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Eikonal approximation for composite Coulomb scattering
Volume 75A, number 1,2
PHYSICS LETTERS
24 December 1979
EIKONAL APPROXJMATION FOR COMPOSITE COULOMB SCATTERING T. 1SHIHARA Institute ofApplied Physics, University of Tsukuba, Ibaraki 300-31, Japan
and
H. NARUMI Department of Physics, Faculty of Science, Hiroshima University, Hiroshima 730, Japan Received 15 August 1979
A realistic approach to the eikonal approximation for composite Coulomb scattering is proposed by examining num~rically the potential scattering for a Coulomb plus short-range interaction.
An extension of the Glauber approximation has already been made to electron—ion scattering. The results are not quite satisfactory in the low-energy region [l—4]. Although the Glauber amplitude is known to be exact for pure Coulomb scattering, its applicability to problems with a Coulomb plus shortrange interaction has not yet been made clear. By cxamining this point in the case of potential scattering, an eikonal approximation useful for this type of problems is proposed in this paper. Consider the scattering of a particle of unit mass by the central potential V(,~= _Z/r + V1 (r) and assume that the eikonal approximation is applicable to the short-range potential V1(r). We use atomic units unless otherwise stated. The phase shift for angular momentum 1 is written ~ = ~i + 6~,where Ui is the Coulomb phase shift. The following expression for ~i is obtained if we evaluate it in the JWKB approximation to first order in V1: 2/r2 11/2 = dr V1 ~r)/[2E + 2Z/r (/1- 1 /2) b 1 2 + 2E(1 + 1/2)2] ‘/2}/(2E) is (I) where b1 = {—Z + [Z the distance of closest approach for the Coulomb scattering and F is the projectile energy. Eq. (1) can be interpreted as the integral of— V 1(r)/k(r) along the classical Coulomb trajectory, where k(r)the = (2E + 2 is the local velocity. If we neglect 2Z/r)l/ 38
f
,
Coulomb potential in the denominator of eq. (1), it reduces to the Glauber phase shift for the potential l/~(r),
f
(2) V1 [(h2 + z2)’/21 dz where b = (1 + 1/2)/k is the impact parameter and k = (2E)1/2 is the asymptotic velocity of the projectile. As a numerical example, we take the case of Z = and V 1(r) = —exp(—r). This will be sufficient for our present purpose of finding a practical method to put the Coulomb correction into the Glauber phase shift, although the range and relative strength of V1 may be of some importance for more detailed comparison. We compare, in table 1, the exact il~with c~Fand at E = 20 and SO eV. We can see the importance of the Coulomb effect in the evaluation of !~and it is well taken into account through the classical treatment of~thegives Coulomb scattering by of eq.~(1). is cxpected, an overestimation for As small / ~‘
=
—
~
0
—
values and an underestimation values. high energies, say E 1 100 eV, for the large! agreement be-At tween ~Fand 6~becomes better. Eq. (1), though it gives excellent results, is not in a convenient form for application to practical probhems with composite targets. In order to the potential scattering, it si desirable to go findbeyond a straight-
Volume 75A, number 1,2
PHYSICS LETTERS
Table 1 Comparison of phase shifts in various approximations with exact 1~.For each 1, the first and second lines are forE = 20 eV and E = 50 eV, respectively. C G S I a~ 0
0.4897 0.3874 0.4560
1
0.4954
0.716 1
0.3879 0.4591
0.4865
0.4185
0.3081 0.2707 0.4278
24 December 1979
f1(O) =
0.3591
0.3624
0.3638
0.3429
0.3007 0.2756
0.2935 0.2741
0.2193 0.2519
0.2864 0.2685
3 4
0.1670 0.1940 0.0858 0.1302
0.1620 0.1913 0.0836 0.1282
0.1096 0.1680 0.0533 0.1095
0.1601 0.1893 0.0830 0.1274
5
0.0426
0.0415
0.0254
0.0413
0.0851
0.0839
0.0704
0.0836
0.0011 0.0087
0.0010 0.0085
0.0005 0.0068
0.0010
10
0.0085
line trajectory with a constant velocity as in the case of eq. (2). A straight line through the apse may serve as such a trajectory since the contribution to the integral of eq. (1) comes mostly from the region r b 2 = r2 b~,eq. (1) is1. Introducing a parameter z by z rewritten as
I~ (2! + l)Pz(cos 0)
2i(~l~o)(e2i~iI) —
tamed by using the eikonal phase shift
-
f
or ~ for
/i~and the following approximations: (i) a
y log(kb), (ii)P1(cos 0) ~J0(qb) and 7=—Z/kandq=2ksin(0/2). Consequently we have —
a0
k f~’db, where
2i’Y logkb (e2’~l G,S —1) f?,s(O) = —ik 0 bdb J0(qb)e (7) Fig 1 compares the exact 1f 2 (curve E) with If~(0)I2(curve G) and fS(Q)1(0)1 12 (curve S) for our example at F = 20 and 50 eV. The broken curves are the results 2without calculatedthe by above eq. (6)apprixomations using ~ for (i)that and is, f~(0)I (ii). Comparison of this curve with the curve S shows that (i) and (ii) are good approximations for our problem except for the overall phase. We shall return to this point. The Glauber formula IJ?(0)12, as we can expect from the values of~, underestimates the
f
~,
—
=
(6)
.
