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Dynamical decay of a hot nucleus
Volume 209, number 2,3
PHYSICS LETTERSB
4 August 1988
DYNAMICAL DECAY OF A H O T NUCLEUS M. ABE and N. TAKIGAWA Department of Physics, T6hoku University, Sendai 980, Japan
Received 29 February 1988;revised manuscript received 17 May 1988
Particle decay from a hot compoundnucleus is investigated concerningthe effects of the couplingto low-lyingvibrations and the primary excitation of the residual nucleus. Theystronglyaffectthe energyspectrum at low energies. The peak of the spectrum is shifted to a lower energy compared with that of the conventionalstatistical model. The thermal fluctuation of nuclear surface amplifies the dynamical effects.
Decay of metastable states at finite temperature is a challenging problem in diverse fields of physics: the loss of magnetic flux through SQUID in low-temperature physics, the decay of metastable vacuum in cosmological phase transitions, the nucleation of bubbles or droplets in the liquid-gas mixture and the decay of an excited nucleus produced in heavy-ion collisions. Recently, Grabert et al. [ 1 ] unified two extreme models for the decay of metastable states by using Langer's trick. One is the Kramers dynamical model for the classical decay at high temperature and the other is the Gamow quantum tunneling model at low temperature. In ref. [ 1 ], the coordinate for the decay process dynamically couples to intrinsic degrees of freedom, which are presented by oscillator heat bath. The decay thus becomes a multidimensional problem. In nuclear physics, the multidimensional quantum tunneling has been extensively discussed to explain the large enhancement of the heavy-ion-fusion cross section below the Coulomb barrier [2]. It is not difficult to imagine that similar dynamical effects play an important role also in the decay of a compound nucleus, i.e. in fission and in evaporation. In fact, the Kramers model and its multidimensional extension have been used as dynamical models for fission [ 3 ]. These models, however, cannot be applied to fission at low excitation energies, where quantum tunneling dominates the decay process. In this letter, we study dynamical effects on the particle evaporation from a hot compound nucleus. We argue that the use of the Weisskopf formula for particle evaporation in the usual computer code of statistical models is inadequate in two respects. Firstly, the conventional statistical models take into account intrinsic degrees of freedom only through the level density and totally ignore their effects on the transmission coefficient. It is, however, known that the latter play a significant role in enhancing the heavy-ion-fusion cross section at subCoulomb energies. Secondly, the transmission coefficient is in practice evaluated from the fusion cross section between cold nuclei as the inverse process of the evaporation or from the optical model for elastic scattering. There is, however, a decisive difference between the formation of a compound nucleus and its decay: the former is the reaction between cold nuclei, while the latter is the reaction between hot nuclei. In fusion, the colliding nuclei are initially in their ground states, while evaporated particles escape from an initial state embedded in excited intrinsic states. Therefore, it would be essential to consider the dynamical effects of the intrinsic degrees of freedom by taking into account the fact that the evaporation residue is primarily highly excited. We adopt the following model hamiltonian for particle evaporation from a compound nucleus by assuming that the particle couples to collective vibrations of the residual nucleus:
p2 /~= ~ + V(R)+ E, o~,y,(R)(~,*+,~,) + E, h¢o,(~,~,+ ½). 0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
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Volume 209, number 2,3
PHYSICS LETTERS B
4 August 1988
Here, R is the distance between the centers of the evaporated particle and the residual nucleus, P its conjugate m o m e n t u m , / t the reduced mass of the decay channel. The interaction potential between the fragments contains the nuclear and the Coulomb parts. ~,~ (di) is the creation (annihilation) operator of the ith vibrational boson of the residual nucleus with the excitation energy ho)i, c~, the amplitude of the corresponding zero-point vibration in units of the nuclear radius. The macroscopic coupling form f a c t o r f (R) consists of the nuclear and the Coulomb components,
V(R)
3
(e~c)) 2i
Z4Z~e~22~+ ~ ~Ra,+
f(R)=
R~N) dUN]
dR J'
(2)
where Z~ (ZR) is the atomic number of the evaporated particle (residual nucleus), 2i the multipolarity of the intrinsic excitation ~, R }N) (R ~c)) the nuclear (Coulomb) radius of the evaporation residue and UN the nuclear potential between the fragments. We note that low-lying modes among various vibrational excitations play dominant roles in enhancing sub-Coulomb fusion cross sections [4,6]. For simplicity, we assume that only one low-lying phonon gives a dominant effect on the decay and that it is populated according to the canonical distribution of temperature T. We denote the n-phonon state by [n ) , its energy by e~ and the channel S matrix element by S,,~ (E), E being the kinetic energy of the decaying particle. The path-integral representation of the inclusive transmission coefficient for the dynamical decay is then given by
t(E, T ) = ,,,,,,~IS,,,,(E)I:exp(-e'/T)Z~"V- s~mslim~Rf~ ±~o /l 2
dr exp(iEr) 0
R~
dr' e x p ( - i E r ' ) 0
R~
× ; DR(t)
f DR'(t') exp((i/h){S[R(t)]-S[R'(t')]})~,
(3)
S[R(t)]
where is the action of the decay coordinate in the absence of the coupling. The Z~nv and/~ are the partition function of the intrinsic states and the influence functional, respectively. They are given by /env =
2
exp(-e,/r),
(4)
and exp(-en/T)
P= ,,.,n ~ (mld(R(t);T,O)ln)(nld*(R'(t');r',O)lm)
Zenv
'
(5)
where !
