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An asymptotic expression for the eigenvalues of the normalization Kernel of the resonating group method
Volume 65B, number 2
PHYSICS LETTERS
8 November 1976
AN ASYMPTOTIC EXPRESSION FOR THE EIGENVALUES OF THE NORMALIZATION KERNEL OF THE RESONATING GROUP METHOD J. LOMNITZ-ADLER and D.M. BRINK
Department of Theoretical Physics, Oxford University, Oxford, England Received 15 June 1976 A generating function for the eigenvalues of the RGM Normalization Kernel is expressed in terms of the diagonal matrix elements of the GCM Overlap Kernel. An asymptotic expression for the eigenvalues is obtained by using the Method of Steepest Descent.
The Resonating Group Method (RGM) has been used extensively to describe the scattering of light nuclei, and recently both the RGM and the Generator Coordinate Method (GCM) have been applied to scattering problems involving heavier nuclei (Friedrich [1], Kamimura and Matsuse [2], Beck et al. [5] and Canto (to be published)). The RGM and GCM form a satisfactory starting point for the development of microscopic theories of interactions between heavy nuclei because they include explicitly the effects of antisymmetry of the wave functions. On the other hand they are laborious to use because of the complexity of the Hamiltonian and Overlap Kernels. It is therefore interesting to look for simple asymptotic approximations which are valid for pairs of heavy nuclei. If we describe internal wave functions of the colliding nuclei by Slater Determinants of Harmonic Oscillator wave functions with the same oscillator parameter co, there exists a simple procedure to obtain the Overlap Kernel in the GCM from the Normalization kernel in the RGM [1,2, 5] g l ( a , p) = j r ( ~ , x ) A r(a,
(x, x r) r + ( x ,, p) d 3 x d3x '
x) = (rr/23,) -3/4
exp [-7(ec - x) 2 ]
(1)
where A, B are the mass numbers of target and projectile nuclei, m is the nucleon mass, and where
106
(3)
with M related to the antisymmetrization operator by a constant, ~bint are the internal wave functions of the clusters, and IqS~) is the antisymmetrized eigenfunction o f A + B nucleons in two equal Harmonic Oscillator wells separated by a distance ll. To obtain A (x, x') from I ( a , I~), however is more of a problem, since to do so one needs to invert the operator P, a procedure which has numerical difficulties [1]. This is important because, while the GCM Overlap Kernel is easier to calculate than the RGM Normalization kernel, the RGM wave function is the easier one to interpret. It is known that A(x, x') has the Harmonic Oscillator wave functions as eigenfunctions [ 2 - 4 ] ,
Xiyk(r) = BiBjB k exp [--'yr2 ] Hi(x~-2-7r 1) X H / ( x / ~ r 2 ) H k ( x / ~ r3) with
(4)
B i = (2-},/rr)1/4(2 ii!)-l/2, and where rl, r 2, r 3 are the rectangular coordinates of r, and H n is the nth order Hermite polynomial. These functions satisfy
f Xijk(r)A( r, r')xi'j'k'(r') d3r d3r ' =l.tN6ii,~)jj'~kk', (5)
3' = (2AB/A + B) mw/l~,
A ( x , x ' ) = (6(x -r)q~intl _q{{6(x' - r)~bint}),
I(0t, p) = (qba[ qb[i),
(2)
and the eigenvalue/.l N depends only on the principal quantum number N = i +j + k [4]. Using (4) and (5), and the generating function for Hermite Polynomials
Volume 65B, number 2
PHYSICS LETTERS
8 November 1976
oa
~D eq
Hn(x)tn/n! = exp[-t 2 + 2xt].
tea
(6)
n=0
vO
ee~
O'x
o
© ©
oo
c5
,o
~i
Horiuchi and Suzuki have found a generating function for the eigenvalues/l N o f A ( x , x') [4]. We follow a very similar procedure which gives a generating function for/a N in terms of the diagonal of the GCM Overlap l(at, ,,). F r o m eq. (5) we have
VtD ee~
I © ,ga
eq
eq
eq
t--
~,.2i~.2j,.Zk,, /;';'k! =f ~[(i'i"/'/'k'k"} ijk ~'1 ~2 "3 t~N/"J . . . . . . .
1/2 (7)
o
oo
×
oo e~
o o~
eq oo O
o
"2, o,
and using eqs. (4), (6) the right hand side of (7) is
~(UN/i!j!k!)x2ixZ]x2k=f
g, e-I
(Tr/zv)-3/2 { e x p [ - x
i/k + 2V/Tx'(r +r') -- 3,(r2 +r'2)]
~,o
kO
tg)
tt3
a.