e TheX eikonal approximation formula forf (0) is ob-
1
2
—~----
2ik ~
2
If,(O)I
(au) 10
V~(r)g(r)dz,
(3)
0
where g(r) = [(r + b
2r + k2b 1)/(k
1
+
2Z)] 1f2~Since
ed V1(r) at zis=of 0. short We thus range, have g(r) a straight.line in eq. (3) may eikonal be evaluatphase shift
10 5 ~,5
f
v1[(b~+z2)1/2] dz (4) 2 +Z/b 112. The numerical where = (k are also 1) shown in table 1. values k1 of ~= 1/g(b1) for example They agree very well with ~ except for very small 1. ~=_~
10_i
The scattering amplitude is written by 2~of f(0) =f~(0)+ e 1(0)
(5)
107
30
60
90
120
150
180
@(deg)
where f~(0)is the amplitude for the pure Coulomb potential —Z/r, 0 is the scattering angle and
Fig. 1. Comparison of LfF(O)12 (curve G) and f~(O)I2(curve
S) with the exact li (8)12 (curve E). The broken curve represents the results obtained with eq. (6) using 6~for 5~.
39
Volume
PHYSICS LETTERS
75A, number 1,2
correction to the approximation (i) is, neglecting the
argf,(O) (rad) ~1~
-
F
5), given by contribution of O(’y z~a 1(a1 00) ‘ylog(/+~)
—~
-H2
~
2~
for sufficiently large 1, where C is the Euler constant,
~1
-
‘~
1
F 60
90 6(deg
12C
102
yC- ~3
1= ~1
/3~Q(/ 2)
(8)
The convergence of ~al to a constant value is, in fact, very fast. The broken curve in fig. ~ is arg 1(°)+ 2~a,where ~u is the constant phase correction of
-~
1
3D
24 December 1979
180
1
Fig. 2. Comparison of argf~(o) (curve G) and argf~(O) (curve 5) with the exact argf1(O) (curve E). The broken
curve is the result of curve S plus the constant Coulomb phase correction 2~a.
eq. (8). The agreement with the exact curve E is excellent. In conclusion, our proposed eikonal amplitude ij(0) of eq. (7) with the correction of a constant phase factor exp(2i~o)is shown to be useful for scattering by a Coulomb plus short-range potential. It miproves the Glauber amplitude J?(0) appreciably in the low-energy region. We expect that the Glaubertype approximation based on the present straightline trajectory would be of practical use for problems with composite targets.
forward peak and overestimates the large-angle tail. 2gives generally The present eikonah formula If~(0)l good results, and especially, it reproduces the I’or-
References
ward peak very well. We next examine the phase angle of the amplitude
[11 H. Narumi and A. Tsu~i.Prog. Theor. Phys. 53 (1975) 671.
f 1(0). Comparison of arg4’(O) (curve G) and argf~(0)(curve S) with the exact argf1 (0) (curve E) is made in fig. 2. The errors both in G and S are not small, It turns out that this is mainly caused by the approximation (i) for the Coulomb phase shift. The
40
121 T. Ishihara and J.C.Y. (‘hen. .1. Ph~s.B8 (1975) L417. 131 B.K. Thomas and V. Franko. l’hvs. Rev. Al 3(1976) 2004. [41 11. Narumi and A. Tsuji, 6th Intern. Conf. on Atomic physics (Riga, 1978) Ab. 247.
