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Gamma decay of the giant quadrupole resonance in 208Pb
Volume 148B, number 1,2,3
PHYSICS LETTERS
22 November 1984
GAMMA DECAY OF THE GIANT QUADRUPOLE RESONANCE IN 208pb P.F. BORTIGNON 1
University o f Padova, Zh'partt'rnento di t~sica Galileo Galilei, Padua, Italy and INFN, L.N. Legnaro, Italy
R.A. BROGLIA The Niels Bohr Institute, University o f Copenhagen, DK-2100 Copenhagen ~, Denmark and G.F. BERTSCH Department o f Physics and Cyclotron Laboratory, Michigan State University, East Lansing, M148824, USA Received 11 June 1984 Revised manuscript received 18 August 1984
The gamma-decayof the isoscalar GQR and of the GHR (~ 11 MeV) of 2°8pb to the lowqyingoctupole vibration is calculated in the surface coupling model. A marked quenching of the transitions is observed arising from the correlation between the particle and the hole participating in the vibration, and from the couplingto the GDR. The total F3,(GQR--*3D is found to be only a few percent of F.r(GQR~ gs) consistent with the experimental data. The compound nucleus gammadecay is estimated to be negligible.
The decay of giant resonances is still not a wellunderstood subject in nuclear spectroscopy, giving motivation to the testing of models against available experimental data. The main mechanism damping giant resonances appears to be the excitation of lowlying surface vibrations, induced by the interaction of the nuclear surface with the particle and hole of the resonance (cf. ref. [ 1] and references therein). For example, a major role in the damping of the giant resonances in 208pb is played by the 3 - state at 2.6 MeV excitation. This might be observable in the gamma-decay of the giant resonances, which should have a branch to the octupole vibration as a result of the coupling. Also in the particle decay populating multiplets based on the 3 - state [2]. In general, electric dipole transitions are the most likely to be observed experimentally. This fact restricts the multipolarity of the giant resonances which may be studied through gamma-decay. Another important 1 Partially supported by MPI of Italy. 20
feature restricting the usefulness of these studies is the energy of the mode. This is because for states below the giant dipole resonance (GDR), the E1 effective charge becomes very smaU, while for transitions having similar or larger energies than the GDR the situation is reversed. An excellent candidate for gamma-decay studies is the isovector giant quadrupole resonance (GQR) which can decay through enhanced transitions to the lowlying octupole vibrations as well as the GDR which can decay to the low-lying quadrupole vibration again with enhanced strength. The isoscalar GQR can also decay to the low-lying 3 - but with a much reduced transition strength as compared with the single-particle transition. Recently an experiment has been performed in 208pb where the photon decay from the energy regions associated with the isoscalar GQR, GMR and GDR was studied. The results have been reported in ref. [3]. In the present paper we use the surface model developed in ref. [4] to calculate the decay associated 0370-2693/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 148B, number 1,2,3
PHYSICS LETTERS
with the isoscalar GQR as well as the ~ V = 2 component of the isoscalar giant hexadecapole resonance (GHR). Both modes lie in the same energy region and have been observed in hadron experiments (cf. ref. [5] and references therein). The properties of these vibrations are shown in table 1. Also shown are the corresponding results of an RPA calculation making use of a Hartree-Fock single-particle basis calculated with a Skyrme Ill interaction and a separable force where the radial dependence is parameterized
Table 1 Properties of the isoscalar GQR and AN = 2 component o f giant hexadecapole resonance [5]. The predictions of the surface coupling model [4] are also shown. j~r
2+ 4+
E(MeV)
r(MeV)
EWSR(%)
th.
exp.
th.
exp.
th.
exp.
