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Complex-generator-coordinate technique in resonating-group calculations employing flexible fragment wave functions
Volume 69B, number 1
PHYSICS LETTERS
COMPLEX-GENERATOR-COORDINATE CALCULATIONS
18 July 1977
TECHNIQUE IN RESONATING-GROUP
EMPLOYING FLEXIBLE FRAGMENT
WAVE FUNCTIONS*
D.R. THOMPSON 1 lnstitut ffir Theoretische Physik der Universitat Tubingen, Tubingen, Germany
and M. LeMERE and Y.C. TANG School of Phystcs and Astronomy, Umversity o f Mmnesota, Minneapohs, Minnesota 55455, USA
Recewed 5 April 1977 It is shown that the complex-generator-coordinate techmque can be used to alleviate the computational difficulties assocmted with resonating-group calculations for relatively heavy systems, even when the fragments are described by rather general wave functions The case of n + 160 elastic scattering Is presented as an illustrative example.
Resonating-group calculations, which employ fully antisymmetrlc many-nucleon wave functions and a reasonable nucleon-nucleon potential, have been quite successful in predicting the properties of light nuclear systems [1 ]. Because of the comphcations introduced by the microscopic nature of these calculations, apphcation to systems containing more than about ten nucleons was, until rather recently, not attempted. During the past few years, however, the resonating-group method (RGM) has begun to receive an increasing amount of attention, due m most part to the development of computational procedures [1, 2] which extend its usefulness to include nuclear systems which have heretofore been examined only by macroscopic, phenomenological means. The Munster group [3, 4], for example, has made use of the correspondence between the RGM and the generator-coordinate method [2] to calculate the elastic scattering of such rather heavy systems as 160 + 160 and 160 + 40Ca. This procedure is suitable for numerical computations, but requires the use of rather large integration step size; consequently, the problem of numerical accuracy may become quite severe especially in heavy systems [3]. In addition, it has the disadvantage that its extension to scattering * Work supported m part by the U.S. Energy Research and Development Admimstratlon under Contract No. AT(11-1)1764. 1 Alexander yon Humboldt Stiftung Fellow.
problems involving fragments characterized by unequal width parameters and to reaction problems may not be straightforward. Another somewhat different procedure has been used by the group at Tubingen [5, 6] and Minnesota [ 7 - 9 ] . This procedure, called the complex-generatorcoordinate technique, is described in some detail in ref. [9] (to be referred to as TT) In connection with the problem o f N + 160 scattering. It has, m fact, been used recently in calculations of a + 40Ca [6] and a + 160 [7] scattering where a proper consideration of the various fragment sizes was included. Up to now, however, this technique has only been used in calculations where the internal function describing the fragment was chosen as an antlsymmetrized product of harmonic-oscillator functions in the lowest shell-model configuration. It is the purpose of the present note to show that the extension to include more general fragment wave functions is straightforward, and that the complex-generator-coordinate technique can still significantly simplify RGM calculations in heavy systems even when such general fragment wave functions are required. In the following we shall first give a very brief description of the complex-generator-coordinate technique. Then we shall show how the generalization is accomplished and Illustrate using the n + 160 scattering problem as an example. In the single-channel RGM, the wave function for the system has the form [1]
Volume 69B, number 1
