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On the distribution of angular momentum absorbed by compound nuclei in heavy-ion fusion reactions
Volume 138B, number 1,2,3
PHYSICS LETTERS
12 April 1984
ON THE DISTRIBUTION OF ANGULAR MOMENTUM ABSORBED BY COMPOUND NUCLEI IN HEAVY-ION FUSION REACTIONS S. LANDOWNE Technische Universitiit Munchen, D-8046 Garching, West Germany and C.H. DASSO Nordita, Blegdamsve] 17, DK-2100 Copenhagen O, Denmark Received 30 December 1983
It is shown that the coupling effects required to explain the magnitude of the sub-barrier fusion cross section of 4°Ar + 122Sn also predict that large angular momenta should be observed in 122Sn (4°At, xn3,) reactions at energies above the barrier.
In the general problem of barrier penetration with couplings to additional degrees of freedom, off-diagonal couplings enhance the transmitted flux when the incident energy is below the barrier and reduce the transmission when the energy is above it [ 1,2]. The former mechanism provides an understanding of why lowenergy fusion cross sections are enhanced by orders of magnitude compared to ordinary potential barrier penetration calculations [ 1 - 3 ] . It also appears to explain the irregular variations in the sub-barrier cross sections which have been observed when different isotopes are fused together [3,4]. In this work we want to draw attention to effects caused by off-diagonal couplings on the fusion reaction process at energies above the Coulomb barrier. In particular, it has recently been noted that the coupling has an important role in determining the distribution of angular momentum which is absorbed by the compound nucleus [5]. This type of effect has also been known to follow as a consequence of the coupling to collective surface modes (cf. ref. [6] and references therein). The angular momentum distributions can, in principle, be reconstructed from 3,-ray multiplicity measurements and statistical model analyses. In the present contribution we show that the coupling interaction required for explaining the sub-barrier fusion 32
rates also gives rise to specific effects on the angular momentum distributions predicted for 7-ray emission at higher bombarding energies. Sub-barrier fusion cross sections have been reproduced in ref. [3] using a simple approximation [2] to take the effects of channel coupling into account. The expression for the cross section used in ref. [3] is
of(E) = 7rr~(e/E) ~
I(01m)[ 2
m
× ln{1 + exp[(E - Vb0 --
Xm)/el}.
(1)
Here the states Ira)are eigenstates of the coupling matrix H c evaluated at the position r b of the s-wave barrier, Hclm)
= )t m
[m).
The height of the s-wave barrier is V0 and e is a measure of its thickness. The state 10)represents the ground state of the separated nuclei. It is well known that the form ofeq. (1) results from using a parabolic approximation for the penetration factors of the various partial waves [7]. Thus we may also write
0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 138B, number 1,2,3
PHYSICS LETTERS
12 April 1984 I
o f ( E ) = /ZJ o / ( E ) , Ol(E ) = (lr/k2)(21 + 1)
X~
~.
