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Transition probabilities for electron capture
Coleman,
J. P.
McDowell,
Van den Bos, De Heer,
Physica
32
1164-l
169
M. R. C. J.
1;. J.
1966
TRANSITION
PROBABILITIES
by J.
P. COLEMAN
FOR ELECTRON
and M. R. C. MCDOWELL
LIepartment, University
(Mathematics
and J. VAN DEN (F.o.hl. Laboratoriunl
CAPTURE
BOS
of Durham,
and F. J.
vocr Massascheiding,
England)
DE
HEER
Arnstrrdam,
Nedcrland)
Synopsis A comparison
is made
approximation processes babilities
and
the
both
parameter
+ H(d) by the former
where
given
approximation
it is pointed
the probability 2s state
for
methods
nl =
yield approximately
2s,
holds
impact
The
smaller
impact
the impulse
while
capture
transition
than
those
(within
is less than
parameter,
for small
29.
approximation
into the 2p state
of the
this only
Is,
between
for the electron
are always
out that in the impulse of capture
all values
approximation
formulation,
approximation
+ Hf,
of validity) Kramers
in an impact
Brinkman-Kramers
H+ + H(ls)
latter. Furthermore, into
the
that
in the
parameters.
proof the
its range
for capture BrinkmanNevertheless
the same ratio for the total 2s and 2p cross sections.
1. Irttrodz~tion. Predictions of rates of electron from neutral atoms H++X+H(nz)+X+
capture
by fast protons (1)
are often made in the first order Brinkman-Kramers approximation (henceforth referred to as the OBK approximationl)) in order to avoid the great mathematical complexity of higher order approximations. Again, interpretation of the dominance of particular final orbital angular momenta I in experiments on capturez) may be aided by impact parameter arguments based on the OBK approximation. In this paper we compare some such predictions with those of a second order (“impulse”) approximation for the processes H+ + H(ls)
-+ H(nZ) + Hf
where nz = Is, 2s, 2$. 2. Transition
probability.
(4 (3)
The cross section for process (2) may be written +I
in the usual notationz) -
1164
-
TRANSITION
PROBABILITIES
FOR ELECTRON
1165
CAPTURE
where
Rif = <$f Iv,l y;>
(5)
and vf
=
vl2
+
v23,
(6)
The term Vi2 does not contribute to zero order in (m/M), except perhaps at relativistic energies, so that we may write & Expanding
= <+f
iv231
uli+>.
(7)
!Pi in terms of the resolvent 1
Goz =
Et - K -
where K is the kinetic energy operator,
Rif =
(#f
Vl3
+
(8)
is
we have to second order 1
Iv231[ 1 +
K
_
v13
+
is
‘13-j b)
(9)
where the first term in (9) is the Brinkman-Kramers approximation, and both terms constitute the impulse approximation, when Et = E, for the mth member of the complete set implied by the summation (9). Turning now from a wave treatment to an impact parameter treatment, we write m
QI~-~z(~~) = 2s b(pM2 0
where a(p)nl and velocity functions,
p dpnai
(10)
is the (complex) transition amplitude at impact parameter p vi for capture into the &state. If we expand Rtf in Legendre
c
Z=O
(2 + 1)Cd%)~z(cos0)
and assume a classical relation between angular momentum meter
(11)
and impact para-
1 = ktp then replacing
the summation
by an integration 44
and
(12) it is easily shown that
= a(p),l,
a, (13)
where p = kr kind.
kf, and JO(X) is the zero order Bessel function
of the first
1166 J. P. COLEMAN, M. R. C. MCDOWELL, We have used (13) to compute
J. VAN DEN BOS AND F. J. DE HEER
a(p),2 in the impulse approximation
the matrix elements calculated by Coleman and McDowells); (10) provided a check on the numerical work. The OBK transition ties are readily calculated initially by M ay 4) : &)7&z =
analytically
the
following
equation probabiliresult
used
cc.00
(2:)s viss
e ik?&(kz,
=
from
from
--oo
k,, c( -
iv,) !&(kz,
k,, c( + *vi) dk, dk,
(14)
-cm
where a = AEdf/vz and G-(x, y, z) is the Fourier transform with respect to Y = (x, y, z) of the function @and qiis the Fourier transform of -Q/r. Expressions for a(p),l, nl = Is, 2s, 2p,, 2p, in terms of modified Bessel functions of the second kind are given in the appendix. This result was obtained independently by one of us (J. v. d. B.). 3. Results and discussion. In fig. la and 1b we show the quantity p ia( as a function of p for the symmetric resonance case at 25 keV and 100 keV,
Fig. Fig. 1. a) Transition
probabilities I) III)
Fig.
1. b). Transition
OBK Born
probabilities I) III)
OBK Born
1.
b
for H+ + H( 1s) j H( 1s) + H+. II) Impulse IV) for
E = 25 keV.
McCarroll
H+ + H( 1s) --f H(ls) II)
Impulse
IV)
McCarroll.
+
Ii-‘.
