsstrahlung experiments of Fuller and Hayward [ref. ““)I, and Tominasu 31) have detected a non-Lorentzian shape to the giant resonance on the low-energy side; that of Goryachev et al. 13) has detected structure on both the low-energy and high-energy sides of the main peak. For comparison. the results of Harvey et al., Goryachev et al. and Beil et al. are presented in fig. 8.
GIANT DIPOLE RESONANCE
659
A determination of the photoneutron spectrum obtained by the irradiation of lead by bremsstrahlung of maximum energy 31 MeV, performed by McNeil1 et al. 32) has detected structure superimposed on the evaporation neutron spectrum. He has correlated this structure with the peaks in the (y, n) cross section detected by Beil et al. A comparison of the experimental results with those of the eigenchannel calculation reveals that the theoretically predicted position of the giant resonance is too low by 0.6 MeV. This position, however, depends on the residual interaction strength chosen, which for reasons of computing time could not be optimised. The calculated height of the giant resonance is found to be 3.2 b and the FWHM to be 0.70 MeV. These values may be compared with the experimental values of 0.64 b and 4.05 MeV respectively. It is a characteristic feature of cross sections calculated using a continuum shell model in which only the most important lp-lh configurations are included, that the calculated resonances are too high and too narrow. For example, in the lp-lh shellmodel calculation for 40Ca [ref. “)] the discrepancy in the calculated and measured height of the main giant resonance exceeds a factor of five. It is expected that the inclusion of higher-order np-nh configurations should lower and broaden the resonances. Buck and Hill “) have simulated the effect of the omitted configurations in a coupled channel calculation for I60 by including an imaginary absorption term in the Hartree-Fock potential. This had the effect of bringing the results of their ~al~ulation into better agreement with experiment. A similar effect could be expected in the case of “‘Pb. Danos and Greiner 33) have calculated the damping of the giant resonance in heavy nuclei and obtained spreading widths in the range 0.5 to 2.5 MeV by examining the coupling of the lp-lh configurations to 2p-2h configurations. The coupling of the p-h configurations to collective states such as surface quadrupole and octupole vibrations 34) may also contribute to a spreading of the giant resonance. Examining fig. 6, it may be seen that the most pronounced of the minor peaks falls into the region in which structure is obtained in the measurements of Goryachev et al. and Beil et al. More experimental information is required, however, before any definite identification of peaks can be made. The integrated cross section up to 16 MeV was obtained by calculating the area under the curve in fig. 6. This was found to be 3.9 MeV - b. It can be compared with the experimental values of: (i) 2.91 MeV - b (integrated up to 28 MeV) obtained by Harvey et al. 29); (ii) 4.0 MeV * b (integrated up to 18.5 MeV) obtained by Goryachev et al. 13); (iii) 3.48 MeV * b (’in t egrated up to 25 MeV) obtained by Veyssiere et al. 12). The classical sum rule yields a value of 2.98 MeV * b. The results of the eigenchannel continuum calculations thus agree with the results of the bound-state calculations of Gillet et al. lo) in that the total integrated cross section for 208Pb is in agreement with experiment, whilst for 160, for example, the theoretical integrated cross section is about twice that of experiment.
R. F. BARRETT AND P. P. DELSANTO
660
4.3. PARTIAL CROSS SECTIONS AND ANGULAR
DISTRIBUTIONS
Once all of the eigenphases have been obtained, the partial photo cross sections are easily calculated using the technique described by Wahsweiler et al. in an earlier paper 2’). The results of such a calculation for “‘Pb are shown in fig. 9. In the energy range considered, the photo-proton partial cross sections are negligible. The predominant contribution to the giant resonance is seen to come from the (y, n,) reaction, while the peak at 13.75 MeV is due mainly to the (y, ns) reaction with contributions from the (y, nr) and (y, n,) reactions. A comparison of these results with experiment is not possible because of a lack of experimental data.
10
16 Excitadf (4
Energy C&V)
1)I/’
OL
10
I
,\,
12 Excitation Energy’?MeV)
tb)
(4
16
GIANT DIPOLE RESONANCE
Fig. 9. The partial photoneutron
661
cross sections for 208Pb as calculated reaction theory.
from the eigenchannel
TABLE 3
The potential parameters used in this calculation to reproduce the single-particle energy level scheme of zosPb Proton levels
Y
Neutron levels
lj*
-44.79
37.28
%
-44.4
30.49
-45.3
23.9
-44.94
26.11
-45.47
0.0
-45.3
23.9
li+
-51.98
37.36
2ge
- 59.28
27.40
3p+
-60.44
13.86 13.86
Y
3p3
- 60.44
24 li+
-59.71
20.68
-51.98
31.36
3% % 4% 3%
2f$
-59.71
20.68
1jy
-44.79
37.28
lhe
-59.53
33.68
li+
-45.0
33.86
3%
-60.28
0.0
2%
-44.94
26.11
2d+
-59.71
22.46
3p3
-44.38
32.54
lh jl
-59.53
33.68
26
-43.81
25.17
3dt
-59.71
22.46
32.54
lgt
-59.28
27.40
3p+ li+
-44.38 -45.0
33.86
26
-43.81
25.17
LhS
-44.4
30.49
R. = 7.52 fm,
R, = 7.0 fm,
t = 0.7 fm
662
R. F. BARRETT AND P. P. DELSANTO
The angular distributions may also be easily calculated for the individual partial reactions once the eigenphases are known. If the angular distribution W(e) of the emitted neutrons is expanded in terms of 0 where 0 is the angle made with the beam direction, we have W(8) = a, + a2 P,(cos q, (4.3.1) where P, is the nth order Legendre polynomial. The ratio az/ao is plotted over the giant resonance for the partial cross sections (y, n,) and (y, n,) in fig. IO. Measurements of the angular distribu~ons of photoneutrons from 208Pb have been made by several authors 3 5- “). These measurements have the common feature that they were made using a bremsstrahlung photon source and a threshold neutron detector, so that only neutrons of energy greater than a certain value were detected. Usually peak bremsstrahlung energies of 20-30 MeV are chosen with neutron detection thresholds of x 5 MeV. A list of such experiments is given in ref. ’ “).
11 Fig, 10. Anguiar
Excitotk
Energy
fMeV
14
15
)
distribution of the reaction products as calculated for the zosPb(y, no) and 2osPb(y, II,) reactions from the eigenchannel reaction theory.
The greatest contribution of neutrons comes from the giant resonance region of the cross section, and the use of a 5 MeV threshold detector means that only the (y, no) and (y, nl) partial reactions contribute any appreciable fraction of the neutrons detected. The largest contribution comes from the (y, n,) reaction. The values of at/a0 obtained experimentafly are seen from refs. 3 5-37) to lie mainly between -0.3 and -0.5. This is in agreement with the results of the eigenchannel calculation shown in fig. 10 to the limits of the experimental accuracy. More precise experiments are, however, needed. 5. Conclusion The total and partial photo cross sections for ‘08Pb have been calculated in the Ip-lh continuum shell-model approximation using the eigenchannel reaction theory.
