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Microscopic calculation of the coupling between relative motion and quadrupole deformation in heavy-ion collisions
VoIume lOOB, number 2
PHYSICS LETTERS
MICROSCOPIC CALCULATION
26 March 1981
OF THE COUPLING BETWEEN RELATIVE MOTION
AND ,QUADRUPOLE DEFORMATION
IN HEAVY-ION COLLISIONS
H. TRICOIRE Division de Physique Nuckaire,
Institut de Physique Nuclehire, 91406 Orsay-Cidex,
France
and H. FLOCARD and D. VAUTHERIN Division de Physique Thheon’que ‘, Institut de Physique Nuckaire,
91406 Orsay-Ce’dex, France
Received 30 December 1980
Using a microscopic formalism and taking fully into account the Pauli principle, we compute the interaction potential between two 4oCa nuclei as a function of their interdistance and their deformation. We attempt an analysis of our results within the proximity formalism and we point out some difficulties. We also extract the coupling parameter between the radial motion of the ions and the excitations of their giant quadrupole modes.
Over the last years the importance of collective deformations in the description of heavy-ion reactions has been stressed several times [l] . As a consequence an accurate classical description of these reactions must introduce along with the relative distance d of the two ions, coordinates characterizing the most important collective modes of the fragments. In this letter we investigate the dependence of the heavy-ion petential on the quadrupole deformation of projectile and target, from which we deduce the coupling factor between the interdistance d and the quadrupole modes of the fragments in the limit of an infmitesimal excitation of these modes. These potential and coupling factors are essential to describe collisions with either the adiabatic time dependent Hartree-Fock theory [2] (ATDHF)or the coherent surface excitation model [I] . We also compare our results to those obtained by means of the proximity potential [3] and we point out some differences. The heavy-ion system that we consider is 4oCa + 4OCa.At large distances it is described as two Slater determinants constructed from harmonic oscillator wave-functions centered at x = ti/2 (Xtiw= 10.4 MeV). ’ Laboratoire associe au CNRS.
106
The quadrupole deformation of each nucleus is described by a scaling parameter K which transforms the coordinates measured with respect to their center of mass as X’ = j+/3x,
y’ = &by,
Zr = j$/‘5,.
0)
From these formulae it is seen that we restrict ourselves to configurations where the two fragments are axially symmetric with their axis aligned with the collision axis, although the numerical code allows the study of less symmetric configurations [4]. In addition we chased the same scaling factor K for both nuclei. The value of the quadrupole moment Q of a 4o Ca nucleus induced by the transformation (1) is given by Q = (801tt/m.~~)K-“~(K - 1).
(2)
At shorter distances, when the two nuclei overlap, we take into account the Pauli principle by antisymmetrizing the wave functions. We thus obtain a Slater determinant \k(d, K) for the total system and compute the expectation value of the total hamiltonian E(d, K) in the state \k. The two body potential part of this hamiltonian uses the BKN interaction [S] as the nucleon-nucleon interaction. The nuclear collective potential I/N@, K) is then defined by
0 03 1-9163/gl/OOOO-0000/$02.50
0 North-Holland Publishing Company
26 March 1981
PHYSICS LETTERS
Volume lOOB, number 2
like those used in this letter there is a priori some arbitrariness as to the precise definition of the several quantities appearing in formula (4). This has been investigated in ref. [3] and in what follows we shall use the prescription given, in chapter 4.2 of this reference, namely for spherical nuclei s=d-
-601
5
d
(fm)
(3)
h practice it suffices to investigate vN for distances d less than 12 fm. In fig. 1 we display the potentials vN(d, K) for several values of K varying from 0.9 to 1.5 (-33 fm2 < Q < 138 fm2). As expected the nuclear potential extends further out for large values of K (prolate nuclei). One notes that all potential curves exhibit a minimum around the interdistance d = 6 fm and look almost proportional one to another over the range d < 10 fm. We do not have a simple explanation (even by scaling arguments) of this behaviour that results from a delicate balance between internal kinetic and potential terms. One of the most successful and simple parametrization of the heavy-ion potential relies on the idea of the proximity force [3]. In this formalism the important parameters are the interdistance s between the surfaces of the ions, the average curvature radius a and the thickness b of the surfaces of the two colliding ions. The most remarkable property of the proximity potential VP is that it factorizes into a universal function of the quantity s/b multiplied. by the factor Rb. More precisely one has
VP = LFnyl?b@(s/b)
(5)
C, = 3.575 fm, iT, = CJ2, b, = 1.05 fm. The values of the constant r and the analytic expression of the function @are also taken from ref. [3]. When the scaling factor K is introduced, the quantities w and b at the point of closest approach vary as
Fig. 1. Nuclear interaction potentials of two 40Ca nuclei for different values of the deformation parameter K.
