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Real and imaginary parts of the nucleus-nucleus interaction using a microscopic approach
Volume 73B, number 4, 5
PHYSICS LETTERS
13 March 1978
REAL AND IMAGINARY P A R T S O F T H E N U C L E U S - N U C L E U S INTERACTION USING A MICROSCOPIC APPROACH S.K. GUPTA and S. KAILAS Nuclear Physics Division, Bhabha Atomic Research Centre, Bombay 400085, India Received 21 September 1977 Revised 9 January 1978
The energy density formalism is extended into the complex domain to calculate both the real and imaginary parts of the ion-ion potential. Recent nucleon-nucleus potentials calculated by Jeukenne, Lejeune and Mahaux using Reid's hard core nucleon-nucleon interaction are employed to generate the complex potential energy densities required in the calculation. The computed potentials are in good agreement with the phenomenological values.
Introduction. Optical potentials are very useful in describing the elastic scattering and reactions of colliding nuclei. These potentials are usually obtained from phenomenological analyses of nucleus-nucleus scattering data. This procedure in many cases leads to ambiguities in the values of real and imaginary parts of the potentials. However, a parameter-free theoretical calculation of the i o n - i o n potentials may yield more unique values for the potentials and hence will be better suited for transfer reaction calculations which is a further application of these potentials. The present calculation of the i o n - i o n potential is entirely parameter-free and in essence starts from the basic n u c l e o n - n u c l e o n interaction. Formalism and procedure. Using Brueckner's energy density formalism [1] many calculations [2] of the nucleus-nucleus potential have been reported. In this formalism the nucleus-nucleus potential U (taken as real) as a function of separation distance R between their centres is given by the expression
u(R) =f [H(p) -
H ( p 1) - H(P2)] d r ,
(1)
where H(p), H ( P l ) and H(P2) are the energy densities o f the composite system and individual nuclei, respectively. Here P, Pl and P2 are their local nuclear-matter densities and are functions of r. The geometrical rela398
1
2
Fig. 1. Geometrical relationships for colliding nuclei. tionship between r and R is shown in fig. 1. The energy density H ( p ) is written as t-h2/2,,,~ 3 (2,r2~2/3gn5/3 + ^5/3~ H(p) = ~ "'J "g ~" J "J"n Vp ]
(2)
+ ( h 2 / 7 2 m ) [(Vp)2/p] + p v ( p , a ) ,
where On and pp are neutron and proton densities, u is the nucleon-nucleus potential as a function of density p and a = (On - P p ) / P " H ( P l ) and H(P2) are also given by similar expressions. The first term in expression (2) is the kinetic energy density and is the same as given by the T h o m a s - F e r m i approximation. The second term is the inhomogeneity correction to the kinetic energy [1]. The third term is the potential energy density and the single-particle potential v is
Volume 73B, number 4, 5
PHYSICS LETTERS
chosen to be a real self-consistent Hartree-Fock potential. The nucleus-nucleus optical potential can also be visualised as the difference in the sum of kinetic energy and self-energy of the interacting nuclei before and after the interaction. The self-energy for pdr nucleons can be set equal to pu(p)dr, where u is the self-energy (mass operator) of a single nucleon propagating in nuclear matter of density p. This self-energy for a single nucleon has been identified as the single-nucleon optical potential by Hufner and Mahaux [3]. It has also been shown by Kohn and coworkers [4] that for a system of fermions u is a unique functional of the ground state density p. With these considerations, the energy density formalism can be extended into the complex domain by replacing v in expression (1) by the complex nucleon-nucleus optical, potential
u(p, ~, e) -u(p, ~, e) + iw(p, e~,e),
(3)
where e is the nucleon energy. Now U becomes complex with U R and U I as its real and imaginary parts. The real part U R has practically the same meaning as discussed earlier by treating U as real because the real part of the nucleon-nucleus optical potential is quite close to the Hartree-Fock potential. The interpretation for the imaginary part U I can be understood as follows. Writing U I as UI(R ) = f