tl
e
{i°12fdt, fdt2f( (t,,)f(R(tz))sin(o)(t2-t,)))l~exp(-i~ot/h)"
G~ = e x p ~ - ~ -
0
.
(6)
0
In eq. (6), ~o is the hamiltonian of the vibrational boson and
l~=exp[a(~ i dtlf(R(tl) ) exp(-ia)t,))-a* (~ 2 dt, f(R(tl) ) exp(imh))] . 0
(7)
0
We refer to ref. [7] for the other notations in eqs. ( 3 ) - ( 7 ) . The inclusive transmission coefficient can be ex~ The original three-dimensionalityof the decay coordinate R and the angular momentum algebra associatedwith the finite multipolarity 2, are treated by the method of refs. [4,5 ], which eliminates the angular parts of R by introducing a scaled couplingstrength. 150
Volume 209, number 2,3
PHYSICS LETTERS B
4 August 1988
plicitly evaluated in the degenerate approximation [ 7 ], which is valid when the energy of the vibrational excitation is much smaller than the curvature of the Coulomb barrier. The result reads
t(E, T) =
~
e x p ( - x Z / 2 ) to(E, V(R) + x ~ T f ( e ) ),
(8)
where to(E, V(R)) denotes the transmission coefficient for the one-dimensional potential V(R). The O~T is given by o~2 =o~2[ 1 + 2nB(o), T) ] ,
(9)
with the Bose distribution 1
nB(co, T) = exp(ho)/T) - 1
(10)
As seen in eq. (6), the inclusive transmission coefficient is given by a gaussian average of the one-dimensional transmission coefficient to(E, Veer(R, x) ) through an effective potential Vcff(R, x) = V(R) + AVdyn(R,x), A Vdyn(R, X) being xc~xf(R) ~2. The origin of the dynamical potential A Voynis twofold: the quantum zero-point fluctuation and the thermal fluctuation of the intrinsic phonon. The former dominates at low temperatures, while the latter becomes comparable at T -~ hto. The strength of the dynamical coupling a~ increases with temperature [see eqs. (9) and (10) ]. The dynamical effect thus becomes more significant for the decay of a compound nucleus in higher excitation, though the present harmonic approximation to a vibrational mode of nuclear excitation becomes less adequate for higher excitation. Fig. 1 shows the evaporation spectrum of light particle A, A being a, 160 and 4°Ca, from the compound nucleus A + ~sspt. It was calculated according to do"
d-E
=const. ~ (2l+l)tl(E, T) e x p ( - E / T o ) "7"
(11)
where tz(E, T) is the transmission coefficient for the partial wave l. We consider only the s-wave component in the following. The ordinate in fig. 1 is an arbitrary scale, while the abscissa the energy of the evaporated particle is relative to the height of the s-wave potential barrier VB. The slope parameter To was assumed to be 1 MeV for all lines in fig. 1. We calculated the transmission coefficient in three different ways. The line ( 1 ) was obtained by ignoring the dynamical effects on the transmission coefficient, and by adopting the Akyuz-Winther potential [ 8 ] for UN(R). This corresponds to the one-dimensional potential model. The transmission coefficient for the line (2) was calculated by incorporating the effects of coupling to the low-lying 2 + state of l SSpt, whose excitation energy h0)=0.266 MeV is much smaller than the curvature of the s-wave Coulomb barrier. The ax was assigned to be the value at T = 0 a = 0.0513, which was obtained from the data of B (E2). This will correspond to the energy spectrum calculated from the Weisskopf formula by using the reaction data of cold nuclei as the inverse process of the evaporation. The line (3) was calculated by simultaneously taking into account the dynamical coupling and the thermal fluctuation by assuming T = To = 1 MeV. We see that the dynamical coupling to the low-lying vibrational boson significantly affects the evaporation spectrum below the Coulomb barrier. The effect of coupling becomes much larger when the primary excitation of the residual nucleus is considered. Also, the dynamical enhancement is much larger for heavy-cluster evaporation than for ct evaporation. This is because the effective coupling around the Coulomb barrier is proportional to the product of the charges of two fragments. Fig. 2 shows the temperature dependence of the inclusive transmission coefficient for the evapora~2The extension to the case whenthere exist N vibrational modesis straightforward. Especiallyif the couplingformfactorhas a common radial dependence, the N-dimensional integrals over their amplitudes reduce to a one-dimensionalone. The resulting expression for thetransmissioncoefficientisthesameaseq.