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A simple theoretical description of coulomb polarization effects on the coulomb modified scattering length
Volume 182, number
PHYSICS
2
LETTERS
B
18 December
A SIMPLE THEORETICAL DESCRIPTION OF COULOMB ON THE COULOMB MODIFIED SCATTERING LENGTH Gy. BENCZE
POLARIZATION
1986
EFFECTS
’ and C. CHANDLER
Department qf Physics and Astronomy, Umversity qf New Mexico. Albuquerque, NM 87131, LISA Received
23 July 1986
A simple formula is derived for the Coulomb modified s-wave scattering length function in the presence of long-range polarization potentials. When applied to a model of low-energy p-d scattering, the formula reproduces behavior previously described by Kvitsinskii and Merkuriev [Sov. J. Nucl. Phys. 41 (1985) 4121.
For some years it has been known that, in the presence of long-range polarization forces, the standard Coulomb-modified scattering length is infinite if the Coulomb interaction is repulsive [ l-31. Since this is so one must ask how to interpret the finite values that are routinely obtained in both experimental and theoretical studies. For low-energy p-d scattering this question was addressed by Kvitsinskii and Merkuriev [ 21. In their paper an important role was played by a certain approximation (cf. eq. (9) below) which, for energies of experimental interest, gives the wrong sign and the wrong absolute value (by a factor of 2-8). The discovery,that this is so prompted this paper, in which we reexamine and generalize the arguments of Kvitsinskii and Merkuriev and present results of numerical experiments. Our main results are a simple formula (eq. (7) below) for the Coulomb modified s-wave scattering length function in the presence of long-range potentials and, in the case of p-d scattering, a conlirmation of the general conclusions of Kvitsinskii and Merkuriev [ 21. We assume that we are dealing with an effective two-body problem, and we ignore spin (for simplicity). We denote the reduced mass of the system by p and the energy by E. The effective potential V(r) has the form ’ Permanent
adress: Central Research P.O.B. 49, H-1525 Budapest, Hungary.
0370-2693/86/$ (North-Holland
Institute
V(r) =h+
(1)
)
where 11>0 and satisfies
V(v)
is a central
potential
that
CQ s
drrlU(r)I
(2a)
0
and sup r’+’ O
IWr)I
(2b)
for some E> 0. We define k= (2,uE) ‘i2/A and denote by 6 the s-wave scattering phase shift (relative to the pure Coulomb phase shift) produced by U(r) . The usual Coulomb modified scattering length function A(k) is defined [ I] by A(k) = - [KC;(k)cot
d+lh(k)]
-’ .
(3)
Here C,(k)=(n1lk)[exp(nllk)-11-r, h(k) =Re y(iX2k)
-ln(A/2k)
,
where I+Ydenotes the digamma function [4]. When U(r) falls off exponentially as r+co, then the zero energy limit of A(k) is the usual scattering length. When V(r) has a long-range attractive tail that falls off only as some inverse power of r as r--rcq then [l-3,5] the zero energy limit ofA is --CO. To isolate the effects of the long-range tail of the
for Physics,
03.50 0 Elsevier Science Publishers Physics Publishing Division)
U(r)
B.V.
121
Volume 182, number 2
potential,
PHYSICS
we define a “polarization”
V,(r) =O, =U(r),
potential
LETTERS
R
.
b(x)=3~x*(l +2x21m
(5)
(6)
’
1+As(kVG(klf(k)
(7)
>’
We emphasize that the formula given in (11) is quite general and does not depend on any specific asymptotic behavior of the potential P’,(r). We infer that Kvitsinskii and Merkuriev [ 21 used the following (in our notation):
122
1- Cf,,/AskG)l >
results for the potential
(8)
R
of eq. (10) with R = 5 fm. A,= 5 fm for the calculation
e(keV)
&
1 3 5 10 25 40 100 400 800 1000
0.4815x 10-9 0.8743x IO-* 0.4130x 1om7 0.4005x 10-b 0.4750x 10-5 0.1208x 1O-4 0.4422x 1O-4 0.1268x lo-) 0.1553x10-3 0.1577x 10-J
t9b)
(10)
where A=O.O4298 fin-’ and a,=0.632 fm3, values appropriate to the p-d system. We used both R==5 fm and R = 50 fm in our calculations. Over the energy range l- 1000 keV an “exact” value fE( k) of the function f(k) was calculated in by using the IMSL program DGEAR to solve the variable phase equations [ 61. In addition, the first Born approximation fs( k) and the Kvitsinskii-Merkuriev approximation fKM(k) were calculated. As is illustrated by the selected results reproduced in table 1, the first Born approximation reproduces the exact value to within 0.01% over the energy range l-l 000 keV. The approximation (9) changes to the wrong sign at about 30 keV and is in absolute value wrong by factors of 2-8 over the range of experimental interest (400- 1000 keV [ 71). We calculated the function B(k) for various scattering lengths a and for various methods of comput-
where the notation f(k) = tan 6,,, has been introduced. All of the low-energy effects of the polarization potential V,(r) are now isolated in tan 6,,,. Since we are only interested in the limit of zero energy, where h(k) vanishes, we set h(k) equal to zero. The result, which we denote by B(k), is
l- If(k)4(kW’3k)l
nx)+2x
I&( 1 +ix) - 1/3x.
