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A method for analytically computing the resonating-group Kernel function and its application to α+6Li scattering
Volume 82B, number l
PHYSICS LETTERS
12 March 1979
A METHOD FOR ANALYTICALLY COMPUTING THE RESONATING-GROUP KERNEL FUNCTION AND ITS APPLICATION TO at + 6Li SCATTERING W. S[JNKEL
Institut )~r TheoretischePhysik, UniversitiitTi~bingen,Germany Received 7 November 1978
A method is proposed which reduces the computation of the analytical form of the kernel function in the resonatinggroup method (RGM) to the numerical evaluation of determinants. As an example, this method is applied to the case of a + 6 Li scattering.
In a quantitative description of nuclear structure and reaction, the use of a cluster ansatz for the wave function ~ in the projection equation (8~IH-E[$)
=0,
(1)
leads to a coupled set of integrodifferential equations for the relative-motion functions of the clusters [I ]. By choosing a superposition of harmonic-oscillator functions for the cluster internal functions and Gaussian form factors for the nucleon-nucleon potential, the integration over internal coordinates can be analytically carried out and, consequently, the kernels which appear in these equations become analytical functions of the relative coordinates between the clusters and have the general form
K(R', R") = ~ Pv(R', R") exp [Qv(R', R " ) I ,
(2)
where R' and R " represent collectively the Jacobi pat t t rameter-coordinate sets (R1, R 2 ..... R n-- 1 ) and (R'I', ¢! t! R2, ... Rn_l), with n being the number of clusters in the many-nucleon system. In the purpose of our later discussion, the important point to note here is that this analytical structure of K(R', R") has the property of being characterized by a finite number of constants which are the coefficients occuring in the polynomial factors Pv(R', R") and the quadratic functions G ( R ' , R"). Because o f the use of totally antisymmetric functions in RGM calculations, it requires generally con-
siderable effort to derive the kernel functions in the form of eq. (2), if the number of nucleons contained in the system is large. On the other hand, as has been discussed by some authors (see, e.g. ref. [2] ), the numerical evaluation of the kernels at discrete sets of parameter-coordinate points is a much simpler task. We must point out, however, that by such numerical procedure, the insight into the qualitative structure of K(R', R") is entirely lost and individual contributions of various nucleon-exchange terms, which correspond to different exponential factors in eq. (2), cannot be identified. This is, in fact, a rather severe disadvantage, because it has been shown [3] that an understanding o f the importance of different exchange terms can lead to the construction o f microscopically substantiated phenomenological models which will be useful in systematically analyzing the behaviour o f complicated nuclear systems involving large number o f nucleons. In this communication, we discuss a method to compute the kernel function in its analytical form by taking advantage of the above-mentioned simplicity in connection with its numerical evaluation. As will be seen, this method is very easy to apply, which means that there is now the possibility of extending the RGM to study in the future even rather h e a w systems described by the motion of a substantial number of nucleon clusters. Before we discuss this method, we should mention some essential points concerning the derivation of K(R', R"). As has been shown [3,4], an n-cluster wave 17
Volume 82B, number 1
PHYSICS LETTERS
function can be expressed as an integral of antisymmetrized products of single-particle wave functions by the introduction of n generator coordinates S1, S 2 .... ..., S n , collectively denoted as S. Then, one derives first in this generator-coordinate space a kernel function A'(S', S") which is simply related to K(R', R") by an integral transformation. This kernel function is given by a sum of determinants, the elements of which are integrals over single-particle functions of two-or less-particle operators. If one chooses the type of cluster internal functions and nucleon-nucleon potential as mentioned above, then these integrals can be calculated in closed form and are analytical functions of the form
M(S', S") =/~(S', S") exp [q(S', S")] ,
(3)
where/3 and ~ are polynomial and quadratic functions in S' and S", respectively. In the situation where a cluster contains more than four nucleons with spin and isospin saturation, the requirement of the Pauli principle demands that excited oscillator states must be occupied. Thus, the maximum degree of the polynomial/~ (S', S") is determined by the highest excitation of the nucleons in various clusters. In the kernel function • K ~ ( S ,' S") o f a N - n u c l e o n s y stem, we must therefore calculate the determinants of N × N matrices with elements of the type given by eq. (3) and obtain the following analytical form: / ~ ( S ' , S " ) = Z ) pv(S - " ' , S") exp [Qv(S ~ ' , S")] .