[pw(p) - PlW(Pl ) - P2w(P2 )] dr
(4)
(the variables a and e have been suppressed in the above expression). The number of nucleons in an infinitesimal volume dr is represented by p dr and pw(p)dr stands for absorption due to p dr nucleons. But with the first term alone fictitious self-absorption occurs even when nucleons in the colliding nuclei do not interact absorptively. The other t w o terms therefore subtract this self-absorption. Further we use the sudden approximation for the composite density, i.e. P = Pl + P2 • Jeukenne et al. [5] (JLM) have recently computed the complex nucleon-nucleus potential u(p, a, e) as a function of p, a and e, using Reid's hard core n u c l e o n nucleon interaction by numerically solving the B e t h e Goldstone equation in infinite nuclear matter. We can use u = ~ as the first approximation. This is called the local density approximation (LDA). Some improved approximations over LDA have been discussed by
13 March 1978
JLM. These correct for inhomogeneous densities occurring in the nuclei in contrast to the infinite nuclear matter. We use one of their expressions and set
u(p) = ~ u@)/p, where ~ = (bv~)-3fp(r ') exp - ( I t -r'12/b2)dr with b = 1 fm. In the computation of u the corresponding nucleon energy e at a given point is computed by first subtracting the Coulomb energy from the relative energy of the interacting ions in the centre-of-mass system and distributing the net energy to the nucleons of two nuclei according to the momentum carried by them. For the composite system, e is expressed as e = (Plel
+P2e2)/(pl +p2 ) •
The proton density distribution pp was obtained by correcting the charge distribution data given by Uberall [6] for finite proton size. Further, the neutron density is assumed as Npp/Z. With these inputs, expression (1) (which is now complex) has been evaluated by numerical integration and both real and imaginary parts of the i o n - i o n potential have been generated as a function of inter-ion distance for many pairs of nuclei colliding at various relative energies. As illustrative examples, the potentials for 160 + 160 and 40Ca + 160 systems are shown in fig. 2 along with the phenomenological values [7]. For both systems the agreement between the phenomenological values and the calculation is good. The energy dependence of both the calculated real and imaginary potentials is less than 0.3% per MeV change in the incident energy in c.m. system. It should be mentioned that our calculation of the potentials will not be reliable at inter-ion distances less than the sum of the half-density radii of the colliding ions due to the inadequacy of the sudden approximation. It is well known [8] that for heavy ions only the surface region, of both real and imaginary components of the potential, plays a dominant role. Hence the abovementioned deficiency of our calculation is not an impediment in its further use. In effect, as we use the results of JLM, our calculation starts from the nucleon-nucleon interaction, uses an improved form of the local density approximation applied to the colliding nuclei and finally uses the energy density formalism in the complex domain with the sudden approximation for the composite system density to calculate the i o n - i o n optical potential not 399
Volume 73B, number 4, 5
PHYSICS LETTERS
i0 ~
13 March 1978 102
10 2
•,k(a)
160
÷ 160
-
(b)
,:40
MeV __
16 0
.\
CALCULATED
÷
40CA
Ecru= 30 MeV
'----PHENOMENO, I SHALLOW L "t . . . . . PHENOMENO,I ~
~.
10!
O~E~ J / .
%\ BECCHETTI ¢l.al.
101
100
1o 0
l
VR (MeV)
V[ {MeV)
16:
I
6
,
I
.
7 B 9 I0 1'1' 5 R (fro)
\
i
VR
":I O
\\
°
I',,
v,
(MeV)
\
162
-,--
10 !
/'
162
\
163
7 8 9 10 1'1
i, ,2"';';,'0
\ ,', 12 J,e
R(fm)
Fig. 2. Calculated potentials along with the phenomenological ones for (a) 160 + 160 and (b) 4°Ca + 160 systems. involving any parameter at any stage and thus the calculation is entirely microscopic. However, the c o m p l e x energy density formalism need not always use the sudden a p p r o x i m a t i o n or the microscopic potential energy densities.