11.2 11.0
11.4 12.2
2.6 5.3
2.7 4.5
72 18
70 30
.~.~-x X'~
22 November 1984
~= GDR
(A) X-3-,
X---
X
--
X--
I X =GQR
(B)
4,
X----(
- -X
+
I I GQR
GQR
GQR .(C)
(D)
(E)
Fig. 1. Processes associated with the decay of the giant quadrupole resonance into an octupole vibration. Lines label fermion fields while wavy lines label boson fields. A horizontal dotted line ending in a cross represents an external electric dipole field. In (A) we show the processes leading to the def'mition o f an effective charge described by a dashed circle on which the external field acts. The ~-ray can interact directly with a particle-hole vertex or it can first excite the giant dipole resonance ((;DR) which decays into a particle-hole pair. Thus the second graph in (A) where the phonon has been drawn as a horizontal wavy line stands for all possible time orderings associated with the action of the external field. A similar discussion applies to the process described by the second graph in (B). A similar group o f diagrams to both (A) and (B) are obtained interchanging the role o f particles and holes [graphs (C) and (D) ]. The graphs in the lhs o f (A) have a very similar structure to the diagrams entering in the calculation o f the strength function of the GQR [cf. graphs (E)].
21
PHYSICS LETTERS
Volume 148B, number 1,2,3
according to R,aU/ar (cf. ref. [4] for more details). The lowest order diagrams describing the decay processes are shown in figs. IA and 1B. A similar set of diagrams exists where the octupole vibration in graph (A) or the external field and the GDR in graph (B) operates on the hole state [cf. figs. 1C and ID]. The first graphs in figs. IA and IB are evaluated by the following expressions, with h = 2, A’ = 3 and X” = 1:
jlhJ.3
,E
[7,(j2) -(N--2)/A] ~~ __
x AxA,[(2h+
1)/(2h’+ l)] l/2
= -4e
; (
X M(jlj3; Wftjjl;
i”; i”I’ 1
A’) (1)
,
X ci~ll~(El)II12)/[DE~(E+iA/2)]o and <3-llm(El)llGQR(E)>(~) = -fejzis
]rZ(i;)~
-W+W4
x ~~~~~(-1)*“+h+iz+~3
[(2~+1)(2~‘ti)]1/2
22 November 1984
-iebz0.2)-
V-z)/-41
iA/2)lcB).
(2)
In the above equation M(13;X)= (jll!Ro(aU/&+)Y,llj3) with U(r) the single-particle potential, while A1 is the particle-vibration coupling constant determined by the normalization of the vibration [6]. The energy denominators in eq. (1) and (2) are given by [DEN@ t iA/2)] (Aj=(E+iA’/2-e13) X [E t i A/2 - (fic+t + ~23)] ,
(3)
and [DEN@ t iA/2)](BJ=(E+iA’/2
-e13)@+~
-~23).
(4) The giant resonances have a small spurious component in the shell model representation which is eliminated by using the operator
(5)
for the El transition charge. Both energy denominators include fmite imaginary parts. The practical reason for this is that the energy denominator can vanish otherwise. This happens in the graphs (A) of fig. 1 at energies corresponding to real transitions to the intermediate states. In (4) the energy denominator vanishes at energies of particlehole states with the resonance quantum numbers. From a theoretical point of view, the quantities A and A’ take account of the coupling to more complicated configurations not included in our space. The average transition strength will depend on the magnitude of A and A’, with less transition strength the larger these quantities are. No explicit calculations of these quantities are available. Empirically, the smoothness of the giant resonance peaks is consistent with A in the range 0.5-I .O MeV. We have chosen A = 1 MeV in our calculation in keeping with ref. [4]. The external field is partly screened by the mediation of the giant dipole resonance, as shown in the second graphs in figs. 1A and 1B. The sum of all time orderings of the dipole propagator in these graphs leads to a simple renormalization of the coupling (cf. eq. 6-330 of ref. [7]). This is given by
eeff=eP+xW31 X ~i~llrtl(El)llj~)/[DEN(E+
,
(6)
,
where the dipole polarizability
is
X(AE) = -0.76(L~,)~ X [(fiwD t AEtirD/2)(RwD
-AE-irD/2)1-‘. (7)
The quantity AE = E - Tzo,- is the energy of the dipole transition from the GQR at an energy E to the octupole state. In the calculations the experimental values Rw, = 14 MeV, I’, = 4 MeV have been used. At energies of interest, the cancellation between the two terms in eq. (6) is very strong, and the main contribution comes from the imaginary part of X with 11t~12~0.07atE=8.5MeV. Assuming that the strength of the giant resonance at the RPA level is concentrated in a single state, the formula for the gamma-width associated with the transition to the octupole state is dr,(GQR(E)
+ 3-)/dE=
f I1 +x(AE)12(AE,,)3
X 1.05 lu-ilm(E1)IIGQR(E))t~2,aPGQRQeV.