(8)whereo~visreplacedby O~2T= ~,O62[ l+2nB(o)i, T) 1. 151
Volume 209 number 2.3
PHYSICS LETTERS B
4 August 1988
1 X}
E U~'
j 0°
0.~
/;
......... Igi
q
10 ~ 0.2 ]0
.~-~-~--~ I
~
I
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'
'
. . . .
~ ,-~--~
lSO
J
@ ()
7 ,,
77
7 ,t
i
76
7S
~0
I",{MeV] Fig. 2. Comparison of the bare s-wave transmission coefficient to dynamical transmission coefficients for J 6 0 + 'ssPt. The height of the s-wave potential barrier is VB= 74.2 MeV.
80
l{i
(3)
0
._
J -
--f
~\
-
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-
-
-
-
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,5 Vt~
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.
.
.
.
.
.
[MoV] I
Fig. 1. Comparison of the bare energy spectrum to dynamically calculated energy spectra.
.flu]
Fig. 3. The bare versus effective potentials for " 0 +
'~Pt.
tion of 160 from the compound nucleus 160+ 'sspt. The lines ( 1 ) - ( 3 ) correspond to those of the same number in fig. 1. The line (4) was obtained in the same way as the line (3), but for T = 4 MeV. The crosses are also for T= 4 MeV, but without the Coulomb coupling. The dynamical coupling for T= 1 MeV reduces the transmission coefficient above the Coulomb barrier and enhances it below the barrier. The enhancement of the transmission coefficient in the sub-Coulomb region for T= 4 MeV is almost equal to that for T= 1 MeV, despite the fact that the effective coupling strength C~Tis twice as large as that for T= 1 MeV and that the increased c~x clearly more hinders the transmission coefficient above the barrier. This unexpected behavior at the low-energy side originates from the nature of the macroscopic coupling form factor. The Coulomb and the nuclear coupling in eq. (2) cancel each other near the Coulomb barrier. In order to visualize this effect, we compare in fig. 3 four potential barriers. The dots represent the bare potential barrier. The dashed and the solid lines are Veff(R, x = 3 ) for T= 1 and 4 MeV, respectively. The destructive interference between the nuclear and the Coulomb excitation can be clearly seen if we compare the solid line with the crosses, which were obtained by switching off the Coulomb excitation. The region of strong Coulomb excitation extends outside the Coulomb barrier for T= 4 MeV. The Coulomb excitation thus cancels the strong enhancement of the transmission coefficient due to the nuclear excitation as clearly seen in the difference between the line (4) and the crosses in fig. 2. This is why the transmission coefficient for T = 4 MeV at sub-barrier energies does not increase so much as we expect. The destructive interference in the coupling form factor contrasts the problems of nuclear systems with those in low152
Volume 209, number 2,3
PHYSICS LETTERS B
4 August 1988