PHYSICS LETTERS
24 December 1979
EIKONAL APPROXJMATION FOR COMPOSITE COULOMB SCATTERING T. 1SHIHARA Institute ofApplied Physics, University of Tsukuba, Ibaraki 300-31, Japan
and
H. NARUMI Department of Physics, Faculty of Science, Hiroshima University, Hiroshima 730, Japan Received 15 August 1979
A realistic approach to the eikonal approximation for composite Coulomb scattering is proposed by examining num~rically the potential scattering for a Coulomb plus short-range interaction.
An extension of the Glauber approximation has already been made to electron—ion scattering. The results are not quite satisfactory in the low-energy region [l—4]. Although the Glauber amplitude is known to be exact for pure Coulomb scattering, its applicability to problems with a Coulomb plus shortrange interaction has not yet been made clear. By cxamining this point in the case of potential scattering, an eikonal approximation useful for this type of problems is proposed in this paper. Consider the scattering of a particle of unit mass by the central potential V(,~= _Z/r + V1 (r) and assume that the eikonal approximation is applicable to the short-range potential V1(r). We use atomic units unless otherwise stated. The phase shift for angular momentum 1 is written ~ = ~i + 6~,where Ui is the Coulomb phase shift. The following expression for ~i is obtained if we evaluate it in the JWKB approximation to first order in V1: 2/r2 11/2 = dr V1 ~r)/[2E + 2Z/r (/1- 1 /2) b 1 2 + 2E(1 + 1/2)2] ‘/2}/(2E) is (I) where b1 = {—Z + [Z the distance of closest approach for the Coulomb scattering and F is the projectile energy. Eq. (1) can be interpreted as the integral of— V 1(r)/k(r) along the classical Coulomb trajectory, where k(r)the = (2E + 2 is the local velocity. If we neglect 2Z/r)l/ 38
f
,
Coulomb potential in the denominator of eq. (1), it reduces to the Glauber phase shift for the potential l/~(r),
f
(2) V1 [(h2 + z2)’/21 dz where b = (1 + 1/2)/k is the impact parameter and k = (2E)1/2 is the asymptotic velocity of the projectile. As a numerical example, we take the case of Z = and V 1(r) = —exp(—r). This will be sufficient for our present purpose of finding a practical method to put the Coulomb correction into the Glauber phase shift, although the range and relative strength of V1 may be of some importance for more detailed comparison. We compare, in table 1, the exact il~with c~Fand at E = 20 and SO eV. We can see the importance of the Coulomb effect in the evaluation of !~and it is well taken into account through the classical treatment of~thegives Coulomb scattering by of eq.~(1). is cxpected, an overestimation for As small / ~‘
=
—
~
0
—
values and an underestimation values. high energies, say E 1 100 eV, for the large! agreement be-At tween ~Fand 6~becomes better. Eq. (1), though it gives excellent results, is not in a convenient form for application to practical probhems with composite targets. In order to the potential scattering, it si desirable to go findbeyond a straight-
Volume 75A, number 1,2
PHYSICS LETTERS
Table 1 Comparison of phase shifts in various approximations with exact 1~.For each 1, the first and second lines are forE = 20 eV and E = 50 eV, respectively. C G S I a~ 0
0.4897 0.3874 0.4560
1
0.4954
0.716 1
0.3879 0.4591
0.4865
0.4185
0.3081 0.2707 0.4278
24 December 1979
f1(O) =
0.3591
0.3624
0.3638
0.3429
0.3007 0.2756
0.2935 0.2741
0.2193 0.2519
0.2864 0.2685
3 4
0.1670 0.1940 0.0858 0.1302
0.1620 0.1913 0.0836 0.1282
0.1096 0.1680 0.0533 0.1095
0.1601 0.1893 0.0830 0.1274
5
0.0426
0.0415
0.0254
0.0413
0.0851
0.0839
0.0704
0.0836
0.0011 0.0087
0.0010 0.0085
0.0005 0.0068
0.0010
10
0.0085
line trajectory with a constant velocity as in the case of eq. (2). A straight line through the apse may serve as such a trajectory since the contribution to the integral of eq. (1) comes mostly from the region r b 2 = r2 b~,eq. (1) is1. Introducing a parameter z by z rewritten as
I~ (2! + l)Pz(cos 0)
2i(~l~o)(e2i~iI) —
tamed by using the eikonal phase shift
-
f
or ~ for
/i~and the following approximations: (i) a
y log(kb), (ii)P1(cos 0) ~J0(qb) and 7=—Z/kandq=2ksin(0/2). Consequently we have —
a0
k f~’db, where
2i’Y logkb (e2’~l G,S —1) f?,s(O) = —ik 0 bdb J0(qb)e (7) Fig 1 compares the exact 1f 2 (curve E) with If~(0)I2(curve G) and fS(Q)1(0)1 12 (curve S) for our example at F = 20 and 50 eV. The broken curves are the results 2without calculatedthe by above eq. (6)apprixomations using ~ for (i)that and is, f~(0)I (ii). Comparison of this curve with the curve S shows that (i) and (ii) are good approximations for our problem except for the overall phase. We shall return to this point. The Glauber formula IJ?(0)12, as we can expect from the values of~, underestimates the
f
~,
—
=
(6)
.
