- Email: [email protected]
Hybrid model and angular distributions in preequilibrium reactions
Volume 63B, number 1
PHYSICS LETTERS
5 July 1976
HYBRID MODEL AND ANGULAR DISTRIBUTIONS IN PREEQUILIBRIUM REACTIONS G. MANTZOURANIS
Max-Planckdnstitutft~r Kernphysik, Heidelberg, W. Germany Received 21 October 1975 We present a model based on an extension of the hybrid model, from which both angular distributions and spectra of preequilibrium nucleons can be calculated successfully. Connections between the extended hybrid model and the master equation approach are discussed. Recently a modified master equation was presented to explain the experimental angular distributions of nucleons emitted in preequilibrium reactions [ 1]. The method was based on the assumption that the reaction proceeds through a series of two-body collisions, where the incident fast particle gradually loses its energy and memory of the initial direction. We present here a simplified model based on the hybrid model [2] which predicts the absolute values of the differential cross section. The states of the compound system are grouped (as in ref. [1]) into classes labelled by (n, ~2) with n the number of excitons and ~2 the direction of the fast particle. Let us suppose that through the first interaction between target and projectile a state with n o excitons was created with occupation probability Pno(~2) given as
Pno(~2) = rr-1 cos 0 0 ( ½ n - O ) ,
(1)
under the assumption of isotropy of the free nucleonnucleon scattering cross section. The occupation probability pn(~2) for class (n, ~2) may be written as
complexity (see fig. 1). The quantities pn(~2) must be compared with the quantities pn(~2) must be compared with the quantities ~n(~) = fTe~Pn(I2 , t)dt where Pn(~2, t) is the time-dependent occupation probability which obeys the generalized master equation proposed in ref. [1], and Teq is the equilibrium time. As indicated in fig. 1, the pn(~2) give a very good approximation to the angle dependence Of~n(I2), especially for high excitation energies, when the transitions to states of higher complexity become much stronger than the transitions to states with lower exciton number. We have to note that because of the level densities [3] used in the master equation approach we have fd~2~n+2(~2)/fdI2~n(~2 ) X a(n + 3)/(n +1)while in the frame of the hybrid model the corresponding ratio is equal to one. Therefore, and since the cross section is a weighted sum of the pn(~) or pn(~), respectively, one cannot draw any conclusion from fig. 1 on the agreement of the angular distributions obtained in both models. We write the differential cross section as
D. Pn-l(U)
d2o p. (n) = fdS u
u-,
1
1 (2)
with/a = (n - n0)/2 the number of the two-body collisions needed for the creation of a state with n excitons and with W(I2~I2')= r-d
do -' ~-d do (S~'-, ~ ) ,
the probability for the transition of the fast particle. from the direction ~2 to ~2', where do/d~2 is the free differential nucleon-nucleon cross section [1 ]. The functions pn(I2) are axially symmetric [1 ] and they become quickly angle-independent for states of high
de dI2 = °cg n =no An=+2
Xc(e)
on(E) [ d2o ~
(3)
× Xc(e)+•+(e ) + ~ded~2] H.F. ' where o c is the cross section for the formation of the compound-nucleus, fi the mean number of excitons at equilibrium, g is the single-particle level density, Pn(E) the density of the states with n excitons at excitation energy E, U is the energy of the residual nucleus, D n the depletion factor [2], and nfx is the fraction of nucleons x (protons or neutrons) available with in an n-exciton state. The quantities ~c(e) and X+(e) are the 25
Volume 63B, number 1
I
I
PHYSICS LETTERS
I
I
I
I
5 July 1976
I
10 o
-, ik
r-1
5QCo(~t,p ) \\
10 0
IE •
"~ 10-1 -~ ~
I
_
]
, x'X-"---'-
e~
10-1
i
lz/8
,
I
t
t
t
"rt/2
transition rate for the emission of one nucleon x into the c o n t i n u u m and the rate for internuclear transitions, respectively. The last term on the r.h.s, of eq. (3) is
[ I ~ I ' i i I
1SITs(or, p)
lOo r-1
lO 1
10-1
\\
~
lO-2 ~ ,~
x\\
(b)
30 40 E (MeV)
10-3
Fig. 2. The angle-integrated spectrum (a) and the angular distributions (b) of protons from the reaction 181Ta (a, p) at 54.8 MeV bombarding energy as calculated from the hybrid model (dashed curves) and the master equation (solid curves). The points are the data of ref. [4]. 26
the Hauser-Feshbach contribution taken to be isotropic. Since fd~Pn(~2 ) = 1, the angle-integrated spectrum predicted by eq. (3) is identical with the hybrid model predictions. The predictions of the model are shown in figs. 2 , 3 for the reactions 181Ta(a, p) and 59Co(a, p) at 54.8 MeV bombarding energy. The effects due to the nuclear geometry are discussed in ref. [1]. We have also tested the predictions of the model for several reactions in comparison with the results according to the master equation approach. In general, both models are in good agreement. The hybrid model predicts, in general, a stronger forward peak than the master equation calculations. Thus, we see that the hybrid model leads to results which are, in general, equivalent to the results obtained from the approach discussed in ref. [ 1], but it avoids the (especially at excitation energies > 40 MeV) timeconsuming calculations needed for the integration of the generalized master equation, and it predicts the absolute values of the differential cross sections. The author is grateful to Prof. H.A. Weidenmiiller for valuable discussions and a critical reading of the manuscript.