t e m p e r a t u r e physics, where the decay c o o r d i n a t e is assumed to bilinearly couple to intrinsic oscillator degrees o f freedom. Before concluding the paper, we r e m a r k that the authors o f ref. [ 9 ] claim that the experimental energy spect r u m o f ct e v a p o r a t i o n cannot be r e p r o d u c e d b y the statistical m o d e l with conventional values o f parameters. They phenomenologically fit the d a t a by changing the level-density p a r a m e t e r s or the potential p a r a m e t e r s for calculating transmission coefficients a n d argue physical implications o f these changes. The present study suggests that the d y n a m i c a l effect dealt with in this work is much less than the effects observed experimentally for proton a n d ct particle spectra. On the other hand, the d y n a m i c a l effect m a y be particularly significant for heavier cluster emission a n d m a y account for earlier observations o f very-low-energy fragment emission [ 10]. We investigated the d y n a m i c a l effects o f a low-lying collective v i b r a t i o n on the particle e v a p o r a t i o n from a hot nucleus. The heavy-cluster e v a p o r a t i o n is m o r e strongly influenced by the d y n a m i c a l coupling than ct evaporation. The low-energy part o f the e v a p o r a t i o n spectrum is enhanced, while the high-energy part is h i n d e r e d by the coupling. The d y n a m i c a l effects b e c o m e m o r e significant when the p r i m a r y excitation o f the residual nucleus is taken into account. The nuclear-Coulomb interference o f the macroscopic coupling form factor causes an unexpected b e h a v i o r in the t e m p e r a t u r e d e p e n d e n c e o f the d y n a m i c a l effect. The d y n a m i c a l e n h a n c e m e n t o f the low-energy e v a p o r a t i o n spectrum clarified in this p a p e r would be a useful tool to experimentally extract thermal properties o f hot nuclei, especially those o f low-lying collective m o d e s o f excitation in hot nuclei. The authors are i n d e b t e d to Professor A.B. Balantekin for useful discussions at the early stage o f this work. They are grateful also to Professor W.A. F r i e d m a n for useful discussions. This work was s u p p o r t e d by a grant u n d e r the M o n b u s h o International Scientific Research P r o g r a m No. 63044014 from the Japanese Ministry o f Education, Science and Culture.
Note added. After having s u b m i t t e d this paper, we learnt that ref. [ 11 ] treats intimately related subjects. Differently from ours, however, the authors discuss the effect o f the fluctuation o f the saddle-point configuration. Also, the theoretical f r a m e w o r k is very different from ours.
References [ 1] H. Grabert, P. Olschowski and U. Weiss, Phys. Rev. B 36 (1987) 1931, and references therein. [2] S.G. Steadman and M.J. Rhodes-Brown, Annu. Rev. Nucl. Sci. 36 ( 1987 ) 649. [ 3 ] P. Grange, L. Jun-Qing and H. Weidenm~iller,Phys. Rev. C 27 ( 1983 ) 2063; H. WeidenmiJller and Z. Jing-Shang, Phys. Rev. C 28 (1983) 2190, and references therein. [4] H. Esbensen, J.-Q. Wu and G.F. Bertsch, Nucl. Phys. A 411 (1983) 275. [ 5 ] N. Takigawa and K. Ikeda, Proc. Symp. on the Many facets of heavy ion fusion reactions, eds. W. Henning et al., ANL-PHY-86-1 (1986) p. 613. [6] N. Takigawa, Heavy ion fusion reactions, eds. K. Furuno and T. Kishimoto (World Scientific, Singapore, 1984) p. 20. [7] A.B. Balantekin and N. Takigawa, Ann. Phys. (NY) 160 (1985) 441. [8] R.A. Broglia and A. Winther, Heavy ion reactions, Vol. 1. Elastic and inelastic reactions, (Benjamin/Cummings, Menlo Park, 1981) pp. 104-115. [9] L.C. Vaz and J.M. Alexander, Z. Phys. A 305 (1982) 313; A 318 (1984) 231; G. Nebbia, K. Hagel, D. Fabris, Z. Majka, J.B. Natowitz, R.P. Schmitt, B. Sterling, G. Mouchaty, G. Berkowitz, K. Strozewski, G. Viesti, P.L. Gonthier, B. Wilkins, M.N. Namboodiri and H. Ho, Phys. Lett. B 176 (1986) 20. [ 10] A.M. Poskanzer, G.W. Butler and E.K. Hyde, Phys. Rev. C 3 ( 1971 ) 882; G.D. Westfall, R.G. Sextro, A.M. Poskanzer, A.M. Zebelman, G.W. Butler and E.K. Hyde, Phys. Rev. C 17 ( 1978 ) 1368. [ 11 ] L.G. Moretto and D.R. Bowman, Proc. 25th Intern. Winter Meeting on Nuclear physics (Bormio, Italy, January 1987).