PHYSICS LETTERS
18 July 1977
as
qt = f F ( R " ) -¢/' (q~l (41)~b2(~ 2)8 (R - R")X(Rcm)} dR",
(1) where ~1 (41) and ~b2(~2) are antlsymmetric target and projectile wave functions which depend on their respective internal space, spin, and isospin coordinates, collecnvely denoted as ~1 and ~2" The operator ~ ' accounts for the intercluster antlsymmetrization of the wave function. The function ×(Rcm ) describes the center-of-mass motion of the entire system; its choice is arbitrary, since a Gahlean-invariant Hamiltonlan is employed m RGM calculations. The function F(R"), describing the relative motion of the two clusters, is determined from the projectile equation (8xI, l H - E t l q , ) = 0,
(2)
where E t is the total energy and H is the Hamiltonian operator given by fi2 A H=---~V 2Mi=l
2+
A ~ vq-rcm, l
(3)
with Tcm being the total c.m. klnetm-energy operator. Using eqs. (1) and (2), one then finds that F(R") satisfies an integrodifferential equation of the form
R")F(R") dR" = 0,
(4)
with ~g ( R ' , R " ) = (x(R cm)q~1(~1)q~2(~2)~( R - R ' ) I H - Et[
(5) X . ~ ' (~b1 (41)~b2(~2~(R -R")X(Rcm)}} • Because of the presence of the operator s4' in the above equation, the computation of 9~ (R', R") can be quite tedious, especially when the number of nucleons in the system is large. The complex-generator-coordinate technique (see, e.g., the descrlptmn given in TT) can greatly reduce the computational difficulty associated with the evaluation o f ~ (R', R"). Briefly, what one does is to replace the delta functions in eq. (5) by their spectral representation [10]. Then, if the fragment wave functions q~l and ~2 are chosen as antisymmetrized products of harmonic,oscillator functions in their respective lowest shell-model configuration and if a suitable choice for ×(Rcm ) is made, one can rewriteg~ ( R ' , R " )
/
,S ;R S , \ ~ . ~q(G;R ,S)IH-E,[
~(R',R") = "
×~
i=1
A
~(rfiR",S")
\ t = l
}t
dS'dS",
-
,6,
with ~ being the full A-particle antisymmetrization operator .1 . The important point to note here is that, in the expression for ~ ( R ' , R " ) given by eq. (6), there appear only products of complex functions ~Pi,each depending on a single-particle (space, spin, and lsospin) coordinate ft. Thus, the antisymmetrization procedure is now straightforward, and many of the well-known shell-model techniques can be used to facilitate the computation o f ~ (R', R"). Since the step from eq. (5) to eq. (6) cannot be easily accomplished unless ~b1 and 4)2 are chosen to be simple oscillator functions as described above, it would appear that the complex-generator-coordinate technique is not useful when a more general description of the fragment nuclei is required. This, however, is not the case, and in what follows we shall show that the extension of this technique to include more general fragment wave functions is in fact not difficult. To illustrate this, we shall use the problem of n + 160 scattering as a specific example, since for this case many of the formulae given in TT are directly applicable. For the n + 160 system, we write
q, = f F(R") ~ ' ( ¢'16°~(s17)v(t17)
(7) X 8(R - R")x(R cm )} dR", where the notation is the same as that of TT and N q~16 =/~=1 cI)/,
(8)
with
#1 If 01 and 02 are chosen as harmonic-oscillator shell-model functions of the same width parameter, then ~oi in eq (6) will no longer depend on R' or R". As a consequence, the calculation becomes somewhat simplified.
Volume 69B, number 1
PHYSICS LETTERS
°°°2 ,oo _-
I
0
18 July 1977
n.:O 13.18 MeV
\
I
20
I
40
I
60
I
80
I
I00
[
t20
I
140
[
160
180
8 (deg) Fig. 1. C o m p a r B o n o f c a l c u l a t e d a n d e x p e r i m e n t a l d i f f e r e n t i a l cross s e c t i o n s for n + 16 0 s c a t t e r i n g at a c.m. e n e r g y o f 13 18 MeV.
Experimental data shown are those of ref. [13].