663
GIANT DIPOLE RESONANCE
The calculated position of the giant dipole resonance is in reasonable agreement with experiment, supporting the conclusions of Perez 14) that the use of Woods-Saxon wave functions is necessary in “‘Pb shell-model calculations. The value of the total integrated cross section obtained is also in agreement with experiment. The identification of the various subsidiary resonances as bound states embedded in the continuum has been unambiguously accomplished. The configurations primarily responsible for this intermediate structure are: i,‘h,n;
h;‘g$n;
g;‘f+pr.;
h$lg$pr.
and
i,‘j,n.
It was seen that due to the high Coulomb and centrifugal barriers in the case of 208Pb, there exists at low energies a number of sharp resonances which carry little dipole strength. As is characteristic of the continuum shell-model calculations performed to date, the giant dipole resonance state is too high and too narrow compared with experimental measurements. The inclusion of additional np-nh configurations is expected to spread the dipole state, producing better agreement with experiment. The lp-lh continuum calculation described here represents only a first step towards the understanding of the 208Pb photo cross section. The inclusion of extra configurations, such as those describing collective correlations, or the use of a more sophisticated nuclear model, might be expected to contribute further to the explanation of the structure found experimentally. Such calculations have not yet been performed in a framework which correctly treats the continuum. The authors would like to thank Professor W. Greiner for the hospitality offered by him at Frankfurt, and for his encouragement and assistance. They would also like to thank Prof. M. Danos, Dr. P. Antony-Spies and Dr. A. Rabie for helpful discussions. The provision of computing facilities by the Z.R.I., Frankfurt is gratefully acknowledged. References 1) L. C. Biedenharn, M. Danos, P. P. Delsanto, W. Greiner and H. G. Wahsweiler, Rev. Mod. Phys., to be published 2) M. Marangoni and A. M. Saruis, Nucl. Phys. Al32 (1969) 649 3) B. Buck and A. D. Hill, Nucl. Phys. A95 (1967) 271 4) J. Raynal, M. A. Melkanoff and T. Sawada, Nucl. Phys. Al01 (1967) 369 5) J. Eichler, Nucl. Phys. 56 (1964) 577 6) M. S. Weiss, Phys. Lett. 19 (1965) 393 7) A. M. Saruis and M. Marangoni, Nucl. Phys. Al32 (1969) 433 8) D. A. Bromley and A. Weneser, Comments on Nucl. and Part. Phys. 2 (1968) 151 9) V. V. Balashov, V. G. Shevchenko and N. P. Yudin, JETP (Sov. Phys.) 14 (1962) 1371 10) V. Gillet, A. M. Green and E. A. Anderson, Nucl. Phys. 88 (1966) 321 11) G. E. Brown and M. Bolsterli, Phys. Rev. Lett. 3 (1959) 472 12) H. Beil, R. Berg&e, P. Carlos and A. VeyssiBre, 269 (1969) 216; A. VeyssiBre, H. Beil, R. Berg&e, P. Carlos and A. Lepetre, to be published 13) B. I. Goryachev, V. S. Ishkanov, I. M. Kapitanov and V. G. Shevchenko, JETP Lett. (Sov. Phys.) 7 (1968) 161
664 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37)
R. F. BARRETT AND P. P. DELSANTO S. M. Perez, Phys. Lett. 33B (1970) 317 T. T. S. Kuo, J. Blomqvist and G. E. Brown, Phys. Lett. 31B (1970) 93 R. F. Barrett and P. P. Delsanto, Phys. Lett. 34B (1971) 110 C. Mahaux and H. A. Weidenmtiller, Shell-model approach to nuclear reactions (North-Holland, Amsterdam, 1969) M. Danos and W. Greiner, Phys. Rev. 146 (1966) 708 H. A. Weidenmiiller, Comments on Nucl. and Part. Phys. 3 (1969) 112 H. A. Weidenmiiller, Proc. Int. Conf. on properties of nuclear states, Montreal, 1969, p. 529 H. G. Wahsweiler, W. Greiner and M. Danos, Phys. Rev. 170 (1968) 893 P. P. Delsanto, M. F. Roetter and H. G. Wahsweilet, 2. Phys. 222 (1969) 67 C. Ellegaard, J. Kantele and P. Vedelsby, Phys. Lett. 25B (1967) 512 J. Bardwick and R. Tickle, Phys. Rev. 171 (1968) 1305 B. L. Cohen, S. Mayo and R. E. Price, Nucl. Phys. 20 (1960) 360 S. Hinds, R. Middleton, J. H. Bjerregaard, 0. Hansen and 0. Nathan, Phys. Lett. 17 (1965) 302 J. Blomqvist and S. Wahlborn, Ark. Fys. 16 (1959) 545 R. Hofstadter, Rev. Mod. Phys. 28 (1956) 214 R. R. Harvey, J. T. Calwell, R. L. Bramblett and S. C. Fultz, Phys. Rev. 136 (1964) B126 E. G. Fuller and E. Hayward, Nucl. Phys. 33 (1962) 431 T. Tomimasu, J. Phys. Sot. Jap. 25 (1968) 655 K. G. McNeill, J. W. Jury and J. S. Hewitt, Can. J. Phys. 48 (1970) 950 M. Danos and W. Greiner, Phys. Rev. 138 (1965) B876 D. Drechsel, J. B. Seaborn and W. Greiner, Phys. Rev. 162 (1967) 983 H. Tsubota, N. Fujiwara, E. Tanaka, T. Aizawa, M. Kanazawa and N. Mutsuro, J. Phys. Sot. Jap. 26 (1969) 1 K. G. McNeill, J. S. Hewitt and J. W. Jury, Can. J. Phys. 46 (1968) 1974 R. G. Baker and K. G. McNeill, Can. J. Phys. 39 (1961) 1158
2.c
1
(1971) 641-664;
Nuclear Physics Al73 Not to be reproduced
CALCULATION IN A lplh
by photoprint
@ North-Holland
Publishing
Co., Amsterdam
or microfilm without written permission from the publisher
OF THE GIANT DIPOLE RESONANCE
CONTINUUM
SHELL-MODEL
APPROXIMATION
FOR *08Pb R. F. BARRETT t Institut fiir Theoretische
Physik
der Universitiit
Frankfurt,
Frankfurt/Main,
Germany tt
and P. P. DELSANTO Department
of Physics,
University
of Puerto Rico, Mayaguez,
Puerto Rico, USA 77
Received 22 April 1971 Abstract: The results of a continuum shell-model calculation for ro8Pb are presented and discussed. A method of locating the bound states embedded in the continuum is described which enables the accurate location and identification of resonances in the total photo-absorption cross section. The final cross section is then calculated using the eigenchannel reaction theory. The partial photo cross sections and the angular distribution of the reaction products are also calculated. Results of the calculations are compared with those of experiment and of earlier bound-state calculations.