~ lau(xZ)N/N ! = exZI(x/x/~, x/x/~).
o e,I eo
(8t
ee~
(9)
o
e~
e-I t'-I
A(r, r')} d3r d3r ',
noting (1), (2) this becomes
o
E
2
This result is similar to that obtained by Kamimura and Matsuse [2], who make this expansion implicitly when they pick out the Nth eigenvalue ofA(r, r'). We now proceed to show a way of approximating the eigenvalues/JN" From (9) we have
o
',O
.a
o
e-l t--
o
ell
a.I
/.tu = (d/dx2) N cD
©
o o
e~
t=
o'~
o
~L
o
m~ O ¢--) , CD
i tt')
eq
i
o c~ o
a4
(10)
because I is spherically symmetric its dependence is only on z -= x 2, and for convenience we define ID(Z/7 ) =I(x/x/7, x/x~7), from expression (9) and with the knowledge that/a N -+ 1 as N goes to infinity we see that ID(Z/7) is analytic in the complete z-plane, so that
eZ,o z,
O C~
O ¢O
(eX2[(x/x/~, x/x/~)}
(eZlD(Z/7)}=~gia
zN+I
11,
where the integral is taken along any closed loop which goes around z = 0. This integral can be approximated by the Method of Steepest Descents: If we have an integral of the form
S = feW/(Z)g(z) dz,
(12)
:<
107
Volume 65B, number 2
PttYSICS LETTERS
w e can doform the path o f integration so that it passes through a saddle point o f f ( z ) and so that Re f(z) has a relative maximum while Im f(z) remains constant. If W is large and real the main part of the integral will come from this point, which is found with the condition
df/dzlz=zo = 0.
(13)
Using the Method of Steepest Descent to zero order, with
(X + 1)f(zo) = z 0 - (N + 1) In z0; (14) z 0 = N + 1;
g(z) =Io(z/7 ),
lations by Suzuki for a - 1 6 0 (table 1) and 1 6 0 - 1 6 0 (table 2). As we expected, the agreement is better for larger N, and is also better for 1 6 0 - 1 6 0 than a - 1 6 0 . Following the methods used to obtain eq. (9) we can also show that 1 2 + 112)], I(ot, 11) = E/aN/N! (Tot.l~)Nexp [-- 27(0t
N (18) so that we can construct the full overlap kernel with only knowing it's diagonal
I(~t, 1~) "" ~ ID(N + 1/3")((3"~.fJ)N/m !) N
(19)
× exp [ - 717 ( ~ 2 +112)].
eq. (11) becomes N! /aN ~ (N + 1)N (2~r(N + 1))- 1/2 exp IN + 1 ] g(N + 1).
(15) If we use Stirlings' approximation N! ~ (27rN) 1/2 exp [ - N ] N N
8 November 1976
(16)
The authors wish to express their thanks to Dr. N. Takigawa for his corrections to the manuscript as well as for his valuable discussions. This work was supported in part by the Consejo Nacional de Ciencia y Tecnologia de Mexico.
we finally obtain /aN ~ I o ((N + 1)/3') = I ( X / ~ + 1)/3', x/(N + 1)/3'). (17) Because the Method of Steepest Descents is good for large W (= N + 1), and because we have used Stirling's approximation, we expect (17) to be a better approximation for larger N than smaller ones. This leads us to believe that this method gets better as the colliding nuclei get heavier, since in this case there is a large number of redundant states (/aN = 0) which are known in other ways, so that the/a N of interest will have large N. This does seem to be the case when we compare our approximate results with the exact calcu-
108
References
[1] H. Friedrich, Nucl. Phys. A244 (1974) 537. [2] M. Kamimura and T. Matsuse, Prog. Theor. Phys. 51 (1974) 438; Prog. Theor. Phys. 47 (1973) 1765. [3] H. Horiuchi, Prog. Theor. Phys. 47 (1972) 1058; Prog. Theor. Phys. 50 (1973) 529. [4] Y. Suzuki, Prog. Theor. Phys. 50 (1973) 1302. H. Horiuchi and Y. Suzuki, Prog. Theor. Phys. 49 (1973) 1974. [5] R. Beck, J. Borysowicz, D.M. Brink and M.V. Mihailovid, Nucl. Phys. A244 (1975) 58; Nucl. Phys. A244 (1975) 45.