l
NO COUPLING
8
COUPLED
6
~4 I(0Lm)12[1 +exp(x/m)]
1,
m
l Xm
= [vO+ti21(l+l)/21ar2+Xm
E]/e,
0
(2)
where E = (h2k2/2/a). According to eq. (1) there is little effect on the total fusion cross section at energies well above the s-wave barrier due to the coupling. However, eq. (2) shows that this comes about because the transmission is reduced for those partial waves with effective potential barrier heights below the energy E and increased for those whose barriers are above E (see the note below). Thus the coupling alters the distribution of the partial cross sections (see also ref. [5]). In particular, at a given bombarding energy, it increases the probability of forming the compound nucleus with high angular momenta. To see the consequences of these arguments we have calculated the o l distributions according to eq. (2) for the 40Ar + 122Sn reaction at Ecru = 110 MeV. The various parameters are the same as those used in ref. [3] to account for the measured sub-barrier fusion cross section [8]. This distribution is then used in a statistical model evaporation code [9] to calculate the distribution of 7-rays emitted from the neutron decay products. Such analyses have been carried out for 122 Sn (40At, xnT) reactions in ref. [10]. The o l values and the resulting entry populations for the 3n, 4n and 5n evaporation residues are shown in fig. 1, where they are compared to the calculations without coupling. The qualitative difference which the coupling produces in the e l distribution is apparent. The average angular m o m e n t u m changes from 7 = 16~ to 7 = 257i. The total cross sections with and without coupling are similar at 159 mb and 103 mb, respectively. These results indicate that the maximum angular momentum of about 40h obtained from 7-ray multiplicity measurements [10] is accounted for by the coupling interaction. Or, to be more precise, the type of angular m o m e n t u m distribution obtained without coupling would be qualitatively different from those used to analyse the "f-ray data in ref. [10]. The increased probability for absorbing high angular momenta results in a striking shift of the 3n intensity to higher/-values
0
20
40
20
40
I
[
I
[
60
4n v
b"
4 2
0
,
0
20
'40
0
,
20
40
60
Fig. 1. Partial fusion cross section for 4°Ar + 122Sn atEcm = 110 MeV and corresponding "r-rayintensity distributions for the 3n, 4n and 5n evaporation residues. The results including the coupling used to describe the sub-barrier fusion are shown on the right hand side of the figure, while the results without coupling are show onn the left. The s-wave barrier height is 107 MeV. The compound nucleus 162Er is formed at an excitation energy of 51 MeV. The parameters of the calculations are from refs. [3,10]. (compare with fig. 1 o f r e f . [10]). There is also a significant change in the relative 3n and 4n yields. In addition to these main effects there are details which would be interesting to pursue more systematically. N o t e . The fact that off-diagonal couplings lead to enhancements (reductions) in transmission functions at energies below (above) a barrier can be easily demonstrated within the approximation used for eq. (2). In this scheme one needs only to show that the eigenvalues o f H c are distributed around zero. By construction, the matrix element o f H c in the ground state channel is zero since we measure energies and differences in potentials with respect to the initial channel [2]. Thus, by introducing the set o f states which diagonalizes He, we can write
0 = (OIHclO) = ~ ; (OIm)~lHcln) mn
= ~
I<01m>l 2 Xm .
m
33
Volume 138B, number 1,2,3
PHYSICS LETTERS
This shows that e x c e p t for uninteresting limits there are always positive and negative eigenvalues o f H c. We would like to thank J.D. Garrett for advice and help in setting up the evaporation m o d e l calculations. We also t h a n k H. Esbensen for pointing out to us that the effects o f surface deformations on the angular m o m e n t u m distributions were e x p l o r e d in the adiabatic limit in refs. [ 11,12]. One o f us (S.L.) acknowledges the support of the Bundesministerium ffir Forschung und Technologie, Federal Republic o f Germany.
References [1] C.H. Dasso, S. Landowne and A. Winther, Nucl. Phys. A405 (1983) 381. [2] C.H. Dasso, S. Landowne and A. Winther, Nucl. Phys. A407 (1983) 221.
34
12 April 1984
[3] R.A. Broglia, C.H. Dasso, S. Landowne and G. Pollarolo, Phys. Lett. 130 B (1983) 34. [4] R.A. Broglia, C.H. Dasso, S. Landowne and A. Winther, Phys. Rev. C27 (1983) 2433. [5] S. Landowne and S.C. Pieper, submitted to Phys. Rev. C. [6] R.A. Broglia, C.H. Dasso and A. Winther, Proc. Intern. School of Physics "Enrico Fermi", Course LXXVII, eds. R.A. Broglia, C.H. Dasso and R. Ricci (NorthHolland, Amsterdam, 1981). [7] C.Y. Wong, Phys. Rev. Lett. 31 (1973) 766. [8] W. Reisdorf et al., Phys. Rev. Lett. 49 (1982) 1811. [9] J.S. Gilat, GROG12 - A nuclear evaporation computer code - description and user's manual, BNL 50246 (T-580) (U.S. National Technical Information Service, 1970, Springfield, VA). [10] D.L. Hillis et al.,Nucl. Phys. A325 (1979) 216. [ l l ] M.B. Tsang et al., Phys. Rev. C28 (1983) 747. [12] R. Vandenbosch, B.B. Back, S. Gil, A. Lazzarini and A. Ray, Phys. Rev. C28 (1983) 1161.