(4)
v
In other words, the determinants have to be sorted in powers of the generator coordinates S' and S", and in powers of the exponential factors exp [c~(S', S")]. Especially the sorting according to powers of the exponentials is important, because it is equivalent to the decomposition of the kernels in classes of nucleon-exchange between the clusters. As is obvious, this sorting procedure can, in general, be quite difficult, since in a system consisting of n clusters the number of different exponential factors e x p [ ~ ( S ' , S")] is n 2 and tile degree of the polynomial Pv may be quite high. On the other hand, it is well known that the numerical calculation of determinants for fixed values of S ' and S" is very simple and many fast standard computer codes do exist for this purpose. In our method, we make use of this simplicity associated with the numerical evaluation of determinants to obtain the co18
12 March 1979
efficients in the polynomial factors Pv and, thus, avoid the sorting process. Our method is best explained by means of a simple example. Consider an m-degree polynomial m
P(x) =
(5)
a.x", ~a=0
which depends on a single variable x. The coefficients au can be readily evaluated if we numerically calculate P(x) at the points [ 27ri
)
x ° = expl~--~- i p ,
(6)
with p = 0, 1, ..., m. Using the fact that the complex vectors (x~, x x1 ...... x ~x for different values of X are orthogonal to each other, one easily finds that m a~t-
m+l
=
The generalization of this formula for polynomials with more than one variable is straightforward. This means that, by making use of our procedure, the analytical structure of the kernel can be obtained by the numerical calculation of determinants. Mso, we should mention that, by making use of our method, the transformation f r o m / ~ ( S ' , S") to K(R', R") can be greatly facilitated and the determination of the polynomial factors Pv in eq. (2) becomes a rather simple procedure. As an example, we apply our method to the problem of a + 6 Li elastic scattering. In this ten-nucleon problem, we use a trial wave function which describes a three-cluster c~ + ~ + d configuration, i.e., l]/= ~[~Ctl~a2(~d Y(Rc~2-Rd)X(Rcq-RLi)] ,
(8)
where qSal, 4~a2, and Od represent, respectively, the internal wave functions of the incident tx duster, the c~ cluster in 6 Li, and the deuteron cluster. These are chosen to have the lowest configurations in harmonicoscillator well having a common width parameter of 0.35 fm - 2 . The function Y ( R a - R d ) describes the relative motion of the tx and deuteron clusters in 613; it is assumed to be
Y(Rc~2-Rd)= exp [-~fl(R~2-
Rd)2]
,
(9)
with 13taken as 0.25 fm -2 in order to yield approxi-
Volume 82B, number 1 7.49 ~ _ ~ _ -
,
PIIYSICS LETTERS .....
::..
-
. . . . . .