(8) 22
Volume 148B, number 1,2,3
PHYSICS LETTERS
The function P(E) is the probability per unit energy to find the GQR at an energy E. We calculate this using the same energies and particle-vibration couplings as we used for the transition. The main spreading of the P(E) distribution comes from graphs shown in fig. 1E. Details of that calculation are given in ref. [4]. The strength functions associated with the GQR and GHR used in the calculations are displayed in fig. 2. In eq. (8), AE has units of MeV, and the reduced matrix element has units of e 2 fm 2. Despite the strong coupling of the octupole to the giant quadrupole resonance, the other factors in the amplitude combine to produce a rather small width in the end. A
0.3
F\\
(a)
\ \
~O.l
\. w
\
.\ \
0.1
/1"
\.
\
\
/
\.
I
!'o I
I
I
I
17
E (MeV)
I
1
I
/\. ! !
0,3
I
I
I
12.6 dP~(GQR(E) -->3 - ) = --/ dE dE ~ 3.5 eV. (9) 9.8
(b)
We also calculated the dipole width associated with the AN = 2 GHR, which has been observed in the reaction studied in ref. [3], finding
/
i /
0.1
typical particle-hole dipole transition would produce a width of about 1.5 X 103 eV at a transition energy of 8.5 MeV. The Racah coefficient in eqs. (1) and (2) can be estimated from Wigner's semiclassical formula to be of the order of 0.1, so the recoupling times the square root containing the statistical factors reduces the strength by about 1/3. Because of the isovector nature of the operator (5), neutron and proton amplitudes tend to cancel when the states have predominantly isoscalar character. This is the case for the giant quadrupole and the low-lying octupole vibrations, and the calculations show that the interference reduces the width by about a factor of 4 below the width for proton amplitudes alone. Another source of interference is the cancellation between particles and holes, i.e. between the two graphs shown in figs. C and D. This is a general property found in the coupling of density fluctuation modes to other vibrations, and reflects the correlations between the particle and the hole existing in those vibrations (of. refs. [1,8] and references therein). The associated quenching factor is in the present case 1/2-1/3, comparable to what was found for the overall interference in damping. Finally, the effective charge in eq. (6) reduces the width by a factor of 15, to give a width of the order of electron volts. To facilitate comparison with experiment, we evaluate the integrated decay for the energy region 9.8 to 12.6 MeV, t'mding r,(GQR--> 3 i-)
I
I i it."
\..ji
\.\
r~(GHR -~ 3]-)
\
= I
8
I
9
22 November 1984
ilO
I
I
11 t
I
12 13 (MeV)
I
14
15
16
17
Fig. 2. Strength functions calculated in the surface excitation model and associated with the GQR (a) and GHR (b). The associated % of the EWSR associated with these modes are shown in table 1.
~
.6 dr~(GHR(E) -+ 3 - ) dE ~ 2 .6 eV dE 9.8
(10)
The octupole state can also be populated by compound nucleus decay, which will add to the direct decay we have calculated. Fortunately, the level density of the compound system appears to be high enough 23
Volume 148B, number 1,2,3
PHYSICS LETTERS
to make compound decay widths negligible. Using a constant temperature model [9] with parameters fixed by the level density of dipole states at 7-8 MeV [IO], we estimate lo3 .P = 2+ states per MeV at 11 MeV excitation, having an average dipole width of the order of less than 0.1 eV to the octupole state. Experimentally, only an upper limit has been established for the decay to the octupole vibration [ 111, of magnitude of the order of eV. Whether the experiment can be refmed to the point of confirming or refuting eqs. (9) and (10) remains to be seen. Discussions with J. Beene, F. Bertrand, C. Mabaux and F. Zardi are gratefully acknowledged. G.B. received support from the National Science Foundation for this work. References [l] G.F. Bertsch, P.F. Bortignon and R.A. Broglia, Rev. Mod. Phys. 55 (1983) 287.