1. Introduction
**Many shell-model calculations in which the effect of the particle continuum is properly included have already been performed for the lighter magic nuclei i2C [refs. ‘9‘)I, 160 [refs. ‘, 3S4- ‘)I and 40Ca [refs. ‘, ‘)I. Such calculations have the advantage over bound-state shell-model calculations that the widths of resonances in the cross sections may be directly obtained. The doubly magic nucleus, “*Pb, has however not been previously treated in a shell-model calculation which takes proper account of the continuum. This fact is due to the large number of particle-hole configurations which need to be included in any meaningful calculation, and the consequent numerical difficulties and large amount of computing time involved. “*Pb is, however, an interesting nucleus to study in the framework of a lp-lh shellmodel calculation. Whereas previously it had been thought that all shell structure would be washed out by collective effects in such a heavy nucleus, experiments have revealed that this is not the case “). It has been shown for example that the singleparticle and single-hole levels in 20gPb, *07Pb, 2o ‘Bi and ‘07Tl are of a remarkable purity, superior to that of the isotopes contiguous to lighter magic nuclei s). t Postdoctoral fellow of the Alexander v. Humboldt-Stiftung. tt This work has been supported by the Bundesministerium fur Bildung und Wissenschaft Deutsche Forschungsgemeinschaft and by the University of Puerto Rico, Mayaguez. 641
by the
642
R. F. BARRETT
AND P. P. DELSANTO
Previous bound-state calculations 9*1“) for *08Pb have shown that the dipole strength of the unperturbed configurations is largely concentrated by the effect of the residual interactions into one coherent giant resonance state, shifted by several MeV from the others, as predicted by the Brown-Bolster% schematic model ‘l). This effect is more pronounced in lead than in the lighter nuclei because of the existence of a large number of configurations with nearly degenerate unperturbed energies lying between 6 and 9 MeV. The assumptions made in the schematic model are thus more nearly fulfilled. As this coherent state lies in the continuum, an accurate determination of its strength and width can only be accomplished in a calculation which correctly treats the particle continuum. Recent experimental results for *“Pb [refs. ‘*, “)I have revealed structure in the photonuclear cross sections. The purity of the single-particle and single-hole levels in the nuclei *“Pb, *“Pb, *“Bi and *“Tl lead to the speculation that this structure may be described by the shell model. The nucleus *“Pb thus provides a good opportunity for a comparison of the results of a shell-model continuum calculation with those of experiments in the upper region of the periodic table. Such a calculation represents the necessary first step towards a detailed understanding of the structure of the 208Pb photonuclear cross section. The most recent bound-state calculation by Perez 14)obtains a better agreement of the position of the giant dipole resonance with experiment than was achieved in the earlier calculations of Kuo et al. ’ “) and Gillet et al. lo). Perez concludes, however, that the correct treatment of the particle continuum may have an effect on the position of the giant resonance state. The necessity of a continuum shell-model calculation for *‘*Pb is thus indicated. In this paper we present details and the results of a Ip-lh continuum calculation carried out for *‘sPb in the framework of the eigenchannel reaction theory. Preliminary results have been outlined in an earlier letter 16). In sect. 2 of the paper, we describe some features of the eigenchannel technique. Sect. 3 contains details of the configurations considered and potential parameters used in our calculation. The results are presented and discussed in sect. 4. In sect. 5 we summarize our conclusions.
2. Eigenchannel techniques
In this section we describe some of the techniques used in the *‘*Pb eigenchannel calculation. The main features of the eigenchannel method are outlined briefly in subsect. 2.1. In recent years a considerable amount of study has been done on the origin of resonances in the continuum shell model “). It is believed that the origin of resonances lies in the existence of bound states embedded in the continuum (BSEC). A method of locating the BSEC, and thus the positions of resonances in the cross section is discussed in subsect. 2.2.
GIANT DIPOLE RESONANCE
643
In subsect. 2.3 a method of calculating the relative contribution of the various configurations to the cross section in an eigenchannel calculation is explained. The results are used to infer the BSEC origin of the resonances. 2.1. THE EIGENCHANNEL
METHOD
We have used in this calculation the eigenchannel reaction theory developed by Danos and Greiner I’). This method has recently been discussed in detail by Biedenharn et al. ‘) in a review article, in which the formalism of eigenchannel theory is presented, together with the results of the lp-1 h continuum calculations for “C, ’ 6O and 40Ca. In this paper, we present only an outline of the theory. The eigenchannel method is a technique for the calculation of the cross sections of nuclear photoprocesses and particle-particle reactions in which the continuum particle states are properly taken into account. Essentially the method consists of a search for the eigenvalues E, and eigenvectors I’(‘) (the so-called eigenchannels) of the S-matrix. One has SI;‘“’ = s, I/C”), (2.1.1) where E, = exp (2i6,),
(2.1.2)
with eigenphases 6, which are real owing to the unitarity of the S-matrix. Eigenchannels and eigenphases can be obtained directly from a shell-model calculation by using the R-matrix method of separating the configuration space into inner and outer regions at some finite radius RM from the nucleus. All of the nuclear interaction is assumed to take place in the inner region. For a given excitation energy, “natural” boundary conditions are introduced at the matching radius RM in terms of the kinetic energy of the nucleon and the common phase shift 6, for all experimental channels. These boundary conditions result in a discretization of the continuum in the inner region and a normal lp-lh shell-model calculation may then be performed in this region. The eigenvalues EA of the shell-model Hamiltonian, diagonalized on the basis provided by this discretization, are then compared with the original excitation energy. An iteration procedure is used to vary phase 6, until one of the eigenvalues E,, after diagonalization of the shell-model Hamiltonian, coincides with the excitation energy. There exist N different values of 6, where N is the number of open channels at a given excitation energy. Once the eigenchannels and eigenphases are known, it is very simple to calculate the cross sections both for y-nucleon and nucleon-nucleon reactions since the S-matrix is given by S,,, = C VP’ exp (2i6,)VJY. II
(2.1.3)
If it were possible to have a fore-knowledge of the likely position of resonances in the calculated cross sections, the task of scanning the cross section in arbitrary steps to obtain the detailed shape would be greatly simplified.
R. F. BARRETT
644
2.2. BOUND
STATES EMBEDDED
AND P. P. DELSANTO
IN THE CONTINUUM
It is known that sharp resonances in the continuum may be brought about by the coupling of single-particle resonances or bound states to the continuum states 19). As the various holes have different threshold energies, a nuclear excitation E, which is sufficient to lift a particle from a level near the Fermi surface into the continuum may not be sufficient to lift a particle from a deeper lying level into the continuum. At certain sharply defined energies, however, E, may be sufficient to excite the particle from this low-lying level into a higher bound state. Subsequent particle emission may then occur by the coupling of this bound channel to an open continuum channel by means of the residual interaction. The effect of the residual interaction is to shift the energy of the resonance, broaden it and change its character by the admixture of other configurations.