VN(d, K) = E(d, K) - E(=, 1).
2Cs,
(4)
and the properties of the nucleon-nucleon interaction are entirely contained in the coefficient y and the function a. On the other hand the only useful property of the specific heavy-ion system is the average curvature radius of the surfaces. For density distributions
E
~~~~~13,
b =
bsK113, C= CsK1/3.
For any interdistance d and deformation K we can compute the proximity potential using formulae (4), (5) and (6). This is done in fig. 2 for the particular case K = 1. For distances d larger than 8 fm our potential coincides with the proximity potential. However, at shorter distances a large difference arises primarily due to the antisymmetrization introduced in our calcu lation which increases the internal kinetic energy by forcing the promotion of the nucleons into higher orbits [6]. An interesting question arises then as to the possibility of including the antisymmetrization into the proximity formula (4) by an adequate redefinition of the function @(s/b). If such a function exists it
01
-601
-2
-1
;
'
0
1
2
3
I 10
4 stfm)
I dtfm)
Fig. 2. Comparison between our potential I’N (solid curve) and the proximity potential I’p (dashed curve) for spherical ions. The upper scale gives the distance s between the nuclear surfaces calculated according to formula (5).
107
Volume lOOB, number 2
PHYSICS LETTERS
26 March 1981
10
Fig. 3. Influence of the deformation K of the fragments on the quantity -FN/(l?b) plotted as a function of s/b. The &shed curve gives the universal function proposed in ref. [2].
should be proportional to VN/(Eb) where V, is our calculated potential. In fig. 3 we have plotted as a function of s/b the quantities -VN/@b) for three different values of K. We still observe a strong K dependence which rules out a parametrization of our results in the simple form suggested by formula (4). Noticing that the maxima of the three curves occur for different values of s/b we can also eliminate a parametrization of VI,@, K) of the type \k(Eb)@(s/b). Finally in fig. 4 we compare the coupling potential
d(fm)
Fig. 4. Coupling factor between the interdistance of the two ions and their quadrupole moment calculated with our model (solid curve) and the proximity formalism (dashed curve).
In this letter we discussed the influence of quadrupole deformation of two colliding ions on their interaction potential. From our results it seems that antisymmetrization effects on the potential cannot be included into the formalism of proximity. We also calculated in a microscopic way the coupling between the relative motion and the excitation of a collective quadrupole mode of the ions. We believe that this is a significant step towards a description of heavy-ion collisions using the ATDHF theory. References
calculated from our results and the proximity potential. In view of the discussion above the approximate agreement between the two curves for d > 7 fm is certainly a coincidence. One may also note that our results for C(d) rules out a parametrization of V, in the form Q(d)@(K) that would have seemed plausible from a casual observation of fig. 1. Indeed such a factorization would result in a coupling coefficient C(d) proportional to VN(d, 1) in clear contradiction with our results at least ford < 6 fm.
108
[l] R.A. Broglia, C.H. Dasso and A. Winter, preprint Nordita 80/16 (1980), and references quoted therein. 121 M. Baranger and M. V&t&or& Ann. Phys. (NY) 114 (1978) 123. [ 31 J. Blocki, J. Randrup, W.J. Swiatecki and C.F. Tsang, Ann. Phys. 105 (1977) 427. [4] J. Cugnon, H. Doubre and H. Flocard, Nucl. Phys. A331 (1979) 213. [S] P. Bonche, S. Koonin and J.W. Negele, Phys. Rev. Cl3 (1976) 1226. [61 D.M. Brink and F. Stancu, Nucl. Phys. A243 (1975) 175.