PHYSICS LETTERS
8 November 1976
AN ASYMPTOTIC EXPRESSION FOR THE EIGENVALUES OF THE NORMALIZATION KERNEL OF THE RESONATING GROUP METHOD J. LOMNITZ-ADLER and D.M. BRINK
Department of Theoretical Physics, Oxford University, Oxford, England Received 15 June 1976 A generating function for the eigenvalues of the RGM Normalization Kernel is expressed in terms of the diagonal matrix elements of the GCM Overlap Kernel. An asymptotic expression for the eigenvalues is obtained by using the Method of Steepest Descent.
The Resonating Group Method (RGM) has been used extensively to describe the scattering of light nuclei, and recently both the RGM and the Generator Coordinate Method (GCM) have been applied to scattering problems involving heavier nuclei (Friedrich [1], Kamimura and Matsuse [2], Beck et al. [5] and Canto (to be published)). The RGM and GCM form a satisfactory starting point for the development of microscopic theories of interactions between heavy nuclei because they include explicitly the effects of antisymmetry of the wave functions. On the other hand they are laborious to use because of the complexity of the Hamiltonian and Overlap Kernels. It is therefore interesting to look for simple asymptotic approximations which are valid for pairs of heavy nuclei. If we describe internal wave functions of the colliding nuclei by Slater Determinants of Harmonic Oscillator wave functions with the same oscillator parameter co, there exists a simple procedure to obtain the Overlap Kernel in the GCM from the Normalization kernel in the RGM [1,2, 5] g l ( a , p) = j r ( ~ , x ) A r(a,
(x, x r) r + ( x ,, p) d 3 x d3x '
x) = (rr/23,) -3/4
exp [-7(ec - x) 2 ]
(1)
where A, B are the mass numbers of target and projectile nuclei, m is the nucleon mass, and where
106
(3)
with M related to the antisymmetrization operator by a constant, ~bint are the internal wave functions of the clusters, and IqS~) is the antisymmetrized eigenfunction o f A + B nucleons in two equal Harmonic Oscillator wells separated by a distance ll. To obtain A (x, x') from I ( a , I~), however is more of a problem, since to do so one needs to invert the operator P, a procedure which has numerical difficulties [1]. This is important because, while the GCM Overlap Kernel is easier to calculate than the RGM Normalization kernel, the RGM wave function is the easier one to interpret. It is known that A(x, x') has the Harmonic Oscillator wave functions as eigenfunctions [ 2 - 4 ] ,
Xiyk(r) = BiBjB k exp [--'yr2 ] Hi(x~-2-7r 1) X H / ( x / ~ r 2 ) H k ( x / ~ r3) with
(4)
B i = (2-},/rr)1/4(2 ii!)-l/2, and where rl, r 2, r 3 are the rectangular coordinates of r, and H n is the nth order Hermite polynomial. These functions satisfy
f Xijk(r)A( r, r')xi'j'k'(r') d3r d3r ' =l.tN6ii,~)jj'~kk', (5)
3' = (2AB/A + B) mw/l~,
A ( x , x ' ) = (6(x -r)q~intl _q{{6(x' - r)~bint}),
I(0t, p) = (qba[ qb[i),
(2)
and the eigenvalue/.l N depends only on the principal quantum number N = i +j + k [4]. Using (4) and (5), and the generating function for Hermite Polynomials
Volume 65B, number 2
PHYSICS LETTERS
8 November 1976
oa
~D eq
Hn(x)tn/n! = exp[-t 2 + 2xt].
tea
(6)
n=0
vO
ee~
O'x
o
© ©
oo
c5
,o
~i
Horiuchi and Suzuki have found a generating function for the eigenvalues/l N o f A ( x , x') [4]. We follow a very similar procedure which gives a generating function for/a N in terms of the diagonal of the GCM Overlap l(at, ,,). F r o m eq. (5) we have
VtD ee~
I © ,ga
eq
eq
eq
t--
~,.2i~.2j,.Zk,, /;';'k! =f ~[(i'i"/'/'k'k"} ijk ~'1 ~2 "3 t~N/"J . . . . . . .
1/2 (7)
o
oo
×
oo e~
o o~
eq oo O
o
"2, o,
and using eqs. (4), (6) the right hand side of (7) is
~(UN/i!j!k!)x2ixZ]x2k=f
g, e-I
(Tr/zv)-3/2 { e x p [ - x
i/k + 2V/Tx'(r +r') -- 3,(r2 +r'2)]
~,o
kO
tg)
tt3
a.