24
22 November 1984
[2] R.A. Broglia and P.F. Bortignon, in: Giant multipole resonances (Harwood Academic,New York, 1979) p. 317; Phys. Lett. 1OlB (1981) 135. [3] FE. Bertrand, J.R. Beene and T.P. Sjoreen, in: Proc. Symp. on Highly excited states and nuclear structure, eds. N. Marty and N. Van Giai, J. Phys. (Paris) 45 (1984) c4 99. [ 41 P.F. Bortignon and R.A. Broglia, Nucl. Phys. A37 1 (1981) 405. [S ] H.P. Morsch,P. Decowski, M. Rogge,P.Turek,L. Zemto, S.A. Martin, G.P.A. Berg, W. Hurlimann, J. Meissburger and J.G.M. Romer, Phys. Rev. C28 (1983) 1947. [6] P.F. Bortignon,R.A. Broglia,D.R. Bes and R. Liotta, Phys. Rep. 30C (1977) 305. [ 7 ] A. Bohr and B.R. Mottelson, Nuclear structure, Vol. II (Benjamin, New York, 1975). [8] P.F. Bortignon, R.A. Brogha and C.H. Dasso,Nucl. Phys. A398 (1983) 221. [9] D-l. Horen, J.A. Harvey and W. Hill, Phys. Rev. C24 (1981) 1961. [lo] RJ. Holt, HE. Jackson, R.M. Laszewski and J.R. Specht, Phys. Rev. C20 (1979) 93. [ 1 l] J. Beene and F. Bertrand, private communication.
PHYSICS LETTERS
22 November 1984
GAMMA DECAY OF THE GIANT QUADRUPOLE RESONANCE IN 208pb P.F. BORTIGNON 1
University o f Padova, Zh'partt'rnento di t~sica Galileo Galilei, Padua, Italy and INFN, L.N. Legnaro, Italy
R.A. BROGLIA The Niels Bohr Institute, University o f Copenhagen, DK-2100 Copenhagen ~, Denmark and G.F. BERTSCH Department o f Physics and Cyclotron Laboratory, Michigan State University, East Lansing, M148824, USA Received 11 June 1984 Revised manuscript received 18 August 1984
The gamma-decayof the isoscalar GQR and of the GHR (~ 11 MeV) of 2°8pb to the lowqyingoctupole vibration is calculated in the surface coupling model. A marked quenching of the transitions is observed arising from the correlation between the particle and the hole participating in the vibration, and from the couplingto the GDR. The total F3,(GQR--*3D is found to be only a few percent of F.r(GQR~ gs) consistent with the experimental data. The compound nucleus gammadecay is estimated to be negligible.