1
0Den continuum
---------unfilled / levels \
A Fermi surface
Fig. 1. Potential well (schematic) showing possible transitions from a bound level to (a) a higher bound level; (b) a single particle resonance level; (c) the open continuum.
Accepting the origin of sharp resonances to be that described above, we would expect some correspondence between the states found in a bound-state calculation and the peaks obtained in a continuum calculation, although we would not expect them to be at exactly the same energies, even if the same strength and type of residual interaction were used in both cases ‘O). In the eigenchannel method it is known that the basis vectors of the Hilbert space are a function of the eigenphase 6,. This is a result of the dependence of the boundary conditions applied to the wave function at the matching radius on the phase 6 (see subsect. 2.1). At a given excitation energy above the particle emission threshold, some channels may be closed, others may be open but with particle emission restricted by the presence of a Coulomb or centrifugal barrier, while others are completely open. These three situations are illustrated in fig. 1. To obtain the single-particle energies
GIANT DIPOLE RESONANCE
645
and wave functions of the basis states, the Schrodinger equation is solved with the boundary conditions implied by 6 and E, applied at the matching radius. We select a value of the excitation energy which is in the open continuum region for all channels and examine the variation of the eigenvalues of the Schrddinger equation solved in the inner region as a function of the phase 6. The energies of the bound single-particle states lying in the bound region of fig. 1 are independent of any boundary conditions applied at the comparatively large matching radius; for the single-particle resonance states in fig. 1, this is approximately true; while in the open continuum region, the energies of the single-particle states depend directly on the boundary conditions. A typical plot (ideal) of the eigenvalues of the Schrddinger equation as a function of the phase 6 is shown in fig. 2 for given values of the angular quantum numbers I andj.
.;i rll .Z
s
d
3
u”
0
180
6 (degrees)
Fig. 2. A plot (schematic) of the eigenvalues of the Schrbdinger equation as a function of the phase 6 (excitation energy fixed) for arbitrary values of the angular quantum numbers I and j.
It is now instructive to consider the diagonal energies of the shell-model Hamiltonian in the internal region at a given excitation energy as a function of phase 6 when the residual interaction is switched off. The diagonal energies E:“‘(6) for a given configuration L are then simply given by
E:“‘(6)=
P?)(d)-Q,
,
(2.2.1)
where P:“‘(6) are the single-particle energies obtained from fig. 2, and Q, is the threshold energy of the configuration. The threshold energies for the different configurations may be quite different. The dependence of the diagonal energies Ep’ upon phase 6 for the case of “*Pb is shown in fig. 3 for two different values of the excitation energy. The positions of the bound-state and single-particle resonance configurations may be clearly seen. As the excitation energy enters the internal shell-model calculation only through the boundary conditions applied at the matching radius, and as the eigenvalues corresponding to bound states are independent of these boundary conditions, the positions of the bound states in fig. 3 are also independent of the choice of the excitation energy E,. This may be seen from a comparison of figs. 3a and 3b.
R. F. BARRETT AND P. P. DELSANTO
646
PHASE (DEGREES)
(a)
PHASE (DEGREES)
Ib)
Fig. 3. ‘The unperturbed configuration energies as a function of the phase 6 for ‘OsPb with the excitation energy E, equal to (a) 16 MeV; (b) 20 MeV.
We now consider the effect of switching on the residual interaction between the configurations. As the shell-model Hamiltonian cannot have two equal eigenvalues when the off-diagonal terms are non-zero, the channel lines in fig. 3 now do not cross. The residual interaction also shifts the energies of the states (sometimes substantially as in the case of the giant dipole resonance) and changes their character by the ad-
647
GIANT DIPOLE RESONANCE
mixture of different configurations, with the effect, for instance, of concentrating most of the dipole strength in the giant resonance region. The unperturbed diagonal energies of the bound states embedded in the continuum are independent of the phase 6. This tendency persists even after diagonalization in the sense that the corresponding eigenvalues are relatively independent of the phase 6 over a wide range of values of 6. Plots of eigenvalues En(“) after diagonalization versus phase 6 were shown for the case
15
74
13
> f -12 lJ.7
11
10
9
PHASE(DEGREES) (4
PHASE
(DEGREES)
(b)
Fig. 4. The eigenvalues of the shell-model Hamiltonian solved in the inner region as a function of the phase S for zoaPb with the excitation energy E, equal to (a) 16 MeV; (b) 20 MeV.
648
R. F. BARRETT AND P. P. DELSANTO
of “*Pb in fig. 4 for two different values of the excitation energy E,. The positions of the BSEC may be seen. In the region of any BSEC there exists an eigenvalue E$“’ of the shell-model Hamiltonian which is approximately independent of the phase 6 and thus of the boundary conditions at the matching radius. It is also independent of E, for the reasons given above, as may be seen from a comparison of figs. 4a and 4b. The expected positions of the resonances in the calculated cross sections may therefore be obtained by a series of diagonalizations at only one excitation energy. This eliminates the likelihood of narrow resonances, such as those occurring in “‘Pb in the region below 10 MeV, being missed, as could otherwise occur in a search procedure carried out in finite energy steps. The strength and shape of the resonances are found by a detailed determination of the eigenphases at different excitation energies and described briefly in subusing the procedure outlined by Wahsweiler et al. sect. 2.1. As the positions of the resonances are known, excitation energies around these positions may be chosen and the computation of the cross section is greatly facilitated. The origin of a specific resonance can be inferred from the relative contribution of the different configurations to its dipole matrix element. If at the resonance energy, but nowhere else in the immediate neighbourhood, the dipole matrix element contains a large contribution from a bound-state configuration, it means that this bound state is strongly coupled to the continuum by the residual interaction at this excitation energy. The BSEC origin of the resonance is then obvious. A procedure enabling the calculation of the configuration mixing occurring in resonances calculated from the eigenchannel theory is now described. 2.3. CONFIGURATION
MIXING
In a bound-state catculation, a diagonalization of the shell-model ~amiltonian is performed on a basis of particle-hole configurations. After diagonalization, an eigenstate Iv} of the Hamiltonian is represented by a mixture of the basis configurations fen), i.e., (2.3.1) Iv> = c X&n>, c,n where X,, is the amplitude of the basis state Jcn), c represents all quantum numbers except the radial quantum number, and thus characterizes a configuration; n is the radial quantum number. The dipole strength of the state Iv) is proportional to the square of the dipole matrix element between the ground state Ii> and the state Iv): i.e., the dipole strength of Iv> cc ~
GIANT
DIPOLE
RESONANCE
649
In the eigenchannel theory, the basis states are a function of the boundary conditions applied at the matching radius, and are thus different for different eigenphases. Consider the lp-lh basis state wave function I&ln,;j,j+4;
JM) = C(-l)j”-“(ja
j,Mfm-mlJM)
X N,I,SJje M-l- m)*i(lA SA)jA ?n>ri i Ul,,l,i.(Y,)$~lUnAIAjA(rA)IT~ WXrA m:‘>, (2.3.3) see e.g., ref. “I). Uppercase subscripts or superscripts refer to particles; lower-case subscripts or superscripts refer to holes. The kets containing r characterize the charge of the nucleon; II is the radial quantum number. A variation of the boundary conditions at the matching radius results in a different radial wave function for the particle, i.e., ril umAIAjA(rA) is different for different eigenchannels. The angular part of the particle wave function, represented by l(l,s,)j,m) in eq. (2.3.3) is, however, unchanged. The basis state wave functions are of course orthogonal. The orthogonality between different branches of the same configuration, i.e., between states ln,n~;j,j,; JM) which differ only by the quantum number n, arises from the fact that the radial wave function unAIAjA(rA) is an eigenfunction of the radial Schrodinger equation solved in the inner region with the boundary conditions of the vth eigenchannel applied at the matching radius. The orthogonality between configurations i.e., where any quantum number other than n, differs, arises from the inherent orthogonality of the single-particle and single-hole angular wave functions l(i, 5’,)j,m), l(Z=~~)~=~+~>* and the single-hole radial wave functions. This orthogonality is independent of the boundary conditions applied at the matching radius. If we simplify the nomenclature as before by letting c represent all quantum numbers except n, and letting n represent n,, we have the lp-1 h basis wave function in the form lcn’). The v indicates the dependence of lcn’) upon the boundary conditions of the vth eigenchannel. From the arguments of the last paragraph, we have (cW’]cnV) = &cc’(cn’V’]c?zV>.