f ~
x192
Fig. 1. Comparison o f calculated and experimental a(O)/ORuth
(0) for c¢ + 6Li scattering at 99.6 MeV. Experimental data shown are those of Bachelier et al. [6]. mately the correct rms matter radius of 6 Li. The function X(Rc~1- R l i) for the relative motion between the particle and the 6Li nucleus is determined by solving the projection equation (1). For the n u c l e o n - n u c l e o n potential, we use that o f r e f . [5] with 88% Serber and 12% Rosenfeld mixture. This particular exchange mixture is chosen such that the experimental a-particle separation energy in the ground state o f 10B is approximately reproduced. Also, to account crudely for reaction effects, we have multiplied the direct nuclear potential by a factor (1 + i~), with the quantity ~ adjusted to yield a best overall agreement with the experimental differential cross-section result. At an ct + 6 Li c. m. energy of 99.6 MeV where a comparison between calculation and experiment is made, the value of ~ turns out to be equal to 0.5. In fig. 1, we show a comparison between calculated and experimental [6] values of o(0)/ORuth(0 ) at 99.6 MeV. As is seen, the agreement is quite satisfac-
12 March 1979
tory. In particular, the calculation correctly predicts the rapid rise in cross-section at backward angles. This indicates that core-exchange effects are important in this system, in agreement with the conclusion reached by other authors [7,8] from analyzing experimental data at lower energies. Our experience in the c~ + 6 ki calculation shows that the method presented here does yield a significant reduction in computational effort. Thus, it is indeed worthwhile to apply this method to more complicated systems involving a rather large number of nucleons. At present,we are examining the 160 + 20Ne system, treated in a three-cluster ~ + 160 + 160 configuration. The results should be very interesting since, with the analytical form of the kernel function known, we will be able to learn from this calculation the importance of core-exchange effects and the contribution of exchange terms involving nucleons in all three clusters to the binding energies in various states of this system. The author is greatly indebted to Professor K. Wildermuth for valuable advice and continuous encouragement and thanks Professor Y.C. Tang for enlightening discussions and rewriting the English version of the manuscript.
References [1] K. Wildermuth and Y.C. Tang, A unified theory of the nucleus (Vieweg, Braunschweig, 1977). [2] H. Friedrich, Nucl. Phys. A224 (1974) 537. [31 Y.C. Tang, M. LeMere, and D.R. Thompson, Phys. Reports 47, no. 3 (1978). [4] W. Siinkel, Habilitationsschrift, Universit~t Tiibingen (1976). [5] E. Schmid and K. Wildermuth, Nucl. Phys. 26 (1961) 463. [6] D. Bachelier et al., Nucl. Phys. A195 (1972) 361. [7] H. Bohlen et al., Nucl. Phys. A179 (1972) 504. [8] D. Clement, E.J. Kanellopoulos and K. Wildermuth, Acta Phys. Austr. 42 (1975) 29.
19
PHYSICS LETTERS
12 March 1979
A METHOD FOR ANALYTICALLY COMPUTING THE RESONATING-GROUP KERNEL FUNCTION AND ITS APPLICATION TO at + 6Li SCATTERING W. S[JNKEL
Institut )~r TheoretischePhysik, UniversitiitTi~bingen,Germany Received 7 November 1978
A method is proposed which reduces the computation of the analytical form of the kernel function in the resonatinggroup method (RGM) to the numerical evaluation of determinants. As an example, this method is applied to the case of a + 6 Li scattering.
In a quantitative description of nuclear structure and reaction, the use of a cluster ansatz for the wave function ~ in the projection equation (8~IH-E[$)
=0,
(1)
leads to a coupled set of integrodifferential equations for the relative-motion functions of the clusters [I ]. By choosing a superposition of harmonic-oscillator functions for the cluster internal functions and Gaussian form factors for the nucleon-nucleon potential, the integration over internal coordinates can be analytically carried out and, consequently, the kernels which appear in these equations become analytical functions of the relative coordinates between the clusters and have the general form
K(R', R") = ~ Pv(R', R") exp [Qv(R', R " ) I ,
(2)
where R' and R " represent collectively the Jacobi pat t t rameter-coordinate sets (R1, R 2 ..... R n-- 1 ) and (R'I', ¢! t! R2, ... Rn_l), with n being the number of clusters in the many-nucleon system. In the purpose of our later discussion, the important point to note here is that this analytical structure of K(R', R") has the property of being characterized by a finite number of constants which are the coefficients occuring in the polynomial factors Pv(R', R") and the quadratic functions G ( R ' , R"). Because o f the use of totally antisymmetric functions in RGM calculations, it requires generally con-