The decay of giant resonances is still not a wellunderstood subject in nuclear spectroscopy, giving motivation to the testing of models against available experimental data. The main mechanism damping giant resonances appears to be the excitation of lowlying surface vibrations, induced by the interaction of the nuclear surface with the particle and hole of the resonance (cf. ref. [ 1] and references therein). For example, a major role in the damping of the giant resonances in 208pb is played by the 3 - state at 2.6 MeV excitation. This might be observable in the gamma-decay of the giant resonances, which should have a branch to the octupole vibration as a result of the coupling. Also in the particle decay populating multiplets based on the 3 - state [2]. In general, electric dipole transitions are the most likely to be observed experimentally. This fact restricts the multipolarity of the giant resonances which may be studied through gamma-decay. Another important 1 Partially supported by MPI of Italy. 20
feature restricting the usefulness of these studies is the energy of the mode. This is because for states below the giant dipole resonance (GDR), the E1 effective charge becomes very smaU, while for transitions having similar or larger energies than the GDR the situation is reversed. An excellent candidate for gamma-decay studies is the isovector giant quadrupole resonance (GQR) which can decay through enhanced transitions to the lowlying octupole vibrations as well as the GDR which can decay to the low-lying quadrupole vibration again with enhanced strength. The isoscalar GQR can also decay to the low-lying 3 - but with a much reduced transition strength as compared with the single-particle transition. Recently an experiment has been performed in 208pb where the photon decay from the energy regions associated with the isoscalar GQR, GMR and GDR was studied. The results have been reported in ref. [3]. In the present paper we use the surface model developed in ref. [4] to calculate the decay associated 0370-2693/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 148B, number 1,2,3
PHYSICS LETTERS
with the isoscalar GQR as well as the ~ V = 2 component of the isoscalar giant hexadecapole resonance (GHR). Both modes lie in the same energy region and have been observed in hadron experiments (cf. ref. [5] and references therein). The properties of these vibrations are shown in table 1. Also shown are the corresponding results of an RPA calculation making use of a Hartree-Fock single-particle basis calculated with a Skyrme Ill interaction and a separable force where the radial dependence is parameterized
Table 1 Properties of the isoscalar GQR and AN = 2 component o f giant hexadecapole resonance [5]. The predictions of the surface coupling model [4] are also shown. j~r
2+ 4+
E(MeV)
r(MeV)
EWSR(%)
th.
exp.
th.
exp.
th.
exp.
11.2 11.0
11.4 12.2
2.6 5.3
2.7 4.5
72 18
70 30
.~.~-x X'~
22 November 1984
~= GDR
(A) X-3-,
X---
X
--
X--
I X =GQR
(B)
4,
X----(
- -X
+
I I GQR
GQR
GQR .(C)
(D)
(E)
Fig. 1. Processes associated with the decay of the giant quadrupole resonance into an octupole vibration. Lines label fermion fields while wavy lines label boson fields. A horizontal dotted line ending in a cross represents an external electric dipole field. In (A) we show the processes leading to the def'mition o f an effective charge described by a dashed circle on which the external field acts. The ~-ray can interact directly with a particle-hole vertex or it can first excite the giant dipole resonance ((;DR) which decays into a particle-hole pair. Thus the second graph in (A) where the phonon has been drawn as a horizontal wavy line stands for all possible time orderings associated with the action of the external field. A similar discussion applies to the process described by the second graph in (B). A similar group o f diagrams to both (A) and (B) are obtained interchanging the role o f particles and holes [graphs (C) and (D) ]. The graphs in the lhs o f (A) have a very similar structure to the diagrams entering in the calculation o f the strength function of the GQR [cf. graphs (E)].
21
PHYSICS LETTERS
Volume 148B, number 1,2,3
according to R,aU/ar (cf. ref. [4] for more details). The lowest order diagrams describing the decay processes are shown in figs. IA and 1B. A similar set of diagrams exists where the octupole vibration in graph (A) or the external field and the GDR in graph (B) operates on the hole state [cf. figs. 1C and ID]. The first graphs in figs. IA and IB are evaluated by the following expressions, with h = 2, A’ = 3 and X” = 1:
jlhJ.3
,E
[7,(j2) -(N--2)/A] ~~ __
x AxA,[(2h+
1)/(2h’+ l)] l/2
= -4e
; (
X M(jlj3; Wftjjl;
i”; i”I’ 1
A’) (1)
,
X ci~ll~(El)II12)/[DE~(E+iA/2)]o and <3-llm(El)llGQR(E)>(~) = -fejzis
]rZ(i;)~
-W+W4
x ~~~~~(-1)*“+h+iz+~3
[(2~+1)(2~‘ti)]1/2
22 November 1984
-iebz0.2)-
V-z)/-41
iA/2)lcB).