153
PHYSICS LETTERSB
4 August 1988
DYNAMICAL DECAY OF A H O T NUCLEUS M. ABE and N. TAKIGAWA Department of Physics, T6hoku University, Sendai 980, Japan
Received 29 February 1988;revised manuscript received 17 May 1988
Particle decay from a hot compoundnucleus is investigated concerningthe effects of the couplingto low-lyingvibrations and the primary excitation of the residual nucleus. Theystronglyaffectthe energyspectrum at low energies. The peak of the spectrum is shifted to a lower energy compared with that of the conventionalstatistical model. The thermal fluctuation of nuclear surface amplifies the dynamical effects.
Decay of metastable states at finite temperature is a challenging problem in diverse fields of physics: the loss of magnetic flux through SQUID in low-temperature physics, the decay of metastable vacuum in cosmological phase transitions, the nucleation of bubbles or droplets in the liquid-gas mixture and the decay of an excited nucleus produced in heavy-ion collisions. Recently, Grabert et al. [ 1 ] unified two extreme models for the decay of metastable states by using Langer's trick. One is the Kramers dynamical model for the classical decay at high temperature and the other is the Gamow quantum tunneling model at low temperature. In ref. [ 1 ], the coordinate for the decay process dynamically couples to intrinsic degrees of freedom, which are presented by oscillator heat bath. The decay thus becomes a multidimensional problem. In nuclear physics, the multidimensional quantum tunneling has been extensively discussed to explain the large enhancement of the heavy-ion-fusion cross section below the Coulomb barrier [2]. It is not difficult to imagine that similar dynamical effects play an important role also in the decay of a compound nucleus, i.e. in fission and in evaporation. In fact, the Kramers model and its multidimensional extension have been used as dynamical models for fission [ 3 ]. These models, however, cannot be applied to fission at low excitation energies, where quantum tunneling dominates the decay process. In this letter, we study dynamical effects on the particle evaporation from a hot compound nucleus. We argue that the use of the Weisskopf formula for particle evaporation in the usual computer code of statistical models is inadequate in two respects. Firstly, the conventional statistical models take into account intrinsic degrees of freedom only through the level density and totally ignore their effects on the transmission coefficient. It is, however, known that the latter play a significant role in enhancing the heavy-ion-fusion cross section at subCoulomb energies. Secondly, the transmission coefficient is in practice evaluated from the fusion cross section between cold nuclei as the inverse process of the evaporation or from the optical model for elastic scattering. There is, however, a decisive difference between the formation of a compound nucleus and its decay: the former is the reaction between cold nuclei, while the latter is the reaction between hot nuclei. In fusion, the colliding nuclei are initially in their ground states, while evaporated particles escape from an initial state embedded in excited intrinsic states. Therefore, it would be essential to consider the dynamical effects of the intrinsic degrees of freedom by taking into account the fact that the evaporation residue is primarily highly excited. We adopt the following model hamiltonian for particle evaporation from a compound nucleus by assuming that the particle couples to collective vibrations of the residual nucleus:
p2 /~= ~ + V(R)+ E, o~,y,(R)(~,*+,~,) + E, h¢o,(~,~,+ ½). 0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
(1) 149
Volume 209, number 2,3
PHYSICS LETTERS B
4 August 1988
Here, R is the distance between the centers of the evaporated particle and the residual nucleus, P its conjugate m o m e n t u m , / t the reduced mass of the decay channel. The interaction potential between the fragments contains the nuclear and the Coulomb parts. ~,~ (di) is the creation (annihilation) operator of the ith vibrational boson of the residual nucleus with the excitation energy ho)i, c~, the amplitude of the corresponding zero-point vibration in units of the nuclear radius. The macroscopic coupling form f a c t o r f (R) consists of the nuclear and the Coulomb components,
V(R)
3
(e~c)) 2i
Z4Z~e~22~+ ~ ~Ra,+
f(R)=
R~N) dUN]
dR J'
(2)
where Z~ (ZR) is the atomic number of the evaporated particle (residual nucleus), 2i the multipolarity of the intrinsic excitation ~, R }N) (R ~c)) the nuclear (Coulomb) radius of the evaporation residue and UN the nuclear potential between the fragments. We note that low-lying modes among various vibrational excitations play dominant roles in enhancing sub-Coulomb fusion cross sections [4,6]. For simplicity, we assume that only one low-lying phonon gives a dominant effect on the decay and that it is populated according to the canonical distribution of temperature T. We denote the n-phonon state by [n ) , its energy by e~ and the channel S matrix element by S,,~ (E), E being the kinetic energy of the decaying particle. The path-integral representation of the inclusive transmission coefficient for the dynamical decay is then given by
t(E, T ) = ,,,,,,~IS,,,,(E)I:exp(-e'/T)Z~"V- s~mslim~Rf~ ±~o /l 2
dr exp(iEr) 0
R~
dr' e x p ( - i E r ' ) 0
R~
× ; DR(t)
f DR'(t') exp((i/h){S[R(t)]-S[R'(t')]})~,
(3)
S[R(t)]
where is the action of the decay coordinate in the absence of the coupling. The Z~nv and/~ are the partition function of the intrinsic states and the influence functional, respectively. They are given by /env =
2
exp(-e,/r),
(4)
and exp(-en/T)
P= ,,.,n ~ (mld(R(t);T,O)ln)(nld*(R'(t');r',O)lm)
Zenv
'
(5)
where !