[3] [4]
We w o u l d like to thank Dr. M.K. Mehta for his active interest in this w o r k and Drs. B.K. Jain and B.C. Sinha for m a n y useful discussions.
[5]
References
[7]
[1] K.A. Brueckner, J.R. Buchler, G. Jorna and R.J. Lombard, Phys. Rev. 171 (1968) 1188; 181 (1969) 1543. [2] K.A. Brueckner, J.R. Buchler and M.M. Kelly, Phys. Rev. 173 (1968) 944;
[8]
400
[6]
J. Galin, D. Guerreau, M. Lefort and X. Tarrago, Phys. Rev. C9 (1974) 1018; C. Ng6 et al., Nucl. Phys. A240 (1975) 353; C. Ng6 et al., Nucl. Phys. A252 (1975) 237. J. Hfifner and C. Mahaux, Ann. Phys. (NY) 73 (1972) 525. P. Hohenberg and W. Kohn, Phys. Rev. 136B (1964) 864; LJ. Sham and W. Kohn, Phys. Rev. 145 (1966) 561. J.P. Jeukenne, A. Lejeune and C. Mahaux, Phys. Rev. C16 (1977) 80. H. Llberall, Electron scattering from complex nuclei, Vol. A (Academic Press, New York, 1971). J.V. Maher et al., Phys. Rev. 188 (1969) 1665; F.D. Becchetti, P.R. Christensen, V.I. Manko and R.J. Nickles, Nucl. Phys. A203 (1973) 1. G.R. Satchler, Proc. Intern. Conf. on Reactions between complex nuclei, eds. R.L. Robinson, F.K. McGowan, T.B. Ball and T.H. Hamilton (Nashville, 1974) (North-Holland, Amsterdam, 1974) Vol. 2, p. 171.
PHYSICS LETTERS
13 March 1978
REAL AND IMAGINARY P A R T S O F T H E N U C L E U S - N U C L E U S INTERACTION USING A MICROSCOPIC APPROACH S.K. GUPTA and S. KAILAS Nuclear Physics Division, Bhabha Atomic Research Centre, Bombay 400085, India Received 21 September 1977 Revised 9 January 1978
The energy density formalism is extended into the complex domain to calculate both the real and imaginary parts of the ion-ion potential. Recent nucleon-nucleus potentials calculated by Jeukenne, Lejeune and Mahaux using Reid's hard core nucleon-nucleon interaction are employed to generate the complex potential energy densities required in the calculation. The computed potentials are in good agreement with the phenomenological values.