PHYSICS LETTERS
12 April 1984
ON THE DISTRIBUTION OF ANGULAR MOMENTUM ABSORBED BY COMPOUND NUCLEI IN HEAVY-ION FUSION REACTIONS S. LANDOWNE Technische Universitiit Munchen, D-8046 Garching, West Germany and C.H. DASSO Nordita, Blegdamsve] 17, DK-2100 Copenhagen O, Denmark Received 30 December 1983
It is shown that the coupling effects required to explain the magnitude of the sub-barrier fusion cross section of 4°Ar + 122Sn also predict that large angular momenta should be observed in 122Sn (4°At, xn3,) reactions at energies above the barrier.
In the general problem of barrier penetration with couplings to additional degrees of freedom, off-diagonal couplings enhance the transmitted flux when the incident energy is below the barrier and reduce the transmission when the energy is above it [ 1,2]. The former mechanism provides an understanding of why lowenergy fusion cross sections are enhanced by orders of magnitude compared to ordinary potential barrier penetration calculations [ 1 - 3 ] . It also appears to explain the irregular variations in the sub-barrier cross sections which have been observed when different isotopes are fused together [3,4]. In this work we want to draw attention to effects caused by off-diagonal couplings on the fusion reaction process at energies above the Coulomb barrier. In particular, it has recently been noted that the coupling has an important role in determining the distribution of angular momentum which is absorbed by the compound nucleus [5]. This type of effect has also been known to follow as a consequence of the coupling to collective surface modes (cf. ref. [6] and references therein). The angular momentum distributions can, in principle, be reconstructed from 3,-ray multiplicity measurements and statistical model analyses. In the present contribution we show that the coupling interaction required for explaining the sub-barrier fusion 32
rates also gives rise to specific effects on the angular momentum distributions predicted for 7-ray emission at higher bombarding energies. Sub-barrier fusion cross sections have been reproduced in ref. [3] using a simple approximation [2] to take the effects of channel coupling into account. The expression for the cross section used in ref. [3] is
of(E) = 7rr~(e/E) ~
I(01m)[ 2
m
× ln{1 + exp[(E - Vb0 --
Xm)/el}.
(1)
Here the states Ira)are eigenstates of the coupling matrix H c evaluated at the position r b of the s-wave barrier, Hclm)
= )t m
[m).
The height of the s-wave barrier is V0 and e is a measure of its thickness. The state 10)represents the ground state of the separated nuclei. It is well known that the form ofeq. (1) results from using a parabolic approximation for the penetration factors of the various partial waves [7]. Thus we may also write
0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 138B, number 1,2,3
PHYSICS LETTERS
12 April 1984 I
o f ( E ) = /ZJ o / ( E ) , Ol(E ) = (lr/k2)(21 + 1)
X~
~.
l
NO COUPLING
8
COUPLED
6
~4 I(0Lm)12[1 +exp(x/m)]
1,
m
l Xm
= [vO+ti21(l+l)/21ar2+Xm
E]/e,
0
(2)
where E = (h2k2/2/a). According to eq. (1) there is little effect on the total fusion cross section at energies well above the s-wave barrier due to the coupling. However, eq. (2) shows that this comes about because the transmission is reduced for those partial waves with effective potential barrier heights below the energy E and increased for those whose barriers are above E (see the note below). Thus the coupling alters the distribution of the partial cross sections (see also ref. [5]). In particular, at a given bombarding energy, it increases the probability of forming the compound nucleus with high angular momenta. To see the consequences of these arguments we have calculated the o l distributions according to eq. (2) for the 40Ar + 122Sn reaction at Ecru = 110 MeV. The various parameters are the same as those used in ref. [3] to account for the measured sub-barrier fusion cross section [8]. This distribution is then used in a statistical model evaporation code [9] to calculate the distribution of 7-rays emitted from the neutron decay