PHYSICS LETTERS
MICROSCOPIC CALCULATION
26 March 1981
OF THE COUPLING BETWEEN RELATIVE MOTION
AND ,QUADRUPOLE DEFORMATION
IN HEAVY-ION COLLISIONS
H. TRICOIRE Division de Physique Nuckaire,
Institut de Physique Nuclehire, 91406 Orsay-Cidex,
France
and H. FLOCARD and D. VAUTHERIN Division de Physique Thheon’que ‘, Institut de Physique Nuckaire,
91406 Orsay-Ce’dex, France
Received 30 December 1980
Using a microscopic formalism and taking fully into account the Pauli principle, we compute the interaction potential between two 4oCa nuclei as a function of their interdistance and their deformation. We attempt an analysis of our results within the proximity formalism and we point out some difficulties. We also extract the coupling parameter between the radial motion of the ions and the excitations of their giant quadrupole modes.
Over the last years the importance of collective deformations in the description of heavy-ion reactions has been stressed several times [l] . As a consequence an accurate classical description of these reactions must introduce along with the relative distance d of the two ions, coordinates characterizing the most important collective modes of the fragments. In this letter we investigate the dependence of the heavy-ion petential on the quadrupole deformation of projectile and target, from which we deduce the coupling factor between the interdistance d and the quadrupole modes of the fragments in the limit of an infmitesimal excitation of these modes. These potential and coupling factors are essential to describe collisions with either the adiabatic time dependent Hartree-Fock theory [2] (ATDHF)or the coherent surface excitation model [I] . We also compare our results to those obtained by means of the proximity potential [3] and we point out some differences. The heavy-ion system that we consider is 4oCa + 4OCa.At large distances it is described as two Slater determinants constructed from harmonic oscillator wave-functions centered at x = ti/2 (Xtiw= 10.4 MeV). ’ Laboratoire associe au CNRS.
106
The quadrupole deformation of each nucleus is described by a scaling parameter K which transforms the coordinates measured with respect to their center of mass as X’ = j+/3x,
y’ = &by,
Zr = j$/‘5,.
0)
From these formulae it is seen that we restrict ourselves to configurations where the two fragments are axially symmetric with their axis aligned with the collision axis, although the numerical code allows the study of less symmetric configurations [4]. In addition we chased the same scaling factor K for both nuclei. The value of the quadrupole moment Q of a 4o Ca nucleus induced by the transformation (1) is given by Q = (801tt/m.~~)K-“~(K - 1).
(2)
At shorter distances, when the two nuclei overlap, we take into account the Pauli principle by antisymmetrizing the wave functions. We thus obtain a Slater determinant \k(d, K) for the total system and compute the expectation value of the total hamiltonian E(d, K) in the state \k. The two body potential part of this hamiltonian uses the BKN interaction [S] as the nucleon-nucleon interaction. The nuclear collective potential I/N@, K) is then defined by
0 03 1-9163/gl/OOOO-0000/$02.50
0 North-Holland Publishing Company
26 March 1981
PHYSICS LETTERS
Volume lOOB, number 2
like those used in this letter there is a priori some arbitrariness as to the precise definition of the several quantities appearing in formula (4). This has been investigated in ref. [3] and in what follows we shall use the prescription given, in chapter 4.2 of this reference, namely for spherical nuclei s=d-
-601
5
d
(fm)
(3)
h practice it suffices to investigate vN for distances d less than 12 fm. In fig. 1 we display the potentials vN(d, K) for several values of K varying from 0.9 to 1.5 (-33 fm2 < Q < 138 fm2). As expected the nuclear potential extends further out for large values of K (prolate nuclei). One notes that all potential curves exhibit a minimum around the interdistance d = 6 fm and look almost proportional one to another over the range d < 10 fm. We do not have a simple explanation (even by scaling arguments) of this behaviour that results from a delicate balance between internal kinetic and potential terms. One of the most successful and simple parametrization of the heavy-ion potential relies on the idea of the proximity force [3]. In this formalism the important parameters are the interdistance s between the surfaces of the ions, the average curvature radius a and the thickness b of the surfaces of the two colliding ions. The most remarkable property of the proximity potential VP is that it factorizes into a universal function of the quantity s/b multiplied. by the factor Rb. More precisely one has
VP = LFnyl?b@(s/b)
(5)
C, = 3.575 fm, iT, = CJ2, b, = 1.05 fm. The values of the constant r and the analytic expression of the function @are also taken from ref. [3]. When the scaling factor K is introduced, the quantities w and b at the point of closest approach vary as
Fig. 1. Nuclear interaction potentials of two 40Ca nuclei for different values of the deformation parameter K.