(2)
In the above equation M(13;X)= (jll!Ro(aU/&+)Y,llj3) with U(r) the single-particle potential, while A1 is the particle-vibration coupling constant determined by the normalization of the vibration [6]. The energy denominators in eq. (1) and (2) are given by [DEN@ t iA/2)] (Aj=(E+iA’/2-e13) X [E t i A/2 - (fic+t + ~23)] ,
(3)
and [DEN@ t iA/2)](BJ=(E+iA’/2
-e13)@+~
-~23).
(4) The giant resonances have a small spurious component in the shell model representation which is eliminated by using the operator
(5)
for the El transition charge. Both energy denominators include fmite imaginary parts. The practical reason for this is that the energy denominator can vanish otherwise. This happens in the graphs (A) of fig. 1 at energies corresponding to real transitions to the intermediate states. In (4) the energy denominator vanishes at energies of particlehole states with the resonance quantum numbers. From a theoretical point of view, the quantities A and A’ take account of the coupling to more complicated configurations not included in our space. The average transition strength will depend on the magnitude of A and A’, with less transition strength the larger these quantities are. No explicit calculations of these quantities are available. Empirically, the smoothness of the giant resonance peaks is consistent with A in the range 0.5-I .O MeV. We have chosen A = 1 MeV in our calculation in keeping with ref. [4]. The external field is partly screened by the mediation of the giant dipole resonance, as shown in the second graphs in figs. 1A and 1B. The sum of all time orderings of the dipole propagator in these graphs leads to a simple renormalization of the coupling (cf. eq. 6-330 of ref. [7]). This is given by
eeff=eP+xW31 X ~i~llrtl(El)llj~)/[DEN(E+
,
(6)
,
where the dipole polarizability
is
X(AE) = -0.76(L~,)~ X [(fiwD t AEtirD/2)(RwD
-AE-irD/2)1-‘. (7)
The quantity AE = E - Tzo,- is the energy of the dipole transition from the GQR at an energy E to the octupole state. In the calculations the experimental values Rw, = 14 MeV, I’, = 4 MeV have been used. At energies of interest, the cancellation between the two terms in eq. (6) is very strong, and the main contribution comes from the imaginary part of X with 11t~12~0.07atE=8.5MeV. Assuming that the strength of the giant resonance at the RPA level is concentrated in a single state, the formula for the gamma-width associated with the transition to the octupole state is dr,(GQR(E)
+ 3-)/dE=
f I1 +x(AE)12(AE,,)3
X 1.05 lu-ilm(E1)IIGQR(E))t~2,aPGQRQeV.
(8) 22
Volume 148B, number 1,2,3
PHYSICS LETTERS
The function P(E) is the probability per unit energy to find the GQR at an energy E. We calculate this using the same energies and particle-vibration couplings as we used for the transition. The main spreading of the P(E) distribution comes from graphs shown in fig. 1E. Details of that calculation are given in ref. [4]. The strength functions associated with the GQR and GHR used in the calculations are displayed in fig. 2. In eq. (8), AE has units of MeV, and the reduced matrix element has units of e 2 fm 2. Despite the strong coupling of the octupole to the giant quadrupole resonance, the other factors in the amplitude combine to produce a rather small width in the end. A
0.3
F\\
(a)
\ \
~O.l
\. w
\
.\ \
0.1
/1"
\.
\
\
/
\.
I
!'o I
I
I
I
17
E (MeV)
I
1
I
/\. ! !