V,(r)=da,2r“,
kC$A, - (1 +;IhA,)f kC~+[k*C~A,+~h(l+#zA)fl
Table 1 Selected numerical
-coth
Eq. (9)) which follows from the work of Berger and Spruch [ 11, represents an approximation to the Born approximation for the potential given in eq. (10) below. In order to test the formula (7) quantitatively, we used the polarization potential VP(r) consistent with that used by Kvitsinskii and Merkuriev [ 21,
Combining (3) and (5) with the trigonometric identity for the cotangent of the sum of two angles yields
B(K) -A,(k)t
(9a)
and
A,(k)=-[kC~(k)tan6,,,~+~h(k)]-’
B(k) =4(k)
>
&M(k) = (&/2)b(J./2k)
(4)
where the radius R is arbitrary but fixed. By itself the potential VPwould produce a phase shift a,., relative to the pure Coulomb phase shift. The remainder Bcp,s=6 -6,,, of the phase shift 6 is the additional phase shift produced by the short-range potential U(r) - V,(r). Associated with 6,,, is another scattering length function A,(k) defined by
A(k)=
1986
where
P’,(r)
O
I8 December
B
1.0000 1.oooo
1.oooo 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999
1.oooo
1.002 0.9679 0.8079 0.4274 0.0333 -0.1375 -0.5463 -2.321 -5.573 - 7.709
of J3.
_ 113 0.959 0.993 0.998 0.999 0.999 0.999 0.999 1.000
1.000
Volume 182, number
PHYSICS
2
I
-10.0.’
’
’
’ I"II'
ENERGY
IO
i 100
IKEV)
Fig. 1. The function B/A, for f=fe (for the potential of eq. (10) scattering length and A,=4.0 fm (--)(doublet scattering length [ 7]), 11.I fm (------)(quartet scattering length [7]).
ing the function f( k). Those in table 1, for example were calculated usingf,( k) and A,(k) = 4 fm, which is the experimental value of the doublet scattering length [ 71. To an accuracy of 0.2% the values of B( k) are independent of the method used to calculatef( k), the radius R used, which of the eqs. (7) or (8) was used or whether we included the effective range in the low-energy expansion for A,(k). When we restored the function h(k) to the formulas, some new structure appeared in B(k) below 1 keV but there were no essential changes above that energy. At low energies ( 1- 10 keV) there is some dependence on the value of the scattering length, which is illustrated in fig. 1. We conclude that for p-d scattering eq. (9) represents a poor approximation for energies of experimental interest.
LETTERS
B
18 December
1986
Nevertheless, the essential conclusions of Kvitsinskii and Merkuriev [ 2 ] remain valid. The effects of the long-range polarization potential are found only at extremely low energies and by 10 keV the scattering length function B(k) has reached a constant plateau. This plateau covers the range of experimental interest and thus allows a stable extrapolation of experimental results to zero energy. The result of the extrapolation is precisely the long-range modified scattering length a. This general behavior is insensitive to the value of the radius R, to the value longrange modified scattering length a and to the method of calculatingf( k). We believe, on the basis of numerical experiments, that the function B(k) exhibits similar properties for more general long-range potentials. The authors are indebted to the United States National Science Foundation and the Hungarian Academy of Sciences for supporting a cooperative research program which made this work possible. References [ I] R.O. Berger and L. Spruch, Phys. Rev. 138 (1965) B1106. [ 21 A.A. Kvitsinskii and S.P. Mercuriev, Yad. Fiz. 41 (1985) 647 [Sov. J. Nucl. Phys. 41 (1985) 4121. Phys. Lett. B 163 (1985) 2 1. M. Abramowitz and LA. Stegun, Handbook of Mathematical functions (US National Bureau of Standards, Washington, DC, 1972). Gy. Bencze, C. Chandler, J.L. Friar, A.G. Gibson and G.L. Payne, Low energy scattering theory for Coulomb plus longrange potentials, University ofNew Mexico preprint (1986). V.V. Babikov, The method of phase functions in quantum mechanics, 2nd Ed. (Nauka, Moscow, 1976) p. 132 [in Russian]. E. Huttel et al., Nucl. Phys. A 406 (1983) 443.