e TheX eikonal approximation formula forf (0) is ob-
1
2
—~----
2ik ~
2
If,(O)I
(au) 10
V~(r)g(r)dz,
(3)
0
where g(r) = [(r + b
2r + k2b 1)/(k
1
+
2Z)] 1f2~Since
ed V1(r) at zis=of 0. short We thus range, have g(r) a straight.line in eq. (3) may eikonal be evaluatphase shift
10 5 ~,5
f
v1[(b~+z2)1/2] dz (4) 2 +Z/b 112. The numerical where = (k are also 1) shown in table 1. values k1 of ~= 1/g(b1) for example They agree very well with ~ except for very small 1. ~=_~
10_i
The scattering amplitude is written by 2~of f(0) =f~(0)+ e 1(0)
(5)
107
30
60
90
120
150
180
@(deg)
where f~(0)is the amplitude for the pure Coulomb potential —Z/r, 0 is the scattering angle and
Fig. 1. Comparison of LfF(O)12 (curve G) and f~(O)I2(curve
S) with the exact li (8)12 (curve E). The broken curve represents the results obtained with eq. (6) using 6~for 5~.
39
Volume
PHYSICS LETTERS
75A, number 1,2
correction to the approximation (i) is, neglecting the
argf,(O) (rad) ~1~
-
F
5), given by contribution of O(’y z~a 1(a1 00) ‘ylog(/+~)
—~
-H2
~
2~
for sufficiently large 1, where C is the Euler constant,
~1
-
‘~
1
F 60
90 6(deg
12C
102
yC- ~3
1= ~1
/3~Q(/ 2)
(8)
The convergence of ~al to a constant value is, in fact, very fast. The broken curve in fig. ~ is arg 1(°)+ 2~a,where ~u is the constant phase correction of
-~
1
3D
24 December 1979
180
1
Fig. 2. Comparison of argf~(o) (curve G) and argf~(O) (curve 5) with the exact argf1(O) (curve E). The broken
curve is the result of curve S plus the constant Coulomb phase correction 2~a.
eq. (8). The agreement with the exact curve E is excellent. In conclusion, our proposed eikonal amplitude ij(0) of eq. (7) with the correction of a constant phase factor exp(2i~o)is shown to be useful for scattering by a Coulomb plus short-range potential. It miproves the Glauber amplitude J?(0) appreciably in the low-energy region. We expect that the Glaubertype approximation based on the present straightline trajectory would be of practical use for problems with composite targets.
forward peak and overestimates the large-angle tail. 2gives generally The present eikonah formula If~(0)l good results, and especially, it reproduces the I’or-
References
ward peak very well. We next examine the phase angle of the amplitude
[11 H. Narumi and A. Tsu~i.Prog. Theor. Phys. 53 (1975) 671.
f 1(0). Comparison of arg4’(O) (curve G) and argf~(0)(curve S) with the exact argf1 (0) (curve E) is made in fig. 2. The errors both in G and S are not small, It turns out that this is mainly caused by the approximation (i) for the Coulomb phase shift. The
40
121 T. Ishihara and J.C.Y. (‘hen. .1. Ph~s.B8 (1975) L417. 131 B.K. Thomas and V. Franko. l’hvs. Rev. Al 3(1976) 2004. [41 11. Narumi and A. Tsuji, 6th Intern. Conf. on Atomic physics (Riga, 1978) Ab. 247.