tl
e
{i°12fdt, fdt2f( (t,,)f(R(tz))sin(o)(t2-t,)))l~exp(-i~ot/h)"
G~ = e x p ~ - ~ -
0
.
(6)
0
In eq. (6), ~o is the hamiltonian of the vibrational boson and
l~=exp[a(~ i dtlf(R(tl) ) exp(-ia)t,))-a* (~ 2 dt, f(R(tl) ) exp(imh))] . 0
(7)
0
We refer to ref. [7] for the other notations in eqs. ( 3 ) - ( 7 ) . The inclusive transmission coefficient can be ex~ The original three-dimensionalityof the decay coordinate R and the angular momentum algebra associatedwith the finite multipolarity 2, are treated by the method of refs. [4,5 ], which eliminates the angular parts of R by introducing a scaled couplingstrength. 150
Volume 209, number 2,3
PHYSICS LETTERS B
4 August 1988
plicitly evaluated in the degenerate approximation [ 7 ], which is valid when the energy of the vibrational excitation is much smaller than the curvature of the Coulomb barrier. The result reads
t(E, T) =
~
e x p ( - x Z / 2 ) to(E, V(R) + x ~ T f ( e ) ),
(8)
where to(E, V(R)) denotes the transmission coefficient for the one-dimensional potential V(R). The O~T is given by o~2 =o~2[ 1 + 2nB(o), T) ] ,
(9)
with the Bose distribution 1
nB(co, T) = exp(ho)/T) - 1
(10)
As seen in eq. (6), the inclusive transmission coefficient is given by a gaussian average of the one-dimensional transmission coefficient to(E, Veer(R, x) ) through an effective potential Vcff(R, x) = V(R) + AVdyn(R,x), A Vdyn(R, X) being xc~xf(R) ~2. The origin of the dynamical potential A Voynis twofold: the quantum zero-point fluctuation and the thermal fluctuation of the intrinsic phonon. The former dominates at low temperatures, while the latter becomes comparable at T -~ hto. The strength of the dynamical coupling a~ increases with temperature [see eqs. (9) and (10) ]. The dynamical effect thus becomes more significant for the decay of a compound nucleus in higher excitation, though the present harmonic approximation to a vibrational mode of nuclear excitation becomes less adequate for higher excitation. Fig. 1 shows the evaporation spectrum of light particle A, A being a, 160 and 4°Ca, from the compound nucleus A + ~sspt. It was calculated according to do"
d-E
=const. ~ (2l+l)tl(E, T) e x p ( - E / T o ) "7"
(11)
where tz(E, T) is the transmission coefficient for the partial wave l. We consider only the s-wave component in the following. The ordinate in fig. 1 is an arbitrary scale, while the abscissa the energy of the evaporated particle is relative to the height of the s-wave potential barrier VB. The slope parameter To was assumed to be 1 MeV for all lines in fig. 1. We calculated the transmission coefficient in three different ways. The line ( 1 ) was obtained by ignoring the dynamical effects on the transmission coefficient, and by adopting the Akyuz-Winther potential [ 8 ] for UN(R). This corresponds to the one-dimensional potential model. The transmission coefficient for the line (2) was calculated by incorporating the effects of coupling to the low-lying 2 + state of l SSpt, whose excitation energy h0)=0.266 MeV is much smaller than the curvature of the s-wave Coulomb barrier. The ax was assigned to be the value at T = 0 a = 0.0513, which was obtained from the data of B (E2). This will correspond to the energy spectrum calculated from the Weisskopf formula by using the reaction data of cold nuclei as the inverse process of the evaporation. The line (3) was calculated by simultaneously taking into account the dynamical coupling and the thermal fluctuation by assuming T = To = 1 MeV. We see that the dynamical coupling to the low-lying vibrational boson significantly affects the evaporation spectrum below the Coulomb barrier. The effect of coupling becomes much larger when the primary excitation of the residual nucleus is considered. Also, the dynamical enhancement is much larger for heavy-cluster evaporation than for ct evaporation. This is because the effective coupling around the Coulomb barrier is proportional to the product of the charges of two fragments. Fig. 2 shows the temperature dependence of the inclusive transmission coefficient for the evapora~2The extension to the case whenthere exist N vibrational modesis straightforward. Especiallyif the couplingformfactorhas a common radial dependence, the N-dimensional integrals over their amplitudes reduce to a one-dimensionalone. The resulting expression for thetransmissioncoefficientisthesameaseq.(8)whereo~visreplacedby O~2T= ~,O62[ l+2nB(o)i, T) 1. 151
Volume 209 number 2.3
PHYSICS LETTERS B
4 August 1988
1 X}
E U~'
j 0°
0.~
/;
......... Igi
q
10 ~ 0.2 ]0
.~-~-~--~ I
~
I
>:-+~-+-~
'
'
. . . .