= Cl -o{16 [It=~ 1 [ht(rt-R16)~i(st, t ') 1
× exp[-707(r i -R16)2]]
}
(9) •
It is noted that each of the functions q~! is specifically chosen to be a translationally invarlant antisymmetrized product of single-particle wave functions of the lowest configuration in a harmonic-oscillator well of width parameter 07. Nevertheless, if the number of terms N in eq. (8) is chosen large enough, the function q~16 can be made very flexible. With ¢16 gwen by eqs. (8) and (9), eq. (4) becomes N
~ ( R ' , R " ) in the form o f e q . (6). Since this choice involves the oscillator width parameter of the fragment wave function (see eq. (23) of TT), it is not clear in the present case how this choice should be made because different width parameters appear on the bra and ket sides of eq. (11). However, by using the fact that the operator (H - Et) is Galilean-invarlant, one can write eq. (11) as
(X(Rcm)lX(Rcm)) ~ iI(R',R") = (Zi(R cm)lZl(Rcm) ) X (Zt(Rcm)dPt6(R - R')a(Sl7)V(tl7)
(12)
gt' (~]6(R - R")a(s17)v(t17)Z/(Rcm)}). The arbitrary f u n c t i o n Zi(Rcm ) can now be chosen, in X I H - Etl
t,/=l
f ~ ,I(_~',R")F(R") dR" = O,
(10)
accordance with eq. (23) of TT, to be
where
Zl(Rcm ) = e x p [ - 7 (1m + 1)%R2cml
~zI (R',R") = (x(Rcan) d~t~(R - R')~(Sl 7 )V(t l 7 ) l H - Et [ (11)
X ~ ' {rb]5(R - R")o~(Sl7)V(tl7)X(Rcm)}). As mentioned above, a particular form for the function X(Rcm) must be chosen in order to express
(13)
with m = 1 6. With this choice, the matrix element c~l(R', R") can be written in the form of eq. (6) (with different complex single-pamcle functions appearing on the bra and ket sides), and its evaluation can be accomphshed just as in the single-width-parameter case discussed in TT. 3
Volume 69B, number 1
PHYSICS LETTERS
To Illustrate the procedure described above, we have performed a calculation for the n + 160 system. For simphcity, the number o f terms in the fragment function ~b16 o f eq. (8) is limited t o N = 2. The parameters c! and a7 are chosen to yield the experimentally determined rms matter radius of 160 (2.6 fm) and a best over-all agreement with the form-factor data [11] for q2 up to about 7 fm - 2 . These parameters turn out to be c I = 1.0,
c 2 = 1514.0, (14)
~1 -- 0.24 fm - 2 ,
a 2 = 0.34 fm - 2 .
Except for the fact that the 160 cluster is described by the two-width-parameter function discussed above, the present calculation IS performed in the same manner as that described in a previous publication [I 2]. In particular, the central part o f the nucleon-nucleon potential in eq. (3) is given by eqs. ( 9 ) - ( 1 1 ) o f ref. [12]. This potential differs from the one used in TT in that it contains a weakly repulsive core and yields a satisfactory description of not only the low-energy nucleon-nucleon scattering data but also the essential properties of the deuteron, triton, and a particle. The exchange-mixture parameter u in this potential (see eq. (9) of ref. [12] ) is determined by fitting the neutron separation energy o f 3.26 MeV in the l = 0, j~r = 1 + first excited state of 170. The resultant value o f u is 0.945. Also, we have used the same spin-orbit potential as in ref. [12] ; this yields a satisfactory value for the energy splitting between the l = 2, 5/2 + ground and first 3/2 + excited states. Finally, for calculations at energies where reaction channels are open, we have incorporated the phenomenological imaginary potential given by eqs. (17) and (18) o f ref. [12]. The n + 160 differential cross section calculated at a c.m. energy o f 13.18 MeVis shown in fig. 1, where a comparison with experimental data [13] is also made. To obtain the calculated result, we have used an imaginary-potential strength I4)s = 4.7 MeV, the same as that used in ref. [12]. With this choice, the calculated reaction and integrated-elastic cross sections are, respectively, equal to 646 and 856 mb, which agree quite well with measured values. A comparison of fig. 1 with fig. 3b o f ref. [12] shows that both calculations employing, respectively, two-width-parameter and one-width-parameter 160 functions yield similarly good agreement with experi-
18 July 1977
mental results. This indicates that the function ~b16 can, in fact, be taken as a simple harmonic-oscdlator shell-model wave function characterized by a single properly-chosen width parameter and a more general fragment wave function is not needed in this case (see also the discussion given in ref. [14]). In vaew of the doubly-closed-shell nature of 160, such a finding is actually not too surprising. In conclusion, we have shown that it is mdeed possible to generahze the complex-generator-coordinate technique so that flexible fragment wave functions can be employed even in resonating-group calculations for rather heavy systems. This generahzatlon is quite straightforward, and we have illustrated using the n + 160 scattering problem as an example. Although it is found that in this particular example the generalization is not necessary, we do feel certain that for calculations involving non-closed-shell nuclei, this procedure will definitely be important. For instance, it is well-known that the nucleus 6Li is rather diffuse and cannot be adequately described by a simple harmonic-oscillator shell-model wave function. Thus, for a proper consideration of the inelastic process 6Li(p, p')6 Li* and the reaction 6 Li(p, 3 He)4 He, it will be necessary to adopt a flexible function for 6L1 and, consequently, utilize the complex-~nerator-coordlnate technique outlined in this note *~. ,2 It is clear of course that the techmque outhned here can also be used to handle reaction problems where all fragments involved are characterized by different width parameters.