0,3
I
I
I
12.6 dP~(GQR(E) -->3 - ) = --/ dE dE ~ 3.5 eV. (9) 9.8
(b)
We also calculated the dipole width associated with the AN = 2 GHR, which has been observed in the reaction studied in ref. [3], finding
/
i /
0.1
typical particle-hole dipole transition would produce a width of about 1.5 X 103 eV at a transition energy of 8.5 MeV. The Racah coefficient in eqs. (1) and (2) can be estimated from Wigner's semiclassical formula to be of the order of 0.1, so the recoupling times the square root containing the statistical factors reduces the strength by about 1/3. Because of the isovector nature of the operator (5), neutron and proton amplitudes tend to cancel when the states have predominantly isoscalar character. This is the case for the giant quadrupole and the low-lying octupole vibrations, and the calculations show that the interference reduces the width by about a factor of 4 below the width for proton amplitudes alone. Another source of interference is the cancellation between particles and holes, i.e. between the two graphs shown in figs. C and D. This is a general property found in the coupling of density fluctuation modes to other vibrations, and reflects the correlations between the particle and the hole existing in those vibrations (of. refs. [1,8] and references therein). The associated quenching factor is in the present case 1/2-1/3, comparable to what was found for the overall interference in damping. Finally, the effective charge in eq. (6) reduces the width by a factor of 15, to give a width of the order of electron volts. To facilitate comparison with experiment, we evaluate the integrated decay for the energy region 9.8 to 12.6 MeV, t'mding r,(GQR--> 3 i-)
I
I i it."
\..ji
\.\
r~(GHR -~ 3]-)
\
= I
8
I
9
22 November 1984
ilO
I
I
11 t
I
12 13 (MeV)
I
14
15
16
17
Fig. 2. Strength functions calculated in the surface excitation model and associated with the GQR (a) and GHR (b). The associated % of the EWSR associated with these modes are shown in table 1.
~
.6 dr~(GHR(E) -+ 3 - ) dE ~ 2 .6 eV dE 9.8
(10)
The octupole state can also be populated by compound nucleus decay, which will add to the direct decay we have calculated. Fortunately, the level density of the compound system appears to be high enough 23
Volume 148B, number 1,2,3
PHYSICS LETTERS
to make compound decay widths negligible. Using a constant temperature model [9] with parameters fixed by the level density of dipole states at 7-8 MeV [IO], we estimate lo3 .P = 2+ states per MeV at 11 MeV excitation, having an average dipole width of the order of less than 0.1 eV to the octupole state. Experimentally, only an upper limit has been established for the decay to the octupole vibration [ 111, of magnitude of the order of eV. Whether the experiment can be refmed to the point of confirming or refuting eqs. (9) and (10) remains to be seen. Discussions with J. Beene, F. Bertrand, C. Mabaux and F. Zardi are gratefully acknowledged. G.B. received support from the National Science Foundation for this work. References [l] G.F. Bertsch, P.F. Bortignon and R.A. Broglia, Rev. Mod. Phys. 55 (1983) 287.
24
22 November 1984
[2] R.A. Broglia and P.F. Bortignon, in: Giant multipole resonances (Harwood Academic,New York, 1979) p. 317; Phys. Lett. 1OlB (1981) 135. [3] FE. Bertrand, J.R. Beene and T.P. Sjoreen, in: Proc. Symp. on Highly excited states and nuclear structure, eds. N. Marty and N. Van Giai, J. Phys. (Paris) 45 (1984) c4 99. [ 41 P.F. Bortignon and R.A. Broglia, Nucl. Phys. A37 1 (1981) 405. [S ] H.P. Morsch,P. Decowski, M. Rogge,P.Turek,L. Zemto, S.A. Martin, G.P.A. Berg, W. Hurlimann, J. Meissburger and J.G.M. Romer, Phys. Rev. C28 (1983) 1947. [6] P.F. Bortignon,R.A. Broglia,D.R. Bes and R. Liotta, Phys. Rep. 30C (1977) 305. [ 7 ] A. Bohr and B.R. Mottelson, Nuclear structure, Vol. II (Benjamin, New York, 1975). [8] P.F. Bortignon, R.A. Brogha and C.H. Dasso,Nucl. Phys. A398 (1983) 221. [9] D-l. Horen, J.A. Harvey and W. Hill, Phys. Rev. C24 (1981) 1961. [lo] RJ. Holt, HE. Jackson, R.M. Laszewski and J.R. Specht, Phys. Rev. C20 (1979) 93. [ 1 l] J. Beene and F. Bertrand, private communication.