~ lau(xZ)N/N ! = exZI(x/x/~, x/x/~).
o e,I eo
(8t
ee~
(9)
o
e~
e-I t'-I
A(r, r')} d3r d3r ',
noting (1), (2) this becomes
o
E
2
This result is similar to that obtained by Kamimura and Matsuse [2], who make this expansion implicitly when they pick out the Nth eigenvalue ofA(r, r'). We now proceed to show a way of approximating the eigenvalues/JN" From (9) we have
o
',O
.a
o
e-l t--
o
ell
a.I
/.tu = (d/dx2) N cD
©
o o
e~
t=
o'~
o
~L
o
m~ O ¢--) , CD
i tt')
eq
i
o c~ o
a4
(10)
because I is spherically symmetric its dependence is only on z -= x 2, and for convenience we define ID(Z/7 ) =I(x/x/7, x/x~7), from expression (9) and with the knowledge that/a N -+ 1 as N goes to infinity we see that ID(Z/7) is analytic in the complete z-plane, so that
eZ,o z,
O C~
O ¢O
(eX2[(x/x/~, x/x/~)}
(eZlD(Z/7)}=~gia
zN+I
11,
where the integral is taken along any closed loop which goes around z = 0. This integral can be approximated by the Method of Steepest Descents: If we have an integral of the form
S = feW/(Z)g(z) dz,
(12)
:<
107
Volume 65B, number 2
PttYSICS LETTERS
w e can doform the path o f integration so that it passes through a saddle point o f f ( z ) and so that Re f(z) has a relative maximum while Im f(z) remains constant. If W is large and real the main part of the integral will come from this point, which is found with the condition
df/dzlz=zo = 0.
(13)
Using the Method of Steepest Descent to zero order, with
(X + 1)f(zo) = z 0 - (N + 1) In z0; (14) z 0 = N + 1;
g(z) =Io(z/7 ),
lations by Suzuki for a - 1 6 0 (table 1) and 1 6 0 - 1 6 0 (table 2). As we expected, the agreement is better for larger N, and is also better for 1 6 0 - 1 6 0 than a - 1 6 0 . Following the methods used to obtain eq. (9) we can also show that 1 2 + 112)], I(ot, 11) = E/aN/N! (Tot.l~)Nexp [-- 27(0t
N (18) so that we can construct the full overlap kernel with only knowing it's diagonal
I(~t, 1~) "" ~ ID(N + 1/3")((3"~.fJ)N/m !) N
(19)
× exp [ - 717 ( ~ 2 +112)].
eq. (11) becomes N! /aN ~ (N + 1)N (2~r(N + 1))- 1/2 exp IN + 1 ] g(N + 1).
(15) If we use Stirlings' approximation N! ~ (27rN) 1/2 exp [ - N ] N N
8 November 1976
(16)
The authors wish to express their thanks to Dr. N. Takigawa for his corrections to the manuscript as well as for his valuable discussions. This work was supported in part by the Consejo Nacional de Ciencia y Tecnologia de Mexico.
we finally obtain /aN ~ I o ((N + 1)/3') = I ( X / ~ + 1)/3', x/(N + 1)/3'). (17) Because the Method of Steepest Descents is good for large W (= N + 1), and because we have used Stirling's approximation, we expect (17) to be a better approximation for larger N than smaller ones. This leads us to believe that this method gets better as the colliding nuclei get heavier, since in this case there is a large number of redundant states (/aN = 0) which are known in other ways, so that the/a N of interest will have large N. This does seem to be the case when we compare our approximate results with the exact calcu-
108
References
[1] H. Friedrich, Nucl. Phys. A244 (1974) 537. [2] M. Kamimura and T. Matsuse, Prog. Theor. Phys. 51 (1974) 438; Prog. Theor. Phys. 47 (1973) 1765. [3] H. Horiuchi, Prog. Theor. Phys. 47 (1972) 1058; Prog. Theor. Phys. 50 (1973) 529. [4] Y. Suzuki, Prog. Theor. Phys. 50 (1973) 1302. H. Horiuchi and Y. Suzuki, Prog. Theor. Phys. 49 (1973) 1974. [5] R. Beck, J. Borysowicz, D.M. Brink and M.V. Mihailovid, Nucl. Phys. A244 (1975) 58; Nucl. Phys. A244 (1975) 45.