Introduction. Optical potentials are very useful in describing the elastic scattering and reactions of colliding nuclei. These potentials are usually obtained from phenomenological analyses of nucleus-nucleus scattering data. This procedure in many cases leads to ambiguities in the values of real and imaginary parts of the potentials. However, a parameter-free theoretical calculation of the i o n - i o n potentials may yield more unique values for the potentials and hence will be better suited for transfer reaction calculations which is a further application of these potentials. The present calculation of the i o n - i o n potential is entirely parameter-free and in essence starts from the basic n u c l e o n - n u c l e o n interaction. Formalism and procedure. Using Brueckner's energy density formalism [1] many calculations [2] of the nucleus-nucleus potential have been reported. In this formalism the nucleus-nucleus potential U (taken as real) as a function of separation distance R between their centres is given by the expression
u(R) =f [H(p) -
H ( p 1) - H(P2)] d r ,
(1)
where H(p), H ( P l ) and H(P2) are the energy densities o f the composite system and individual nuclei, respectively. Here P, Pl and P2 are their local nuclear-matter densities and are functions of r. The geometrical rela398
1
2
Fig. 1. Geometrical relationships for colliding nuclei. tionship between r and R is shown in fig. 1. The energy density H ( p ) is written as t-h2/2,,,~ 3 (2,r2~2/3gn5/3 + ^5/3~ H(p) = ~ "'J "g ~" J "J"n Vp ]
(2)
+ ( h 2 / 7 2 m ) [(Vp)2/p] + p v ( p , a ) ,
where On and pp are neutron and proton densities, u is the nucleon-nucleus potential as a function of density p and a = (On - P p ) / P " H ( P l ) and H(P2) are also given by similar expressions. The first term in expression (2) is the kinetic energy density and is the same as given by the T h o m a s - F e r m i approximation. The second term is the inhomogeneity correction to the kinetic energy [1]. The third term is the potential energy density and the single-particle potential v is
Volume 73B, number 4, 5
PHYSICS LETTERS
chosen to be a real self-consistent Hartree-Fock potential. The nucleus-nucleus optical potential can also be visualised as the difference in the sum of kinetic energy and self-energy of the interacting nuclei before and after the interaction. The self-energy for pdr nucleons can be set equal to pu(p)dr, where u is the self-energy (mass operator) of a single nucleon propagating in nuclear matter of density p. This self-energy for a single nucleon has been identified as the single-nucleon optical potential by Hufner and Mahaux [3]. It has also been shown by Kohn and coworkers [4] that for a system of fermions u is a unique functional of the ground state density p. With these considerations, the energy density formalism can be extended into the complex domain by replacing v in expression (1) by the complex nucleon-nucleus optical, potential
u(p, ~, e) -u(p, ~, e) + iw(p, e~,e),
(3)
where e is the nucleon energy. Now U becomes complex with U R and U I as its real and imaginary parts. The real part U R has practically the same meaning as discussed earlier by treating U as real because the real part of the nucleon-nucleus optical potential is quite close to the Hartree-Fock potential. The interpretation for the imaginary part U I can be understood as follows. Writing U I as UI(R ) = f
[pw(p) - PlW(Pl ) - P2w(P2 )] dr
(4)
(the variables a and e have been suppressed in the above expression). The number of nucleons in an infinitesimal volume dr is represented by p dr and pw(p)dr stands for absorption due to p dr nucleons. But with the first term alone fictitious self-absorption occurs even when nucleons in the colliding nuclei do not interact absorptively. The other t w o terms therefore subtract this self-absorption. Further we use the sudden approximation for the composite density, i.e. P = Pl + P2 • Jeukenne et al. [5] (JLM) have recently computed the complex nucleon-nucleus potential u(p, a, e) as a function of p, a and e, using Reid's hard core n u c l e o n nucleon interaction by numerically solving the B e t h e Goldstone equation in infinite nuclear matter. We can use u = ~ as the first approximation. This is called the local density approximation (LDA). Some improved approximations over LDA have been discussed by