(2.3.4)
The dipole cross section at a specific excitation energy is given in the eigenchannel theory by (2.3.5) where M, = (+)*Ct~fi T
(G)‘l-
Z:-C
(2.3.6)
[see e.g. ref. 21)] where Ii) is the ground state’if the nucleus and Iv> is the vth eigenchannel = ccR X&n”>. The normali~tion of the basis wave functions to unity inside the matching radius is arbitrary. Hence a renormalization must be introduced to normalize the eigenvector of the S-matrix to unit flux. This is represented by the factor cc, (Cc.)” in eq. (2.3.6) [see ref. ““)I.
R. F. BARRETT
650
AND P. P. DELSANTO
Let us define a vector
f-K> = M, c XcnlCfiV)n
(2.3.7)
Comparing eq. (2.3.7) with eq. (2.3.1) we could choose z,, X,‘, to be the percentage contribution of the configuration c to the eigenchannel iv) and 6,
=
L@
h:fl@c 1412~ 0_e_
(2.3.8)
Y
the total contribution of a configuration c to the cross section cr at a given energy. Then P = ~,cF,, For this to be a meaningful definition, we must show that ~nlXcn12 is independent of the change of basis produced by the changing boundary conditions at the matching radius. We have (2.39) We change the basis I&‘) to another complete set Ic’n”“) describing the same space by changing the boundary conditions to those of the vth eigenchannel: (2.3.10)
i.e., = ; [ ;
X~n(C’n’~‘~CnP)](Crn’v’>, (2.3.11)
from eq. (2.3.4), where A’&, the equivalent of A’,,,on the new basis, is thus given by XLn = ; x,(cn’“‘[cI2”>.
(2.3.12)
Similarly (2.3.13)
X,, X~~(cmY~c~~‘v’)(c~‘Y’~cnV)(c~z’Y’~cn’Y’> (2.3.14) =
5 [ ; Xcn(cn'Y'lcnY)][
= ; IXd12~ &,JX,,,Iz is thus independent of any change boundary conditions at the matching radius, to calculate the configuration mixing in the Results of the calculation of contributions are presented and discussed in subsect. 4.1.
C X~~(cmYIcdv’)“j
(2.3.15)
m
(2.3.16)
in the basis produced by the changing and we are justified in using eq. (23.8) eigenchannel theory. from different configurations for *‘*Rb
GIANT DIPOLE RESONANCE
651
3. Choice of parameters iu the *“Pb calculation In this section, we discuss the choice of parameters
in the “*Pb
calculation.
The
lp-1 h configurations considered are listed in subsect. 3.1. Details of the central singleparticle potential for lead are given in subsect. 3.2, and the residual interaction between the lp-lh configurations is detailed in subsect. 3.3. TABLE 1 The l- configurations Neutron configuration
included in the zosPb calculation, Unperturbed energies (MeV)
Proton configuration
11.50 (estimated)
i* -‘_i+ iy-l
3.1. CHOICE
and their unperturbed
energies
Unperturbed energies (MeV) 15.45 (estimated)
14.65 (estimated
7.21
j,
6.51
h+-i ig
7.70
hs-’
ga
9.41
8.46
h+-’ gf ft-’ da
6.92
10.57
6.56
7.44
6-i
dt
5.58
7.73
f*-’ g*
6.51
8.25
P*-i s+
5.49
7.90
p_t-id+
5.99
7.38
P*- ’ s*
6.39
9.05
P+-’ ds
6.89
8.76
P+- i di f;-’ d+
5.91
6.83
f*- i gj;
8.29
f$- i gp
5.80
hq-’ gq
13.40 (estimated) 7.47
7.36
OF CONFIGURATIONS
In the calculation described in this paper, we have included all of the lowest order l- configurations emanating from the last filled shell in “‘Pb. The configurations considered are listed in table 1, and the unperturbed energies shown. In order to diagonalize the shell-model Hamiltonian in the internal region, only a limited number of radial quantum numbers can be considered for each configuration, i.e., the infinite set of basis vectors of the Hilbert space must be arbitrarily truncated. In this calculation, we have considered four branches for each configuration. The
652
R. F. BARRETT AND P. P. DELSANTO
effect of truncation of the Hilbert space in an eigenchannel calculation has been investigated by Delsanto et al. 22). 3.2. SINGLE-PARTICLE
POTENTIAL
is customary in the particle-hole model to represent the average single-particle potential by some form which, although somewhat arbitrarily, reproduces the basic features of the nucleus studied (e.g., the nuclear radius and the single-particle energy level scheme as observed experimentally). In table 2, the single-particle energy level scheme is listed for “‘Pb. The particle and hole energies are known from experiments performed on the nuclei “‘Pb, 2ogBi, “‘Tl and 207Pb. References to such experiments are given in table 2. It
TABLE
The single-particle Ref. ’ 3, neutron particles
energy (MeV)
2h+ 3d3. 2gg 4%
energy (MeV)
proton particles
Ref. *6)
Ref. t5)
Ref. 24)
li+
lj*
2
and single-hole energy level scheme for “*Pb
neutron holes
proton holes
(MN
energy WeV) ______-
34
-
7.38
3%
-
8.05
2%
-
7.95
2d3
-
8.40
lhJ+
-
9.39
3di, lgr;
-
9.72
-1.39
2ge 3pt.