~ ,-~--~
lSO
J
@ ()
7 ,,
77
7 ,t
i
76
7S
~0
I",{MeV] Fig. 2. Comparison of the bare s-wave transmission coefficient to dynamical transmission coefficients for J 6 0 + 'ssPt. The height of the s-wave potential barrier is VB= 74.2 MeV.
80
l{i
(3)
0
._
J -
--f
~\
-
•
t
"
"
•
~
1
"
t
~
t
"
-
-
-
-
~
t
I
z
>
2
t
Jo(,a
~..
---
1[)
"
/0
[
(1)"//"" -.....
~
i
t
h;
f t
.
J0
0
-5 1;)-
,5 Vt~
J0
L0
1
_ ~
.
.
.
.
.
.
[MoV] I
Fig. 1. Comparison of the bare energy spectrum to dynamically calculated energy spectra.
.flu]
Fig. 3. The bare versus effective potentials for " 0 +
'~Pt.
tion of 160 from the compound nucleus 160+ 'sspt. The lines ( 1 ) - ( 3 ) correspond to those of the same number in fig. 1. The line (4) was obtained in the same way as the line (3), but for T = 4 MeV. The crosses are also for T= 4 MeV, but without the Coulomb coupling. The dynamical coupling for T= 1 MeV reduces the transmission coefficient above the Coulomb barrier and enhances it below the barrier. The enhancement of the transmission coefficient in the sub-Coulomb region for T= 4 MeV is almost equal to that for T= 1 MeV, despite the fact that the effective coupling strength C~Tis twice as large as that for T= 1 MeV and that the increased c~x clearly more hinders the transmission coefficient above the barrier. This unexpected behavior at the low-energy side originates from the nature of the macroscopic coupling form factor. The Coulomb and the nuclear coupling in eq. (2) cancel each other near the Coulomb barrier. In order to visualize this effect, we compare in fig. 3 four potential barriers. The dots represent the bare potential barrier. The dashed and the solid lines are Veff(R, x = 3 ) for T= 1 and 4 MeV, respectively. The destructive interference between the nuclear and the Coulomb excitation can be clearly seen if we compare the solid line with the crosses, which were obtained by switching off the Coulomb excitation. The region of strong Coulomb excitation extends outside the Coulomb barrier for T= 4 MeV. The Coulomb excitation thus cancels the strong enhancement of the transmission coefficient due to the nuclear excitation as clearly seen in the difference between the line (4) and the crosses in fig. 2. This is why the transmission coefficient for T = 4 MeV at sub-barrier energies does not increase so much as we expect. The destructive interference in the coupling form factor contrasts the problems of nuclear systems with those in low152
Volume 209, number 2,3
PHYSICS LETTERS B
4 August 1988
t e m p e r a t u r e physics, where the decay c o o r d i n a t e is assumed to bilinearly couple to intrinsic oscillator degrees o f freedom. Before concluding the paper, we r e m a r k that the authors o f ref. [ 9 ] claim that the experimental energy spect r u m o f ct e v a p o r a t i o n cannot be r e p r o d u c e d b y the statistical m o d e l with conventional values o f parameters. They phenomenologically fit the d a t a by changing the level-density p a r a m e t e r s or the potential p a r a m e t e r s for calculating transmission coefficients a n d argue physical implications o f these changes. The present study suggests that the d y n a m i c a l effect dealt with in this work is much less than the effects observed experimentally for proton a n d ct particle spectra. On the other hand, the d y n a m i c a l effect m a y be particularly significant for heavier cluster emission