products. Such analyses have been carried out for 122 Sn (40At, xnT) reactions in ref. [10]. The o l values and the resulting entry populations for the 3n, 4n and 5n evaporation residues are shown in fig. 1, where they are compared to the calculations without coupling. The qualitative difference which the coupling produces in the e l distribution is apparent. The average angular m o m e n t u m changes from 7 = 16~ to 7 = 257i. The total cross sections with and without coupling are similar at 159 mb and 103 mb, respectively. These results indicate that the maximum angular momentum of about 40h obtained from 7-ray multiplicity measurements [10] is accounted for by the coupling interaction. Or, to be more precise, the type of angular m o m e n t u m distribution obtained without coupling would be qualitatively different from those used to analyse the "f-ray data in ref. [10]. The increased probability for absorbing high angular momenta results in a striking shift of the 3n intensity to higher/-values
0
20
40
20
40
I
[
I
[
60
4n v
b"
4 2
0
,
0
20
'40
0
,
20
40
60
Fig. 1. Partial fusion cross section for 4°Ar + 122Sn atEcm = 110 MeV and corresponding "r-rayintensity distributions for the 3n, 4n and 5n evaporation residues. The results including the coupling used to describe the sub-barrier fusion are shown on the right hand side of the figure, while the results without coupling are show onn the left. The s-wave barrier height is 107 MeV. The compound nucleus 162Er is formed at an excitation energy of 51 MeV. The parameters of the calculations are from refs. [3,10]. (compare with fig. 1 o f r e f . [10]). There is also a significant change in the relative 3n and 4n yields. In addition to these main effects there are details which would be interesting to pursue more systematically. N o t e . The fact that off-diagonal couplings lead to enhancements (reductions) in transmission functions at energies below (above) a barrier can be easily demonstrated within the approximation used for eq. (2). In this scheme one needs only to show that the eigenvalues o f H c are distributed around zero. By construction, the matrix element o f H c in the ground state channel is zero since we measure energies and differences in potentials with respect to the initial channel [2]. Thus, by introducing the set o f states which diagonalizes He, we can write
0 = (OIHclO) = ~ ; (OIm)~lHcln)
= ~
I<01m>l 2 Xm .
m
33
Volume 138B, number 1,2,3
PHYSICS LETTERS
This shows that e x c e p t for uninteresting limits there are always positive and negative eigenvalues o f H c. We would like to thank J.D. Garrett for advice and help in setting up the evaporation m o d e l calculations. We also t h a n k H. Esbensen for pointing out to us that the effects o f surface deformations on the angular m o m e n t u m distributions were e x p l o r e d in the adiabatic limit in refs. [ 11,12]. One o f us (S.L.) acknowledges the support of the Bundesministerium ffir Forschung und Technologie, Federal Republic o f Germany.
References [1] C.H. Dasso, S. Landowne and A. Winther, Nucl. Phys. A405 (1983) 381. [2] C.H. Dasso, S. Landowne and A. Winther, Nucl. Phys. A407 (1983) 221.
34
12 April 1984
[3] R.A. Broglia, C.H. Dasso, S. Landowne and G. Pollarolo, Phys. Lett. 130 B (1983) 34. [4] R.A. Broglia, C.H. Dasso, S. Landowne and A. Winther, Phys. Rev. C27 (1983) 2433. [5] S. Landowne and S.C. Pieper, submitted to Phys. Rev. C. [6] R.A. Broglia, C.H. Dasso and A. Winther, Proc. Intern. School of Physics "Enrico Fermi", Course LXXVII, eds. R.A. Broglia, C.H. Dasso and R. Ricci (NorthHolland, Amsterdam, 1981). [7] C.Y. Wong, Phys. Rev. Lett. 31 (1973) 766. [8] W. Reisdorf et al., Phys. Rev. Lett. 49 (1982) 1811. [9] J.S. Gilat, GROG12 - A nuclear evaporation computer code - description and user's manual, BNL 50246 (T-580) (U.S. National Technical Information Service, 1970, Springfield, VA). [10] D.L. Hillis et al.,Nucl. Phys. A325 (1979) 216. [ l l ] M.B. Tsang et al., Phys. Rev. C28 (1983) 747. [12] R. Vandenbosch, B.B. Back, S. Gil, A. Lazzarini and A. Ray, Phys. Rev. C28 (1983) 1161.