-0.15
3P+
-
8.28
-1.44
3p*
-0.67
lig
-
9.02
-1.89
2f+
-0.96
2f;
-
9.73
3dt
-2.37
lip
-2.18
lh%
- 10.85
lj,
-2.51
24
-2.89
Ii+
-3.15
*he
-3.79
2g*
-3.93
-11.53
In this calculation, we have assumed the single-particle potential Y(r) to be of the Woods-Saxon form: V(r) = K [~(~)-r
(~)2(z
- 4 ‘, !$I
+ fLl”i1
(3.2.1)
where V_u*=gJ-i,il-
Ze2 r
9
for
r 5 Rc
for
r > R,
(3.2.2)
and p(r) = [l-i-exp ((r-l-Ro)lt)]-i.
(3.2.3)
GIANT DIPOLE RESONANCE
653
Hzre R. denotes the nuclear radius, Z is the charge of the residual nucleus, M is the particle mass, t is the surface thickness parameter of the Fermi distribution, V, is the depth of the Woods-Saxon well, y is the spin-orbit coupling constant and Rc is the Coulomb radius. Values of R, = 7.52 fm and t = 0.7 fm were chosen in agreement with the parameters used by Blomqvist and Wahlborn “) for a Woods-Saxon potential in the lead region. A value of Rc = 7.0 fm was used in agreement with the equivalent uniform radius obtained from electron scattering data “‘). The parameters Vc and y were then optimized for each level (or doublet) in order to fit the calculated energies of the singleparticle and single-hole states accurately to the experimentally measured values. In this way the symmetry potential (i.e. the part of Vc due to the neutron excess) was automatically included. To discretize the continuum, boundary conditions are applied at an arbitrary “matching radius”, which must be sufficiently large that all nuclear interaction can be considered to take place within it. An eigenchannel matching radius of 17.0 fm was used in this calculation. This value was chosen by comparison of the magnitude of the single-hole wave functions at various radii for “*Pb with those obtained in an earlier eigenchannel calculation for I60 at the I60 matching radius ‘). In order that the nuclear interaction between configurations is negligible at the matching radius, it is necessary that the hole wave functions are vanishingly small there. In the case of 160, it has been shown by trial that the chosen matching radius (12 fm) is adequately large, so we infer that 17.0 fm is a large enough value in the case of lead.
3.3. RESIDUAL
INTERACTION
The effective two-body force which is responsible for the residual interaction is assumed to be of the form T/(1,2) = v0J(r,,)[a0+a,a(1)~0(2)+a,z(1)~~(2)+a,,a(1)~a(2)r(1)~2(2)].
(3.3.1)
In this calculation, we have made similar assumptions to those made in earlier eigenchannel calculations ‘); i.e., we have employed a contact force Jh2)
=
6(rl -r2)
(3.3.2)
and we have taken the exchange mixture to be of the Soper type. Due to the large size of the matrix to be diagonalized in the case of lead, it was not possible to vary the strength of the zero-range force I/, in order to optimize the calculated position of the giant resonance. However, in an existing bound-state calculation, Balashov “) employs a similar zero-range Soper force with a strength I’, of - 1220 MeV * fm3. He obtains the position of the giant resonance at 13.8 MeV in reasonable agreement with experiment. We have used the same value, V, = - 1220 MeV * fm3 in the continuum calculation described here.
654
R. F. BARRETT
4. Presentation
and discussion
AND
P. P. DELSANTO
of the results of the eigenchannel
calculation
In this section, we present and discuss the results of the eigenchannel continuum calculation for “‘Pb. In subsect. 3.1, the results of the continuum calculation for the total cross section are compared with the predictions of bound-state calculations, and 2 5
3) Perez
5,"
I-
I 2
d)
z
2m
Balashov et a1. (1962)
-f
5
;
lcoO-2 a, g d 7
9
10 Excitation
Fig.
5. The results
I
I, 8
of bound-state
11
12
13
li
15
lk
Energy(MeV)
shell-model
calculations
for
“@Pb.
GIANT
DIPOLE
655
RESONANCE
in subsect. 3.2, a comparison with experiment is made. In subsect. 3.3, the calculated partial cross sections are presented and the angular distributions given. 4.1. COMPARISON BOUND-STATE
OF RESULTS OF EIGENCHANNEL CALCULATIONS
CALCULATION
WITH THOSE
OF
Previous bound-state calculations for “‘Pb have been performed by Balashov Gillet et al. lo), Kuo et al. ’ “) and Perez ’ “). The results of their calculations are summarized in fig. 5. The energy of the giant dipole resonance obtained by Balashov et al. is higher than that obtained by Gillet et al. and Kuo et al. This is possibly due to the somewhat higher values of the unperturbed configuration energies employed by him. Kuo et al. comment that the energy of the giant dipole resonance obtained from their calculation is too low by several MeV. They indicate that the use of appropriate Woods-Saxon wave functions instead of oscillator wave functions for the single-particle states could shift the dipole strength to higher energies. This has been verified in the most recent “‘Pb bound-state calculation of Perez. He employs Woods-Saxon wave functions, and obtains the position of the giant dipole resonance in much better agreement with experiment. His results are shown in fig. 5a. et al. 9),
Pb
8 Fig. 6. The total photonuclear
10 Excitation
12 Energy
1L
208
16
(MeV)
cross section of zosPb from the eigenchannel
calculation.
In fig. 6, we present the total photonuclear cross section of “*Pb obtained from the eigenchannel calculation described here. As indicated in subsect. 2.2, a correspondence is expected between the resonances found in a continuum calculation and the states found in a bound-state calculation. From fig. 4, we have seen that BSEC are located at energies of 10.1, 10.85, 11.4, 12.875, 13.8 and 15.35 MeV, as well as at a number of low energies. These energies are therefore the expected positions of resonances in the photo cross sections. A complete determination of the photoabsorption cross section reveals that resonances are indeed located at these positions, with the
656
R. F. BARRETT
AND
P. P. DELSANTO
exception of the one expected at 10.1 MeV. This resonance is too weak to be resolved above the background. The relative strengths of the states in the bound-state calculation by Kuo et al. Is) are in good agreement with the results of the eigenchannel calculation. It must be remembered, however, that in the former calculation, the random-phase approximation has been used. The energies of the states obtained by Kuo et al. are lower than those of the corresponding resonances shown in fig. 6. This has been attributed to the use of harmonic oscillator wave functions by Kuo et al. At energies below IO MeV in the case of ‘08Pb, few channels are really open. Due to the high Coulomb and centrifugal barriers, most channels are of the single-particle resonance type shown in fig. 1. This results in a number of sharp resonances at these energies. We found, however, that these resonances do not carry much dipole strength, a result in agreement with the large number of weak states found in this region by Kuo et al. in the corresponding bound-state calculation. The technique described in subsect. 2.2 was used to accurately locate the positions of these resonances. The positions of the corresponding BSEC may be seen in fig. 4. Due to the extreme narrowness of these low-energy resonances, their location by means of a normal search procedure as used by Wahsweiler et al. ‘I) would have been extremely tedious, if not impossible. An analysis of the con~guration mixing occurring in the major resonances was carried out using the procedure detailed in subsect. 2.3. In fig. 7 are shown the contributions to the cross section from the most important configurations. The BSEC origin of the individual resonances may be deduced to be the fohowing: Tire 10.8 MeV resonance. Predominantly due to the i$‘h, neutron and h;‘g% neutron configurations which have unperturbed configuration energies of 11.5 MeV and 9.4 MeV, respectively. 7%e 11.4 MeV resonance. Due to the g; if, proton configuration turbed energy of 10.55 MeV.