13 March 1978
JLM. These correct for inhomogeneous densities occurring in the nuclei in contrast to the infinite nuclear matter. We use one of their expressions and set
u(p) = ~ u@)/p, where ~ = (bv~)-3fp(r ') exp - ( I t -r'12/b2)dr with b = 1 fm. In the computation of u the corresponding nucleon energy e at a given point is computed by first subtracting the Coulomb energy from the relative energy of the interacting ions in the centre-of-mass system and distributing the net energy to the nucleons of two nuclei according to the momentum carried by them. For the composite system, e is expressed as e = (Plel
+P2e2)/(pl +p2 ) •
The proton density distribution pp was obtained by correcting the charge distribution data given by Uberall [6] for finite proton size. Further, the neutron density is assumed as Npp/Z. With these inputs, expression (1) (which is now complex) has been evaluated by numerical integration and both real and imaginary parts of the i o n - i o n potential have been generated as a function of inter-ion distance for many pairs of nuclei colliding at various relative energies. As illustrative examples, the potentials for 160 + 160 and 40Ca + 160 systems are shown in fig. 2 along with the phenomenological values [7]. For both systems the agreement between the phenomenological values and the calculation is good. The energy dependence of both the calculated real and imaginary potentials is less than 0.3% per MeV change in the incident energy in c.m. system. It should be mentioned that our calculation of the potentials will not be reliable at inter-ion distances less than the sum of the half-density radii of the colliding ions due to the inadequacy of the sudden approximation. It is well known [8] that for heavy ions only the surface region, of both real and imaginary components of the potential, plays a dominant role. Hence the abovementioned deficiency of our calculation is not an impediment in its further use. In effect, as we use the results of JLM, our calculation starts from the nucleon-nucleon interaction, uses an improved form of the local density approximation applied to the colliding nuclei and finally uses the energy density formalism in the complex domain with the sudden approximation for the composite system density to calculate the i o n - i o n optical potential not 399
Volume 73B, number 4, 5
PHYSICS LETTERS
i0 ~
13 March 1978 102
10 2
•,k(a)
160
÷ 160
-
(b)
,:40
MeV __
16 0
.\
CALCULATED
÷
40CA
Ecru= 30 MeV
'----PHENOMENO, I SHALLOW L "t . . . . . PHENOMENO,I ~
~.
10!
O~E~ J / .
%\ BECCHETTI ¢l.al.
101
100
1o 0
l
VR (MeV)
V[ {MeV)
16:
I
6
,
I
.
7 B 9 I0 1'1' 5 R (fro)
\
i
VR
":I O
\\
°
I',,
v,
(MeV)
\
162
-,--
10 !
/'
162
\
163
7 8 9 10 1'1
i, ,2"';';,'0
\ ,', 12 J,e
R(fm)
Fig. 2. Calculated potentials along with the phenomenological ones for (a) 160 + 160 and (b) 4°Ca + 160 systems. involving any parameter at any stage and thus the calculation is entirely microscopic. However, the c o m p l e x energy density formalism need not always use the sudden a p p r o x i m a t i o n or the microscopic potential energy densities.
[3] [4]
We w o u l d like to thank Dr. M.K. Mehta for his active interest in this w o r k and Drs. B.K. Jain and B.C. Sinha for m a n y useful discussions.
[5]
References
[7]
[1] K.A. Brueckner, J.R. Buchler, G. Jorna and R.J. Lombard, Phys. Rev. 171 (1968) 1188; 181 (1969) 1543. [2] K.A. Brueckner, J.R. Buchler and M.M. Kelly, Phys. Rev. 173 (1968) 944;
[8]
400
[6]
J. Galin, D. Guerreau, M. Lefort and X. Tarrago, Phys. Rev. C9 (1974) 1018; C. Ng6 et al., Nucl. Phys. A240 (1975) 353; C. Ng6 et al., Nucl. Phys. A252 (1975) 237. J. Hfifner and C. Mahaux, Ann. Phys. (NY) 73 (1972) 525. P. Hohenberg and W. Kohn, Phys. Rev. 136B (1964) 864; LJ. Sham and W. Kohn, Phys. Rev. 145 (1966) 561. J.P. Jeukenne, A. Lejeune and C. Mahaux, Phys. Rev. C16 (1977) 80. H. Llberall, Electron scattering from complex nuclei, Vol. A (Academic Press, New York, 1971). J.V. Maher et al., Phys. Rev. 188 (1969) 1665; F.D. Becchetti, P.R. Christensen, V.I. Manko and R.J. Nickles, Nucl. Phys. A203 (1973) 1. G.R. Satchler, Proc. Intern. Conf. on Reactions between complex nuclei, eds. R.L. Robinson, F.K. McGowan, T.B. Ball and T.H. Hamilton (Nashville, 1974) (North-Holland, Amsterdam, 1974) Vol. 2, p. 171.