References [1 ] Y.C. Tang and D.R. Thompson, m: Proc. 2nd Intern. Conf. on Clustering phenomena in nuclei, College Park, Maryland, 1975 (National Techmcal Information Service, Springfield, Vlrgima 22161) p. 119, and references contamed therein; K. Wfldermuth and Y C. Tang, A unified theory of the nucleus (Vleweg, Braunschweig, Germany, 1977). [2] D.M. Brink, in. Proc. Intern. Conf. on Clustering phenomena in nuclei, Bochum, Germany, 1969 (IAEA, Vienna, 1969) p. 147. [3] H. Friedrich, Nucl. Phys. A224 (1974) 537, and references contained thereto. [4] H. Friednch, K. Langanke and A. Welguny, Phys. Lett. 63B (1976) 125. [5] W. Sknkel and K. Wlldermuth, Phys. Lett. 41B (1972) 439
Volume 69B, number 1
PHYSICS LETTERS
[6] W. Sunkel, Phys. Lett. 65B (1976)419. [7] M. LeMere, Y.C. Tang and D.R. Thompson, Pliys. Rev. C14 (1976) 23. [8] D.R. Thompson, M. LeMere and Y.C. Tang, Nucl. Phys. A270 (1976) 211, M. LeMere, Y.C. Tang and D.R. Thompson, Phys. Lett. 63B (1976) 1,and Phys. Rev. C14 (1976) 1715. [9] D.R. Thompson and Y.C Tang, Phys. Rev. C12 (1975) 1432 and C13 (1976) 2597.
18 July 1977
[10] H. Honuchl, Progr. Theor. Phys. 47 (1972) 1058. [11] H Crannell, Phys Rev. 148 (1966)1107. [12] D.R. Thompson, M. LeMere and Y.C. Tang, Systematic investigation of scattering problems with the resonatinggroup method, wtll appear in Nucl. Phys. [13] R.W. Bauer, J.D. Anderson and LJ. Chnstensen, Nucl. Phys. 47 (1963) 241. [14] F.S. Chwleroth, Y.C. Tang and D.R. Thompson, Nucl. Phys. A189 (1972) 1.
PHYSICS LETTERS
COMPLEX-GENERATOR-COORDINATE CALCULATIONS
18 July 1977
TECHNIQUE IN RESONATING-GROUP
EMPLOYING FLEXIBLE FRAGMENT
WAVE FUNCTIONS*
D.R. THOMPSON 1 lnstitut ffir Theoretische Physik der Universitat Tubingen, Tubingen, Germany
and M. LeMERE and Y.C. TANG School of Phystcs and Astronomy, Umversity o f Mmnesota, Minneapohs, Minnesota 55455, USA
Recewed 5 April 1977 It is shown that the complex-generator-coordinate techmque can be used to alleviate the computational difficulties assocmted with resonating-group calculations for relatively heavy systems, even when the fragments are described by rather general wave functions The case of n + 160 elastic scattering Is presented as an illustrative example.
Resonating-group calculations, which employ fully antisymmetrlc many-nucleon wave functions and a reasonable nucleon-nucleon potential, have been quite successful in predicting the properties of light nuclear systems [1 ]. Because of the comphcations introduced by the microscopic nature of these calculations, apphcation to systems containing more than about ten nucleons was, until rather recently, not attempted. During the past few years, however, the resonating-group method (RGM) has begun to receive an increasing amount of attention, due m most part to the development of computational procedures [1, 2] which extend its usefulness to include nuclear systems which have heretofore been examined only by macroscopic, phenomenological means. The Munster group [3, 4], for example, has made use of the correspondence between the RGM and the generator-coordinate method [2] to calculate the elastic scattering of such rather heavy systems as 160 + 160 and 160 + 40Ca. This procedure is suitable for numerical computations, but requires the use of rather large integration step size; consequently, the problem of numerical accuracy may become quite severe especially in heavy systems [3]. In addition, it has the disadvantage that its extension to scattering * Work supported m part by the U.S. Energy Research and Development Admimstratlon under Contract No. AT(11-1)1764. 1 Alexander yon Humboldt Stiftung Fellow.