N
I , , I , , ~ I it12 iz O
m/8
¢
~\~4MeV \,
• 20
T~
22NeV"
Io "~ 100
10-1
~/2
Fig. 3. The same as in fig. 2 for the reaction 59Co (~,p). The dashed curves are the predictions of the hybdrid model.
Fig. 1. The functions pn(S2)for no = 3 (dashed curve no. 1) and for n = 5, 7 (dashed curves No. 2, 3) in comparison with the quantities ~n(S2) [ fd~2~n(~2)]-1 (solid lines) calculated from the master equation of ref. [1 ], at an excitation energy of 60 MeV and g = 4 MeV-1 with the initial configuration 2p-lh. The solid curves are not shown, ff they come too close to the dashed ones. The curve No. 3a represents the master equation calculation for n = 7 at 20 MeV excitation energy, while the functions pn(~) of the hybrid model axe independent of energy.
(a)
rt/8
0
-
O
102
22 MeV
\
\~38MeV 10-2
I
.
References [1] G.Mantzouranis, D. Agassi and H.A. Weidenmtlller, Phys. Lett. 57B (i975) 220; and to be submitted to Z. Physik A. [2] M. Blann, Phys. Rev. Lett. 27 (1971) 337. [3] F.C. Williams,Jr., Phys. Lett. 31B (1970) 184. [4] A. Chevarier et al., Phys. Rev. C8 (1973) 2155.
PHYSICS LETTERS
5 July 1976
HYBRID MODEL AND ANGULAR DISTRIBUTIONS IN PREEQUILIBRIUM REACTIONS G. MANTZOURANIS
Max-Planckdnstitutft~r Kernphysik, Heidelberg, W. Germany Received 21 October 1975 We present a model based on an extension of the hybrid model, from which both angular distributions and spectra of preequilibrium nucleons can be calculated successfully. Connections between the extended hybrid model and the master equation approach are discussed. Recently a modified master equation was presented to explain the experimental angular distributions of nucleons emitted in preequilibrium reactions [ 1]. The method was based on the assumption that the reaction proceeds through a series of two-body collisions, where the incident fast particle gradually loses its energy and memory of the initial direction. We present here a simplified model based on the hybrid model [2] which predicts the absolute values of the differential cross section. The states of the compound system are grouped (as in ref. [1]) into classes labelled by (n, ~2) with n the number of excitons and ~2 the direction of the fast particle. Let us suppose that through the first interaction between target and projectile a state with n o excitons was created with occupation probability Pno(~2) given as
Pno(~2) = rr-1 cos 0 0 ( ½ n - O ) ,
(1)
under the assumption of isotropy of the free nucleonnucleon scattering cross section. The occupation probability pn(~2) for class (n, ~2) may be written as
complexity (see fig. 1). The quantities pn(~2) must be compared with the quantities pn(~2) must be compared with the quantities ~n(~) = fTe~Pn(I2 , t)dt where Pn(~2, t) is the time-dependent occupation probability which obeys the generalized master equation proposed in ref. [1], and Teq is the equilibrium time. As indicated in fig. 1, the pn(~2) give a very good approximation to the angle dependence Of~n(I2), especially for high excitation energies, when the transitions to states of higher complexity become much stronger than the transitions to states with lower exciton number. We have to note that because of the level densities [3] used in the master equation approach we have fd~2~n+2(~2)/fdI2~n(~2 ) X a(n + 3)/(n +1)while in the frame of the hybrid model the corresponding ratio is equal to one. Therefore, and since the cross section is a weighted sum of the pn(~) or pn(~), respectively, one cannot draw any conclusion from fig. 1 on the agreement of the angular distributions obtained in both models. We write the differential cross section as
D. Pn-l(U)
d2o p. (n) = fdS u
u-,
1
1 (2)
with/a = (n - n0)/2 the number of the two-body collisions needed for the creation of a state with n excitons and with W(I2~I2')= r-d