siderable effort to derive the kernel functions in the form of eq. (2), if the number of nucleons contained in the system is large. On the other hand, as has been discussed by some authors (see, e.g. ref. [2] ), the numerical evaluation of the kernels at discrete sets of parameter-coordinate points is a much simpler task. We must point out, however, that by such numerical procedure, the insight into the qualitative structure of K(R', R") is entirely lost and individual contributions of various nucleon-exchange terms, which correspond to different exponential factors in eq. (2), cannot be identified. This is, in fact, a rather severe disadvantage, because it has been shown [3] that an understanding o f the importance of different exchange terms can lead to the construction o f microscopically substantiated phenomenological models which will be useful in systematically analyzing the behaviour o f complicated nuclear systems involving large number o f nucleons. In this communication, we discuss a method to compute the kernel function in its analytical form by taking advantage of the above-mentioned simplicity in connection with its numerical evaluation. As will be seen, this method is very easy to apply, which means that there is now the possibility of extending the RGM to study in the future even rather h e a w systems described by the motion of a substantial number of nucleon clusters. Before we discuss this method, we should mention some essential points concerning the derivation of K(R', R"). As has been shown [3,4], an n-cluster wave 17
Volume 82B, number 1
PHYSICS LETTERS
function can be expressed as an integral of antisymmetrized products of single-particle wave functions by the introduction of n generator coordinates S1, S 2 .... ..., S n , collectively denoted as S. Then, one derives first in this generator-coordinate space a kernel function A'(S', S") which is simply related to K(R', R") by an integral transformation. This kernel function is given by a sum of determinants, the elements of which are integrals over single-particle functions of two-or less-particle operators. If one chooses the type of cluster internal functions and nucleon-nucleon potential as mentioned above, then these integrals can be calculated in closed form and are analytical functions of the form
M(S', S") =/~(S', S") exp [q(S', S")] ,
(3)
where/3 and ~ are polynomial and quadratic functions in S' and S", respectively. In the situation where a cluster contains more than four nucleons with spin and isospin saturation, the requirement of the Pauli principle demands that excited oscillator states must be occupied. Thus, the maximum degree of the polynomial/~ (S', S") is determined by the highest excitation of the nucleons in various clusters. In the kernel function • K ~ ( S ,' S") o f a N - n u c l e o n s y stem, we must therefore calculate the determinants of N × N matrices with elements of the type given by eq. (3) and obtain the following analytical form: / ~ ( S ' , S " ) = Z ) pv(S - " ' , S") exp [Qv(S ~ ' , S")] .
(4)
v
In other words, the determinants have to be sorted in powers of the generator coordinates S' and S", and in powers of the exponential factors exp [c~(S', S")]. Especially the sorting according to powers of the exponentials is important, because it is equivalent to the decomposition of the kernels in classes of nucleon-exchange between the clusters. As is obvious, this sorting procedure can, in general, be quite difficult, since in a system consisting of n clusters the number of different exponential factors e x p [ ~ ( S ' , S")] is n 2 and tile degree of the polynomial Pv may be quite high. On the other hand, it is well known that the numerical calculation of determinants for fixed values of S ' and S" is very simple and many fast standard computer codes do exist for this purpose. In our method, we make use of this simplicity associated with the numerical evaluation of determinants to obtain the co18
12 March 1979
efficients in the polynomial factors Pv and, thus, avoid the sorting process. Our method is best explained by means of a simple example. Consider an m-degree polynomial m
P(x) =
(5)
a.x", ~a=0
which depends on a single variable x. The coefficients au can be readily evaluated if we numerically calculate P(x) at the points [ 27ri
)
x ° = expl~--~- i p ,
(6)
with p = 0, 1, ..., m. Using the fact that the complex vectors (x~, x x1 ...... x ~x for different values of X are orthogonal to each other, one easily finds that m a~t-
m+l
=
The generalization of this formula for polynomials with more than one variable is straightforward. This means that, by making use of our procedure, the analytical structure of the kernel can be obtained by the numerical calculation of determinants. Mso, we should mention that, by making use of our method, the transformation f r o m / ~ ( S ' , S") to K(R', R") can be greatly facilitated and the determination of the polynomial factors Pv in eq. (2) becomes a rather simple procedure. As an example, we apply our method to the problem of a + 6 Li elastic scattering. In this ten-nucleon problem, we use a trial wave function which describes a three-cluster c~ + ~ + d configuration, i.e., l]/= ~[~Ctl~a2(~d Y(Rc~2-Rd)X(Rcq-RLi)] ,
(8)
where qSal, 4~a2, and Od represent, respectively, the internal wave functions of the incident tx duster, the c~ cluster in 6 Li, and the deuteron cluster. These are chosen to have the lowest configurations in harmonicoscillator well having a common width parameter of 0.35 fm - 2 . The function Y ( R a - R d ) describes the relative motion of the tx and deuteron clusters in 613; it is assumed to be
Y(Rc~2-Rd)= exp [-~fl(R~2-
Rd)2]
,
(9)
with 13taken as 0.25 fm -2 in order to yield approxi-
Volume 82B, number 1 7.49 ~ _ ~ _ -
,
PIIYSICS LETTERS .....