a n d m a y account for earlier observations o f very-low-energy fragment emission [ 10]. We investigated the d y n a m i c a l effects o f a low-lying collective v i b r a t i o n on the particle e v a p o r a t i o n from a hot nucleus. The heavy-cluster e v a p o r a t i o n is m o r e strongly influenced by the d y n a m i c a l coupling than ct evaporation. The low-energy part o f the e v a p o r a t i o n spectrum is enhanced, while the high-energy part is h i n d e r e d by the coupling. The d y n a m i c a l effects b e c o m e m o r e significant when the p r i m a r y excitation o f the residual nucleus is taken into account. The nuclear-Coulomb interference o f the macroscopic coupling form factor causes an unexpected b e h a v i o r in the t e m p e r a t u r e d e p e n d e n c e o f the d y n a m i c a l effect. The d y n a m i c a l e n h a n c e m e n t o f the low-energy e v a p o r a t i o n spectrum clarified in this p a p e r would be a useful tool to experimentally extract thermal properties o f hot nuclei, especially those o f low-lying collective m o d e s o f excitation in hot nuclei. The authors are i n d e b t e d to Professor A.B. Balantekin for useful discussions at the early stage o f this work. They are grateful also to Professor W.A. F r i e d m a n for useful discussions. This work was s u p p o r t e d by a grant u n d e r the M o n b u s h o International Scientific Research P r o g r a m No. 63044014 from the Japanese Ministry o f Education, Science and Culture.
Note added. After having s u b m i t t e d this paper, we learnt that ref. [ 11 ] treats intimately related subjects. Differently from ours, however, the authors discuss the effect o f the fluctuation o f the saddle-point configuration. Also, the theoretical f r a m e w o r k is very different from ours.
References [ 1] H. Grabert, P. Olschowski and U. Weiss, Phys. Rev. B 36 (1987) 1931, and references therein. [2] S.G. Steadman and M.J. Rhodes-Brown, Annu. Rev. Nucl. Sci. 36 ( 1987 ) 649. [ 3 ] P. Grange, L. Jun-Qing and H. Weidenm~iller,Phys. Rev. C 27 ( 1983 ) 2063; H. WeidenmiJller and Z. Jing-Shang, Phys. Rev. C 28 (1983) 2190, and references therein. [4] H. Esbensen, J.-Q. Wu and G.F. Bertsch, Nucl. Phys. A 411 (1983) 275. [ 5 ] N. Takigawa and K. Ikeda, Proc. Symp. on the Many facets of heavy ion fusion reactions, eds. W. Henning et al., ANL-PHY-86-1 (1986) p. 613. [6] N. Takigawa, Heavy ion fusion reactions, eds. K. Furuno and T. Kishimoto (World Scientific, Singapore, 1984) p. 20. [7] A.B. Balantekin and N. Takigawa, Ann. Phys. (NY) 160 (1985) 441. [8] R.A. Broglia and A. Winther, Heavy ion reactions, Vol. 1. Elastic and inelastic reactions, (Benjamin/Cummings, Menlo Park, 1981) pp. 104-115. [9] L.C. Vaz and J.M. Alexander, Z. Phys. A 305 (1982) 313; A 318 (1984) 231; G. Nebbia, K. Hagel, D. Fabris, Z. Majka, J.B. Natowitz, R.P. Schmitt, B. Sterling, G. Mouchaty, G. Berkowitz, K. Strozewski, G. Viesti, P.L. Gonthier, B. Wilkins, M.N. Namboodiri and H. Ho, Phys. Lett. B 176 (1986) 20. [ 10] A.M. Poskanzer, G.W. Butler and E.K. Hyde, Phys. Rev. C 3 ( 1971 ) 882; G.D. Westfall, R.G. Sextro, A.M. Poskanzer, A.M. Zebelman, G.W. Butler and E.K. Hyde, Phys. Rev. C 17 ( 1978 ) 1368. [ 11 ] L.G. Moretto and D.R. Bowman, Proc. 25th Intern. Winter Meeting on Nuclear physics (Bormio, Italy, January 1987).
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