with an unper-
The 12.875 MeV resonance. The giant dipole resonance, consisting of a mixture of many con~guratioils, the highest contribution from any one configuration being 11 %. The 13.75 MeV resonance. Due predominantly with an unperturbed energy of 13.4 MeV.
to the hi’&
proton configuration
The 15.35 MeV resonance. Due to the ii’j, neutron configuration with an unperturbed energy of 14.65 MeV. Two effects may be noted from these results. Firstly, the effect of the residual interaction is to shift the position of the BSEC by an amount of the order of 1 MeV from their unperturbed positions. Secondly, the giant dipole resonance state is a highly collective mixture of many configurations, as predicted by the Brown-Bolsterli schematic model. This result is also in agreement with the results of the bound-state calculations of Gillet et al. i”).
657
GIANT DIPOLE RESONANCE
In subsect. 4.2, we compare the results of the eigenchannel calculation for “‘Pb with those of experiment. -1
300
200 100
$2%,2
l---.-d h11,2 g%*
P
2cQtoo-
0--
0
n
200-
n
P
-1
92
=7/2
lOI0
loo0 loo 0
41,
ill,2
P
n 2005. &
1000. A too-f-’ 7/2 %A
n
b
10
Fig. 7. A plot of the contributions
-I
16
16
to the total photo cross section from the most important figurations.
4.2. COMPARISON OF THE RESULTS EXPERIMENTAL RESULTS
OF EIGENCHANNEL
CALCULATION
con-
WITH
Fig. 6 shows the total photonuclear cross sections obtained from the eigenchannel calculation. Because the extremely high Coulomb barrier for lead inhibits the emission of protons until excitation energies of about 17 MeV are reached, the predominant contribution to the total cross section comes from the photoneutron reaction.
R. F. BARRETT AND P. P. DELSANTO
658
The (y, n) cross section for lead has been measured by several authors using either quasi-monochromatic y-rays or bremsstrahlung radiation as a source of photons. These measurements are in general agreement as to the position, height and width of the giant resonance. The most recent measurement using the positon-annihilation-inflight technique
by Beil et al. 12) has detected
structure
.-. Excitation
_1_
on the increasing
slope of the
I--I
14 16 Energy(MeV)
18
b)
Excttation
Fig. 8. The experimental
Energy(MeV)
208Pb photoneutron cross section due to (a) Beil et al. I’); (b) Goryachev et al. 13); and (c) Harvey et a/. 29).
giant resonance which was not observed in earlier measurements by Harvey et al. 2 ‘) using the same technique. The bremsstrahlung experiments of Fuller and Hayward [ref. ““)I, and Tominasu 31) have detected a non-Lorentzian shape to the giant resonance on the low-energy side; that of Goryachev et al. 13) has detected structure on both the low-energy and high-energy sides of the main peak. For comparison. the results of Harvey et al., Goryachev et al. and Beil et al. are presented in fig. 8.
GIANT DIPOLE RESONANCE
659
A determination of the photoneutron spectrum obtained by the irradiation of lead by bremsstrahlung of maximum energy 31 MeV, performed by McNeil1 et al. 32) has detected structure superimposed on the evaporation neutron spectrum. He has correlated this structure with the peaks in the (y, n) cross section detected by Beil et al. A comparison of the experimental results with those of the eigenchannel calculation reveals that the theoretically predicted position of the giant resonance is too low by 0.6 MeV. This position, however, depends on the residual interaction strength chosen, which for reasons of computing time could not be optimised. The calculated height of the giant resonance is found to be 3.2 b and the FWHM to be 0.70 MeV. These values may be compared with the experimental values of 0.64 b and 4.05 MeV respectively. It is a characteristic feature of cross sections calculated using a continuum shell model in which only the most important lp-lh configurations are included, that the calculated resonances are too high and too narrow. For example, in the lp-lh shellmodel calculation for 40Ca [ref. “)] the discrepancy in the calculated and measured height of the main giant resonance exceeds a factor of five. It is expected that the inclusion of higher-order np-nh configurations should lower and broaden the resonances. Buck and Hill “) have simulated the effect of the omitted configurations in a coupled channel calculation for I60 by including an imaginary absorption term in the Hartree-Fock potential. This had the effect of bringing the results of their ~al~ulation into better agreement with experiment. A similar effect could be expected in the case of “‘Pb. Danos and Greiner 33) have calculated the damping of the giant resonance in heavy nuclei and obtained spreading widths in the range 0.5 to 2.5 MeV by examining the coupling of the lp-lh configurations to 2p-2h configurations. The coupling of the p-h configurations to collective states such as surface quadrupole and octupole vibrations 34) may also contribute to a spreading of the giant resonance. Examining fig. 6, it may be seen that the most pronounced of the minor peaks falls into the region in which structure is obtained in the measurements of Goryachev et al. and Beil et al. More experimental information is required, however, before any definite identification of peaks can be made. The integrated cross section up to 16 MeV was obtained by calculating the area under the curve in fig. 6. This was found to be 3.9 MeV - b. It can be compared with the experimental values of: (i) 2.91 MeV - b (integrated up to 28 MeV) obtained by Harvey et al. 29); (ii) 4.0 MeV * b (integrated up to 18.5 MeV) obtained by Goryachev et al. 13); (iii) 3.48 MeV * b (’in t egrated up to 25 MeV) obtained by Veyssiere et al. 12). The classical sum rule yields a value of 2.98 MeV * b. The results of the eigenchannel continuum calculations thus agree with the results of the bound-state calculations of Gillet et al. lo) in that the total integrated cross section for 208Pb is in agreement with experiment, whilst for 160, for example, the theoretical integrated cross section is about twice that of experiment.