::..
-
. . . . . .
f ~
x192
Fig. 1. Comparison o f calculated and experimental a(O)/ORuth
(0) for c¢ + 6Li scattering at 99.6 MeV. Experimental data shown are those of Bachelier et al. [6]. mately the correct rms matter radius of 6 Li. The function X(Rc~1- R l i) for the relative motion between the particle and the 6Li nucleus is determined by solving the projection equation (1). For the n u c l e o n - n u c l e o n potential, we use that o f r e f . [5] with 88% Serber and 12% Rosenfeld mixture. This particular exchange mixture is chosen such that the experimental a-particle separation energy in the ground state o f 10B is approximately reproduced. Also, to account crudely for reaction effects, we have multiplied the direct nuclear potential by a factor (1 + i~), with the quantity ~ adjusted to yield a best overall agreement with the experimental differential cross-section result. At an ct + 6 Li c. m. energy of 99.6 MeV where a comparison between calculation and experiment is made, the value of ~ turns out to be equal to 0.5. In fig. 1, we show a comparison between calculated and experimental [6] values of o(0)/ORuth(0 ) at 99.6 MeV. As is seen, the agreement is quite satisfac-
12 March 1979
tory. In particular, the calculation correctly predicts the rapid rise in cross-section at backward angles. This indicates that core-exchange effects are important in this system, in agreement with the conclusion reached by other authors [7,8] from analyzing experimental data at lower energies. Our experience in the c~ + 6 ki calculation shows that the method presented here does yield a significant reduction in computational effort. Thus, it is indeed worthwhile to apply this method to more complicated systems involving a rather large number of nucleons. At present,we are examining the 160 + 20Ne system, treated in a three-cluster ~ + 160 + 160 configuration. The results should be very interesting since, with the analytical form of the kernel function known, we will be able to learn from this calculation the importance of core-exchange effects and the contribution of exchange terms involving nucleons in all three clusters to the binding energies in various states of this system. The author is greatly indebted to Professor K. Wildermuth for valuable advice and continuous encouragement and thanks Professor Y.C. Tang for enlightening discussions and rewriting the English version of the manuscript.
References [1] K. Wildermuth and Y.C. Tang, A unified theory of the nucleus (Vieweg, Braunschweig, 1977). [2] H. Friedrich, Nucl. Phys. A224 (1974) 537. [31 Y.C. Tang, M. LeMere, and D.R. Thompson, Phys. Reports 47, no. 3 (1978). [4] W. Siinkel, Habilitationsschrift, Universit~t Tiibingen (1976). [5] E. Schmid and K. Wildermuth, Nucl. Phys. 26 (1961) 463. [6] D. Bachelier et al., Nucl. Phys. A195 (1972) 361. [7] H. Bohlen et al., Nucl. Phys. A179 (1972) 504. [8] D. Clement, E.J. Kanellopoulos and K. Wildermuth, Acta Phys. Austr. 42 (1975) 29.
19