VN(d, K) = E(d, K) - E(=, 1).
2Cs,
(4)
and the properties of the nucleon-nucleon interaction are entirely contained in the coefficient y and the function a. On the other hand the only useful property of the specific heavy-ion system is the average curvature radius of the surfaces. For density distributions
E
~~~~~13,
b =
bsK113, C= CsK1/3.
For any interdistance d and deformation K we can compute the proximity potential using formulae (4), (5) and (6). This is done in fig. 2 for the particular case K = 1. For distances d larger than 8 fm our potential coincides with the proximity potential. However, at shorter distances a large difference arises primarily due to the antisymmetrization introduced in our calcu lation which increases the internal kinetic energy by forcing the promotion of the nucleons into higher orbits [6]. An interesting question arises then as to the possibility of including the antisymmetrization into the proximity formula (4) by an adequate redefinition of the function @(s/b). If such a function exists it
01
-601
-2
-1
;
'
0
1
2
3
I 10
4 stfm)
I dtfm)
Fig. 2. Comparison between our potential I’N (solid curve) and the proximity potential I’p (dashed curve) for spherical ions. The upper scale gives the distance s between the nuclear surfaces calculated according to formula (5).
107
Volume lOOB, number 2
PHYSICS LETTERS
26 March 1981
10
Fig. 3. Influence of the deformation K of the fragments on the quantity -FN/(l?b) plotted as a function of s/b. The &shed curve gives the universal function proposed in ref. [2].
should be proportional to VN/(Eb) where V, is our calculated potential. In fig. 3 we have plotted as a function of s/b the quantities -VN/@b) for three different values of K. We still observe a strong K dependence which rules out a parametrization of our results in the simple form suggested by formula (4). Noticing that the maxima of the three curves occur for different values of s/b we can also eliminate a parametrization of VI,@, K) of the type \k(Eb)@(s/b). Finally in fig. 4 we compare the coupling potential
d(fm)
Fig. 4. Coupling factor between the interdistance of the two ions and their quadrupole moment calculated with our model (solid curve) and the proximity formalism (dashed curve).
In this letter we discussed the influence of quadrupole deformation of two colliding ions on their interaction potential. From our results it seems that antisymmetrization effects on the potential cannot be included into the formalism of proximity. We also calculated in a microscopic way the coupling between the relative motion and the excitation of a collective quadrupole mode of the ions. We believe that this is a significant step towards a description of heavy-ion collisions using the ATDHF theory. References
calculated from our results and the proximity potential. In view of the discussion above the approximate agreement between the two curves for d > 7 fm is certainly a coincidence. One may also note that our results for C(d) rules out a parametrization of V, in the form Q(d)@(K) that would have seemed plausible from a casual observation of fig. 1. Indeed such a factorization would result in a coupling coefficient C(d) proportional to VN(d, 1) in clear contradiction with our results at least ford < 6 fm.
108
[l] R.A. Broglia, C.H. Dasso and A. Winter, preprint Nordita 80/16 (1980), and references quoted therein. 121 M. Baranger and M. V&t&or& Ann. Phys. (NY) 114 (1978) 123. [ 31 J. Blocki, J. Randrup, W.J. Swiatecki and C.F. Tsang, Ann. Phys. 105 (1977) 427. [4] J. Cugnon, H. Doubre and H. Flocard, Nucl. Phys. A331 (1979) 213. [S] P. Bonche, S. Koonin and J.W. Negele, Phys. Rev. Cl3 (1976) 1226. [61 D.M. Brink and F. Stancu, Nucl. Phys. A243 (1975) 175.