problems involving fragments characterized by unequal width parameters and to reaction problems may not be straightforward. Another somewhat different procedure has been used by the group at Tubingen [5, 6] and Minnesota [ 7 - 9 ] . This procedure, called the complex-generatorcoordinate technique, is described in some detail in ref. [9] (to be referred to as TT) In connection with the problem o f N + 160 scattering. It has, m fact, been used recently in calculations of a + 40Ca [6] and a + 160 [7] scattering where a proper consideration of the various fragment sizes was included. Up to now, however, this technique has only been used in calculations where the internal function describing the fragment was chosen as an antlsymmetrized product of harmonic-oscillator functions in the lowest shell-model configuration. It is the purpose of the present note to show that the extension to include more general fragment wave functions is straightforward, and that the complex-generator-coordinate technique can still significantly simplify RGM calculations in heavy systems even when such general fragment wave functions are required. In the following we shall first give a very brief description of the complex-generator-coordinate technique. Then we shall show how the generalization is accomplished and Illustrate using the n + 160 scattering problem as an example. In the single-channel RGM, the wave function for the system has the form [1]
Volume 69B, number 1
PHYSICS LETTERS
18 July 1977
as
qt = f F ( R " ) -¢/' (q~l (41)~b2(~ 2)8 (R - R")X(Rcm)} dR",
(1) where ~1 (41) and ~b2(~2) are antlsymmetric target and projectile wave functions which depend on their respective internal space, spin, and isospin coordinates, collecnvely denoted as ~1 and ~2" The operator ~ ' accounts for the intercluster antlsymmetrization of the wave function. The function ×(Rcm ) describes the center-of-mass motion of the entire system; its choice is arbitrary, since a Gahlean-invariant Hamiltonlan is employed m RGM calculations. The function F(R"), describing the relative motion of the two clusters, is determined from the projectile equation (8xI, l H - E t l q , ) = 0,
(2)
where E t is the total energy and H is the Hamiltonian operator given by fi2 A H=---~V 2Mi=l
2+
A ~ vq-rcm, l
(3)
with Tcm being the total c.m. klnetm-energy operator. Using eqs. (1) and (2), one then finds that F(R") satisfies an integrodifferential equation of the form
R")F(R") dR" = 0,
(4)
with ~g ( R ' , R " ) = (x(R cm)q~1(~1)q~2(~2)~( R - R ' ) I H - Et[
(5) X . ~ ' (~b1 (41)~b2(~2~(R -R")X(Rcm)}} • Because of the presence of the operator s4' in the above equation, the computation of 9~ (R', R") can be quite tedious, especially when the number of nucleons in the system is large. The complex-generator-coordinate technique (see, e.g., the descrlptmn given in TT) can greatly reduce the computational difficulty associated with the evaluation o f ~ (R', R"). Briefly, what one does is to replace the delta functions in eq. (5) by their spectral representation [10]. Then, if the fragment wave functions q~l and ~2 are chosen as antisymmetrized products of harmonic,oscillator functions in their respective lowest shell-model configuration and if a suitable choice for ×(Rcm ) is made, one can rewriteg~ ( R ' , R " )
/
,S ;R S , \ ~ . ~q(G;R ,S)IH-E,[
~(R',R") = "
×~
i=1
A
~(rfiR",S")
\ t = l
}t
dS'dS",
-
,6,
with ~ being the full A-particle antisymmetrization operator .1 . The important point to note here is that, in the expression for ~ ( R ' , R " ) given by eq. (6), there appear only products of complex functions ~Pi,each depending on a single-particle (space, spin, and lsospin) coordinate ft. Thus, the antisymmetrization procedure is now straightforward, and many of the well-known shell-model techniques can be used to facilitate the computation o f ~ (R', R"). Since the step from eq. (5) to eq. (6) cannot be easily accomplished unless ~b1 and 4)2 are chosen to be simple oscillator functions as described above, it would appear that the complex-generator-coordinate technique is not useful when a more general description of the fragment nuclei is required. This, however, is not the case, and in what follows we shall show that the extension of this technique to include more general fragment wave functions is in fact not difficult. To illustrate this, we shall use the problem of n + 160 scattering as a specific example, since for this case many of the formulae given in TT are directly applicable. For the n + 160 system, we write
q, = f F(R") ~ ' ( ¢'16°~(s17)v(t17)
(7) X 8(R - R")x(R cm )} dR", where the notation is the same as that of TT and N q~16 =/~=1 cI)/,
(8)
with
#1 If 01 and 02 are chosen as harmonic-oscillator shell-model functions of the same width parameter, then ~oi in eq (6) will no longer depend on R' or R". As a consequence, the calculation becomes somewhat simplified.