[ 31 Gy. Bencze and C. Chandler, [4]
[5]
[ 61
[ 71
123
PHYSICS
2
LETTERS
B
18 December
A SIMPLE THEORETICAL DESCRIPTION OF COULOMB ON THE COULOMB MODIFIED SCATTERING LENGTH Gy. BENCZE
POLARIZATION
1986
EFFECTS
’ and C. CHANDLER
Department qf Physics and Astronomy, Umversity qf New Mexico. Albuquerque, NM 87131, LISA Received
23 July 1986
A simple formula is derived for the Coulomb modified s-wave scattering length function in the presence of long-range polarization potentials. When applied to a model of low-energy p-d scattering, the formula reproduces behavior previously described by Kvitsinskii and Merkuriev [Sov. J. Nucl. Phys. 41 (1985) 4121.
For some years it has been known that, in the presence of long-range polarization forces, the standard Coulomb-modified scattering length is infinite if the Coulomb interaction is repulsive [ l-31. Since this is so one must ask how to interpret the finite values that are routinely obtained in both experimental and theoretical studies. For low-energy p-d scattering this question was addressed by Kvitsinskii and Merkuriev [ 21. In their paper an important role was played by a certain approximation (cf. eq. (9) below) which, for energies of experimental interest, gives the wrong sign and the wrong absolute value (by a factor of 2-8). The discovery,that this is so prompted this paper, in which we reexamine and generalize the arguments of Kvitsinskii and Merkuriev and present results of numerical experiments. Our main results are a simple formula (eq. (7) below) for the Coulomb modified s-wave scattering length function in the presence of long-range potentials and, in the case of p-d scattering, a conlirmation of the general conclusions of Kvitsinskii and Merkuriev [ 21. We assume that we are dealing with an effective two-body problem, and we ignore spin (for simplicity). We denote the reduced mass of the system by p and the energy by E. The effective potential V(r) has the form ’ Permanent
adress: Central Research P.O.B. 49, H-1525 Budapest, Hungary.
0370-2693/86/$ (North-Holland
Institute
V(r) =h+
(1)
)
where 11>0 and satisfies
V(v)
is a central
potential
that
CQ s
drrlU(r)I
(2a)
0
and sup r’+’ O
IWr)I
(2b)
for some E> 0. We define k= (2,uE) ‘i2/A and denote by 6 the s-wave scattering phase shift (relative to the pure Coulomb phase shift) produced by U(r) . The usual Coulomb modified scattering length function A(k) is defined [ I] by A(k) = - [KC;(k)cot
d+lh(k)]
-’ .
(3)
Here C,(k)=(n1lk)[exp(nllk)-11-r, h(k) =Re y(iX2k)
-ln(A/2k)
,
where I+Ydenotes the digamma function [4]. When U(r) falls off exponentially as r+co, then the zero energy limit of A(k) is the usual scattering length. When V(r) has a long-range attractive tail that falls off only as some inverse power of r as r--rcq then [l-3,5] the zero energy limit ofA is --CO. To isolate the effects of the long-range tail of the
for Physics,
03.50 0 Elsevier Science Publishers Physics Publishing Division)
U(r)
B.V.
121
Volume 182, number 2
potential,
PHYSICS
we define a “polarization”
V,(r) =O, =U(r),
potential
LETTERS
R
.
b(x)=3~x*(l +2x21m
(5)
(6)
’
1+As(kVG(klf(k)
(7)
>’
We emphasize that the formula given in (11) is quite general and does not depend on any specific asymptotic behavior of the potential P’,(r). We infer that Kvitsinskii and Merkuriev [ 21 used the following (in our notation):
122
1- Cf,,/AskG)l >
results for the potential
(8)
R
of eq. (10) with R = 5 fm. A,= 5 fm for the calculation
e(keV)
&
1 3 5 10 25 40 100 400 800 1000
0.4815x 10-9 0.8743x IO-* 0.4130x 1om7 0.4005x 10-b 0.4750x 10-5 0.1208x 1O-4 0.4422x 1O-4 0.1268x lo-) 0.1553x10-3 0.1577x 10-J
t9b)
(10)
where A=O.O4298 fin-’ and a,=0.632 fm3, values appropriate to the p-d system. We used both R==5 fm and R = 50 fm in our calculations. Over the energy range l- 1000 keV an “exact” value fE( k) of the function f(k) was calculated in by using the IMSL program DGEAR to solve the variable phase equations [ 61. In addition, the first Born approximation fs( k) and the Kvitsinskii-Merkuriev approximation fKM(k) were calculated. As is illustrated by the selected results reproduced in table 1, the first Born approximation reproduces the exact value to within 0.01% over the energy range l-l 000 keV. The approximation (9) changes to the wrong sign at about 30 keV and is in absolute value wrong by factors of 2-8 over the range of experimental interest (400- 1000 keV [ 71). We calculated the function B(k) for various scattering lengths a and for various methods of comput-
where the notation f(k) = tan 6,,, has been introduced. All of the low-energy effects of the polarization potential V,(r) are now isolated in tan 6,,,. Since we are only interested in the limit of zero energy, where h(k) vanishes, we set h(k) equal to zero. The result, which we denote by B(k), is
l- If(k)4(kW’3k)l
nx)+2x
I&( 1 +ix) - 1/3x.