do -' ~-d do (S~'-, ~ ) ,
the probability for the transition of the fast particle. from the direction ~2 to ~2', where do/d~2 is the free differential nucleon-nucleon cross section [1 ]. The functions pn(I2) are axially symmetric [1 ] and they become quickly angle-independent for states of high
de dI2 = °cg n =no An=+2
Xc(e)
on(E) [ d2o ~
(3)
× Xc(e)+•+(e ) + ~ded~2] H.F. ' where o c is the cross section for the formation of the compound-nucleus, fi the mean number of excitons at equilibrium, g is the single-particle level density, Pn(E) the density of the states with n excitons at excitation energy E, U is the energy of the residual nucleus, D n the depletion factor [2], and nfx is the fraction of nucleons x (protons or neutrons) available with in an n-exciton state. The quantities ~c(e) and X+(e) are the 25
Volume 63B, number 1
I
I
PHYSICS LETTERS
I
I
I
I
5 July 1976
I
10 o
-, ik
r-1
5QCo(~t,p ) \\
10 0
IE •
"~ 10-1 -~ ~
I
_
]
, x'X-"---'-
e~
10-1
i
lz/8
,
I
t
t
t
"rt/2
transition rate for the emission of one nucleon x into the c o n t i n u u m and the rate for internuclear transitions, respectively. The last term on the r.h.s, of eq. (3) is
[ I ~ I ' i i I
1SITs(or, p)
lOo r-1
lO 1
10-1
\\
~
lO-2 ~ ,~
x\\
(b)
30 40 E (MeV)
10-3
Fig. 2. The angle-integrated spectrum (a) and the angular distributions (b) of protons from the reaction 181Ta (a, p) at 54.8 MeV bombarding energy as calculated from the hybrid model (dashed curves) and the master equation (solid curves). The points are the data of ref. [4]. 26
the Hauser-Feshbach contribution taken to be isotropic. Since fd~Pn(~2 ) = 1, the angle-integrated spectrum predicted by eq. (3) is identical with the hybrid model predictions. The predictions of the model are shown in figs. 2 , 3 for the reactions 181Ta(a, p) and 59Co(a, p) at 54.8 MeV bombarding energy. The effects due to the nuclear geometry are discussed in ref. [1]. We have also tested the predictions of the model for several reactions in comparison with the results according to the master equation approach. In general, both models are in good agreement. The hybrid model predicts, in general, a stronger forward peak than the master equation calculations. Thus, we see that the hybrid model leads to results which are, in general, equivalent to the results obtained from the approach discussed in ref. [ 1], but it avoids the (especially at excitation energies > 40 MeV) timeconsuming calculations needed for the integration of the generalized master equation, and it predicts the absolute values of the differential cross sections. The author is grateful to Prof. H.A. Weidenmiiller for valuable discussions and a critical reading of the manuscript.
N
I , , I , , ~ I it12 iz O
m/8
¢
~\~4MeV \,
• 20
T~
22NeV"
Io "~ 100
10-1
~/2
Fig. 3. The same as in fig. 2 for the reaction 59Co (~,p). The dashed curves are the predictions of the hybdrid model.
Fig. 1. The functions pn(S2)for no = 3 (dashed curve no. 1) and for n = 5, 7 (dashed curves No. 2, 3) in comparison with the quantities ~n(S2) [ fd~2~n(~2)]-1 (solid lines) calculated from the master equation of ref. [1 ], at an excitation energy of 60 MeV and g = 4 MeV-1 with the initial configuration 2p-lh. The solid curves are not shown, ff they come too close to the dashed ones. The curve No. 3a represents the master equation calculation for n = 7 at 20 MeV excitation energy, while the functions pn(~) of the hybrid model axe independent of energy.
(a)
rt/8
0
-
O
102
22 MeV
\
\~38MeV 10-2
I
.
References [1] G.Mantzouranis, D. Agassi and H.A. Weidenmtlller, Phys. Lett. 57B (i975) 220; and to be submitted to Z. Physik A. [2] M. Blann, Phys. Rev. Lett. 27 (1971) 337. [3] F.C. Williams,Jr., Phys. Lett. 31B (1970) 184. [4] A. Chevarier et al., Phys. Rev. C8 (1973) 2155.