R. F. BARRETT AND P. P. DELSANTO
660
4.3. PARTIAL CROSS SECTIONS AND ANGULAR
DISTRIBUTIONS
Once all of the eigenphases have been obtained, the partial photo cross sections are easily calculated using the technique described by Wahsweiler et al. in an earlier paper 2’). The results of such a calculation for “‘Pb are shown in fig. 9. In the energy range considered, the photo-proton partial cross sections are negligible. The predominant contribution to the giant resonance is seen to come from the (y, n,) reaction, while the peak at 13.75 MeV is due mainly to the (y, ns) reaction with contributions from the (y, nr) and (y, n,) reactions. A comparison of these results with experiment is not possible because of a lack of experimental data.
10
16 Excitadf (4
Energy C&V)
1)I/’
OL
10
I
,\,
12 Excitation Energy’?MeV)
tb)
(4
16
GIANT DIPOLE RESONANCE
Fig. 9. The partial photoneutron
661
cross sections for 208Pb as calculated reaction theory.
from the eigenchannel
TABLE 3
The potential parameters used in this calculation to reproduce the single-particle energy level scheme of zosPb Proton levels
Y
Neutron levels
lj*
-44.79
37.28
%
-44.4
30.49
-45.3
23.9
-44.94
26.11
-45.47
0.0
-45.3
23.9
li+
-51.98
37.36
2ge
- 59.28
27.40
3p+
-60.44
13.86 13.86
Y
3p3
- 60.44
24 li+
-59.71
20.68
-51.98
31.36
3% % 4% 3%
2f$
-59.71
20.68
1jy
-44.79
37.28
lhe
-59.53
33.68
li+
-45.0
33.86
3%
-60.28
0.0
2%
-44.94
26.11
2d+
-59.71
22.46
3p3
-44.38
32.54
lh jl
-59.53
33.68
26
-43.81
25.17
3dt
-59.71
22.46
32.54
lgt
-59.28
27.40
3p+ li+
-44.38 -45.0
33.86
26
-43.81
25.17
LhS
-44.4
30.49
R. = 7.52 fm,
R, = 7.0 fm,
t = 0.7 fm
662
R. F. BARRETT AND P. P. DELSANTO
The angular distributions may also be easily calculated for the individual partial reactions once the eigenphases are known. If the angular distribution W(e) of the emitted neutrons is expanded in terms of 0 where 0 is the angle made with the beam direction, we have W(8) = a, + a2 P,(cos q, (4.3.1) where P, is the nth order Legendre polynomial. The ratio az/ao is plotted over the giant resonance for the partial cross sections (y, n,) and (y, n,) in fig. IO. Measurements of the angular distribu~ons of photoneutrons from 208Pb have been made by several authors 3 5- “). These measurements have the common feature that they were made using a bremsstrahlung photon source and a threshold neutron detector, so that only neutrons of energy greater than a certain value were detected. Usually peak bremsstrahlung energies of 20-30 MeV are chosen with neutron detection thresholds of x 5 MeV. A list of such experiments is given in ref. ’ “).
11 Fig, 10. Anguiar
Excitotk
Energy
fMeV
14
15
)
distribution of the reaction products as calculated for the zosPb(y, no) and 2osPb(y, II,) reactions from the eigenchannel reaction theory.
The greatest contribution of neutrons comes from the giant resonance region of the cross section, and the use of a 5 MeV threshold detector means that only the (y, no) and (y, nl) partial reactions contribute any appreciable fraction of the neutrons detected. The largest contribution comes from the (y, n,) reaction. The values of at/a0 obtained experimentafly are seen from refs. 3 5-37) to lie mainly between -0.3 and -0.5. This is in agreement with the results of the eigenchannel calculation shown in fig. 10 to the limits of the experimental accuracy. More precise experiments are, however, needed. 5. Conclusion The total and partial photo cross sections for ‘08Pb have been calculated in the Ip-lh continuum shell-model approximation using the eigenchannel reaction theory.
663
GIANT DIPOLE RESONANCE
The calculated position of the giant dipole resonance is in reasonable agreement with experiment, supporting the conclusions of Perez 14) that the use of Woods-Saxon wave functions is necessary in “‘Pb shell-model calculations. The value of the total integrated cross section obtained is also in agreement with experiment. The identification of the various subsidiary resonances as bound states embedded in the continuum has been unambiguously accomplished. The configurations primarily responsible for this intermediate structure are: i,‘h,n;
h;‘g$n;
g;‘f+pr.;
h$lg$pr.
and
i,‘j,n.
It was seen that due to the high Coulomb and centrifugal barriers in the case of 208Pb, there exists at low energies a number of sharp resonances which carry little dipole strength. As is characteristic of the continuum shell-model calculations performed to date, the giant dipole resonance state is too high and too narrow compared with experimental measurements. The inclusion of additional np-nh configurations is expected to spread the dipole state, producing better agreement with experiment. The lp-lh continuum calculation described here represents only a first step towards the understanding of the 208Pb photo cross section. The inclusion of extra configurations, such as those describing collective correlations, or the use of a more sophisticated nuclear model, might be expected to contribute further to the explanation of the structure found experimentally. Such calculations have not yet been performed in a framework which correctly treats the continuum. The authors would like to thank Professor W. Greiner for the hospitality offered by him at Frankfurt, and for his encouragement and assistance. They would also like to thank Prof. M. Danos, Dr. P. Antony-Spies and Dr. A. Rabie for helpful discussions. The provision of computing facilities by the Z.R.I., Frankfurt is gratefully acknowledged. References 1) L. C. Biedenharn, M. Danos, P. P. Delsanto, W. Greiner and H. G. Wahsweiler, Rev. Mod. Phys., to be published 2) M. Marangoni and A. M. Saruis, Nucl. Phys. Al32 (1969) 649 3) B. Buck and A. D. Hill, Nucl. Phys. A95 (1967) 271 4) J. Raynal, M. A. Melkanoff and T. Sawada, Nucl. Phys. Al01 (1967) 369 5) J. Eichler, Nucl. Phys. 56 (1964) 577 6) M. S. Weiss, Phys. Lett. 19 (1965) 393 7) A. M. Saruis and M. Marangoni, Nucl. Phys. Al32 (1969) 433 8) D. A. Bromley and A. Weneser, Comments on Nucl. and Part. Phys. 2 (1968) 151 9) V. V. Balashov, V. G. Shevchenko and N. P. Yudin, JETP (Sov. Phys.) 14 (1962) 1371 10) V. Gillet, A. M. Green and E. A. Anderson, Nucl. Phys. 88 (1966) 321 11) G. E. Brown and M. Bolsterli, Phys. Rev. Lett. 3 (1959) 472 12) H. Beil, R. Berg&e, P. Carlos and A. VeyssiBre, 269 (1969) 216; A. VeyssiBre, H. Beil, R. Berg&e, P. Carlos and A. Lepetre, to be published 13) B. I. Goryachev, V. S. Ishkanov, I. M. Kapitanov and V. G. Shevchenko, JETP Lett. (Sov. Phys.) 7 (1968) 161
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