V,(r)=da,2r“,
kC$A, - (1 +;IhA,)f kC~+[k*C~A,+~h(l+#zA)fl
Table 1 Selected numerical
-coth
Eq. (9)) which follows from the work of Berger and Spruch [ 11, represents an approximation to the Born approximation for the potential given in eq. (10) below. In order to test the formula (7) quantitatively, we used the polarization potential VP(r) consistent with that used by Kvitsinskii and Merkuriev [ 21,
Combining (3) and (5) with the trigonometric identity for the cotangent of the sum of two angles yields
B(K) -A,(k)t
(9a)
and
A,(k)=-[kC~(k)tan6,,,~+~h(k)]-’
B(k) =4(k)
>
&M(k) = (&/2)b(J./2k)
(4)
where the radius R is arbitrary but fixed. By itself the potential VPwould produce a phase shift a,., relative to the pure Coulomb phase shift. The remainder Bcp,s=6 -6,,, of the phase shift 6 is the additional phase shift produced by the short-range potential U(r) - V,(r). Associated with 6,,, is another scattering length function A,(k) defined by
A(k)=
1986
where
P’,(r)
O
I8 December
B
1.0000 1.oooo
1.oooo 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999
1.oooo
1.002 0.9679 0.8079 0.4274 0.0333 -0.1375 -0.5463 -2.321 -5.573 - 7.709
of J3.
_ 113 0.959 0.993 0.998 0.999 0.999 0.999 0.999 1.000
1.000
Volume 182, number
PHYSICS
2
I
-10.0.’
’
’
’ I"II'
ENERGY
IO
i 100
IKEV)
Fig. 1. The function B/A, for f=fe (for the potential of eq. (10) scattering length and A,=4.0 fm (--)(doublet scattering length [ 7]), 11.I fm (------)(quartet scattering length [7]).
ing the function f( k). Those in table 1, for example were calculated usingf,( k) and A,(k) = 4 fm, which is the experimental value of the doublet scattering length [ 71. To an accuracy of 0.2% the values of B( k) are independent of the method used to calculatef( k), the radius R used, which of the eqs. (7) or (8) was used or whether we included the effective range in the low-energy expansion for A,(k). When we restored the function h(k) to the formulas, some new structure appeared in B(k) below 1 keV but there were no essential changes above that energy. At low energies ( 1- 10 keV) there is some dependence on the value of the scattering length, which is illustrated in fig. 1. We conclude that for p-d scattering eq. (9) represents a poor approximation for energies of experimental interest.
LETTERS
B
18 December
1986
Nevertheless, the essential conclusions of Kvitsinskii and Merkuriev [ 2 ] remain valid. The effects of the long-range polarization potential are found only at extremely low energies and by 10 keV the scattering length function B(k) has reached a constant plateau. This plateau covers the range of experimental interest and thus allows a stable extrapolation of experimental results to zero energy. The result of the extrapolation is precisely the long-range modified scattering length a. This general behavior is insensitive to the value of the radius R, to the value longrange modified scattering length a and to the method of calculatingf( k). We believe, on the basis of numerical experiments, that the function B(k) exhibits similar properties for more general long-range potentials. The authors are indebted to the United States National Science Foundation and the Hungarian Academy of Sciences for supporting a cooperative research program which made this work possible. References [ I] R.O. Berger and L. Spruch, Phys. Rev. 138 (1965) B1106. [ 21 A.A. Kvitsinskii and S.P. Mercuriev, Yad. Fiz. 41 (1985) 647 [Sov. J. Nucl. Phys. 41 (1985) 4121. Phys. Lett. B 163 (1985) 2 1. M. Abramowitz and LA. Stegun, Handbook of Mathematical functions (US National Bureau of Standards, Washington, DC, 1972). Gy. Bencze, C. Chandler, J.L. Friar, A.G. Gibson and G.L. Payne, Low energy scattering theory for Coulomb plus longrange potentials, University ofNew Mexico preprint (1986). V.V. Babikov, The method of phase functions in quantum mechanics, 2nd Ed. (Nauka, Moscow, 1976) p. 132 [in Russian]. E. Huttel et al., Nucl. Phys. A 406 (1983) 443.
[ 31 Gy. Bencze and C. Chandler, [4]
[5]
[ 61
[ 71
123