Volume 69B, number 1
PHYSICS LETTERS
°°°2 ,oo _-
I
0
18 July 1977
n.:O 13.18 MeV
\
I
20
I
40
I
60
I
80
I
I00
[
t20
I
140
[
160
180
8 (deg) Fig. 1. C o m p a r B o n o f c a l c u l a t e d a n d e x p e r i m e n t a l d i f f e r e n t i a l cross s e c t i o n s for n + 16 0 s c a t t e r i n g at a c.m. e n e r g y o f 13 18 MeV.
Experimental data shown are those of ref. [13].
= Cl -o{16 [It=~ 1 [ht(rt-R16)~i(st, t ') 1
× exp[-707(r i -R16)2]]
}
(9) •
It is noted that each of the functions q~! is specifically chosen to be a translationally invarlant antisymmetrized product of single-particle wave functions of the lowest configuration in a harmonic-oscillator well of width parameter 07. Nevertheless, if the number of terms N in eq. (8) is chosen large enough, the function q~16 can be made very flexible. With ¢16 gwen by eqs. (8) and (9), eq. (4) becomes N
~ ( R ' , R " ) in the form o f e q . (6). Since this choice involves the oscillator width parameter of the fragment wave function (see eq. (23) of TT), it is not clear in the present case how this choice should be made because different width parameters appear on the bra and ket sides of eq. (11). However, by using the fact that the operator (H - Et) is Galilean-invarlant, one can write eq. (11) as
(X(Rcm)lX(Rcm)) ~ iI(R',R") = (Zi(R cm)lZl(Rcm) ) X (Zt(Rcm)dPt6(R - R')a(Sl7)V(tl7)
(12)
gt' (~]6(R - R")a(s17)v(t17)Z/(Rcm)}). The arbitrary f u n c t i o n Zi(Rcm ) can now be chosen, in X I H - Etl
t,/=l
f ~ ,I(_~',R")F(R") dR" = O,
(10)
accordance with eq. (23) of TT, to be
where
Zl(Rcm ) = e x p [ - 7 (1m + 1)%R2cml
~zI (R',R") = (x(Rcan) d~t~(R - R')~(Sl 7 )V(t l 7 ) l H - Et [ (11)
X ~ ' {rb]5(R - R")o~(Sl7)V(tl7)X(Rcm)}). As mentioned above, a particular form for the function X(Rcm) must be chosen in order to express
(13)
with m = 1 6. With this choice, the matrix element c~l(R', R") can be written in the form of eq. (6) (with different complex single-pamcle functions appearing on the bra and ket sides), and its evaluation can be accomphshed just as in the single-width-parameter case discussed in TT. 3
Volume 69B, number 1
PHYSICS LETTERS
To Illustrate the procedure described above, we have performed a calculation for the n + 160 system. For simphcity, the number o f terms in the fragment function ~b16 o f eq. (8) is limited t o N = 2. The parameters c! and a7 are chosen to yield the experimentally determined rms matter radius of 160 (2.6 fm) and a best over-all agreement with the form-factor data [11] for q2 up to about 7 fm - 2 . These parameters turn out to be c I = 1.0,
c 2 = 1514.0, (14)
~1 -- 0.24 fm - 2 ,
a 2 = 0.34 fm - 2 .
Except for the fact that the 160 cluster is described by the two-width-parameter function discussed above, the present calculation IS performed in the same manner as that described in a previous publication [I 2]. In particular, the central part o f the nucleon-nucleon potential in eq. (3) is given by eqs. ( 9 ) - ( 1 1 ) o f ref. [12]. This potential differs from the one used in TT in that it contains a weakly repulsive core and yields a satisfactory description of not only the low-energy nucleon-nucleon scattering data but also the essential properties of the deuteron, triton, and a particle. The exchange-mixture parameter u in this potential (see eq. (9) of ref. [12] ) is determined by fitting the neutron separation energy o f 3.26 MeV in the l = 0, j~r = 1 + first excited state of 170. The resultant value o f u is 0.945. Also, we have used the same spin-orbit potential as in ref. [12] ; this yields a satisfactory value for the energy splitting between the l = 2, 5/2 + ground and first 3/2 + excited states. Finally, for calculations at energies where reaction channels are open, we have incorporated the phenomenological imaginary potential given by eqs. (17) and (18) o f ref. [12]. The n + 160 differential cross section calculated at a c.m. energy o f 13.18 MeVis shown in fig. 1, where a comparison with experimental data [13] is also made. To obtain the calculated result, we have used an imaginary-potential strength I4)s = 4.7 MeV, the same as that used in ref. [12]. With this choice, the calculated reaction and integrated-elastic cross sections are, respectively, equal to 646 and 856 mb, which agree quite well with measured values. A comparison of fig. 1 with fig. 3b o f ref. [12] shows that both calculations employing, respectively, two-width-parameter and one-width-parameter 160 functions yield similarly good agreement with experi-
18 July 1977
mental results. This indicates that the function ~b16 can, in fact, be taken as a simple harmonic-oscdlator shell-model wave function characterized by a single properly-chosen width parameter and a more general fragment wave function is not needed in this case (see also the discussion given in ref. [14]). In vaew of the doubly-closed-shell nature of 160, such a finding is actually not too surprising. In conclusion, we have shown that it is mdeed possible to generahze the complex-generator-coordinate technique so that flexible fragment wave functions can be employed even in resonating-group calculations for rather heavy systems. This generahzatlon is quite straightforward, and we have illustrated using the n + 160 scattering problem as an example. Although it is found that in this particular example the generalization is not necessary, we do feel certain that for calculations involving non-closed-shell nuclei, this procedure will definitely be important. For instance, it is well-known that the nucleus 6Li is rather diffuse and cannot be adequately described by a simple harmonic-oscillator shell-model wave function. Thus, for a proper consideration of the inelastic process 6Li(p, p')6 Li* and the reaction 6 Li(p, 3 He)4 He, it will be necessary to adopt a flexible function for 6L1 and, consequently, utilize the complex-~nerator-coordlnate technique outlined in this note *~. ,2 It is clear of course that the techmque outhned here can also be used to handle reaction problems where all fragments involved are characterized by different width parameters.
References [1 ] Y.C. Tang and D.R. Thompson, m: Proc. 2nd Intern. Conf. on Clustering phenomena in nuclei, College Park, Maryland, 1975 (National Techmcal Information Service, Springfield, Vlrgima 22161) p. 119, and references contamed therein; K. Wfldermuth and Y C. Tang, A unified theory of the nucleus (Vleweg, Braunschweig, Germany, 1977). [2] D.M. Brink, in. Proc. Intern. Conf. on Clustering phenomena in nuclei, Bochum, Germany, 1969 (IAEA, Vienna, 1969) p. 147. [3] H. Friedrich, Nucl. Phys. A224 (1974) 537, and references contained thereto. [4] H. Friednch, K. Langanke and A. Welguny, Phys. Lett. 63B (1976) 125. [5] W. Sknkel and K. Wlldermuth, Phys. Lett. 41B (1972) 439
Volume 69B, number 1
PHYSICS LETTERS
[6] W. Sunkel, Phys. Lett. 65B (1976)419. [7] M. LeMere, Y.C. Tang and D.R. Thompson, Pliys. Rev. C14 (1976) 23. [8] D.R. Thompson, M. LeMere and Y.C. Tang, Nucl. Phys. A270 (1976) 211, M. LeMere, Y.C. Tang and D.R. Thompson, Phys. Lett. 63B (1976) 1,and Phys. Rev. C14 (1976) 1715. [9] D.R. Thompson and Y.C Tang, Phys. Rev. C12 (1975) 1432 and C13 (1976) 2597.
18 July 1977
[10] H. Honuchl, Progr. Theor. Phys. 47 (1972) 1058. [11] H Crannell, Phys Rev. 148 (1966)1107. [12] D.R. Thompson, M. LeMere and Y.C. Tang, Systematic investigation of scattering problems with the resonatinggroup method, wtll appear in Nucl. Phys. [13] R.W. Bauer, J.D. Anderson and LJ. Chnstensen, Nucl. Phys. 47 (1963) 241. [14] F.S. Chwleroth, Y.C. Tang and D.R. Thompson, Nucl. Phys. A189 (1972) 1.