E = 100 keV.
in the OBK and impulse approximations together with the so-called “Born” approximation of Jackson and Schiffs), and the results of a two-state eigenfunction expansion calculation of M c C ar r 0116). It is immediately
TRANSITION PROBABILITIES FOR ELECTRON CAPTURE
1167
10’
1
‘\
10'
;
// I/
c
,P
L_
li'
\
\
ZP
._--__
\
-..
._
\
‘1
‘\ \
10
\
\
2P
,,I
10
Y\
\ \
\ 1
\
\
25
'.\
i_
0
c \\
\
\
10
2
\\ '\\\
',2s
$
“_
_m_?L_, ;
i
4
5
10
Fig. 3. Fig. 2. Relative importance of 2.s and 2fi at E = 25 keV (OBK) and E = 22.7 keV (Impulse) (Is OBK shown for comparison) Impulse -.-.-.OBK Fig. 3. Comparison of 2s, 2p at 50 keV (OBK), 56.2 keV (Impulse) Impulse -.-.-.OBK
-
p(a,)
Fig. 4. Comparison of 2s, 29 at 225 keV (OBK), 185 keV (Impulse) Impulse -.-.-.OBK
clear that the OBK approximation grossly overestimates at all significant impact parameters in this energy range. We now compare in figs. 2,3 and 4 the predictions of the OBK and impulse approximations for capture into the 2s and 2p states at energies in the range
1168
J. P. COLEMAN,
M. R. C. MCDOWELL,
J. VAN
20 to 225 keV. At the lowest energies doubtful
validity)
BOS AND
F. J. DE HEER
(where both approximations
they agree in suggesting P lG)zpl2<
DEN
that
P lG)as)12,
P < PO =
(15)
1
but p
l43)2pl2
>
are of
p l~(p)zsl”,
p >
po =
1,
(16)
where 14f)2P12
=
I+4
2p2i2
+
la(f)2pzi2.
However, at higher energies, while this behaviour OBK approximation with somewhat larger values proximation predicts f
14f)2P12
<
f
(17)
is maintained by the of po, the impulse ap-
14f)2s12.
(18)
Assuming the relationship of eq. (12), simple physical arguments suggest that u(p)sp > u(p)ss for large p and u(p)zp < u(p)zs for small p, as obtained in the OBK approximation, in contrast to the impulse approximation at high energies. Equation (18) implies that at high energies Qrs-ss > Qrs_sp. This behaviour is also predicted by the OBK-approximation, because the significant contribution to the total cross sections comes from the low p-region. The OBK and impulse approximations are also in close agreement in their predictions of the values for the ratio R spof total 2s and 2~ cross sectionss). The fact that Qrs--ns becomes larger than Qrspnp for electron capture at higher energies has been affirmed by experiments of De Heer E.u.~) (He+ incident on He) and of Jaecks E.U. 7) (H+ incident on noble gases). Acknowledgements. One of us (J. P. C.) was supported by a grant from the Science Research Council (United Kingdom). The work of two of us (J. v. d. B. and F. J. d. H) was done as part of the research program of the Dutch Foundation Fundamenteel Onderzoek der Materie (F.O.M.) and was made possible by financial support of the organisation Giver Wetenschappelijk
Onderzoek
(Z.W.O.).
APPENDIX
OBK
expressions
for a (p),z using equation 2
2i u(p)ls
=
y
a(p)zs
=
-
df)zp,
(14)
(1 ;
272
82)
f2 [ 1 +p2
2vi
i1/2 = T
6
~
KZ(fdl
+
K&Z/l
p2)
+ 82) -
f3
1+p2
K2(fdl
+
P2)
1 48
f3 (1 + P2)’
Kdfdl
+ 82)-j
TRANSITION
PROBABILITIES
4fd2p.= where
FOR
ELECTRON
1169
CAPTURE
K3Wl + 82)
(B- 4 P3
g
(1
+
p2)*
b = a + frvg.
Received
5-1 l-65
REFERENCES 1) Mapleton, 2)
De Heer,
3)
R. A., Proc.
Phys. Sm. 85 (1965)
F. J., Wolterbeek
Muller,
1109.
L. and Geballe,
R., Physica
4)
Coleman, J. P. and McDowell, M. R. C., Proc. Phys. Sot. 88 (1965) May, R., Nuclear Fus. 4 (1964) 207, equations 20 and 21.
5)
Jackson,
6)
McCarroll,
7) Jaecks,
J. D. and Schiff, R., Proc. D., Van
Zijl,
Roy.
H., Phys.
31
(1965)
1097.
Rev. 89 (1953) 359.
Sot. A 264 (1961) 547.
B. and Geballe,
R., Phys.
Rev. 137 (1965) A 361.
1745.
J. P.
McDowell,
Van den Bos, De Heer,
Physica
32
1164-l
169
M. R. C. J.
1;. J.
1966
TRANSITION
PROBABILITIES
by J.
P. COLEMAN
FOR ELECTRON
and M. R. C. MCDOWELL
LIepartment, University
(Mathematics
and J. VAN DEN (F.o.hl. Laboratoriunl
CAPTURE
BOS
of Durham,
and F. J.
vocr Massascheiding,
England)
DE
HEER
Arnstrrdam,
Nedcrland)
Synopsis A comparison
is made
approximation processes babilities
and
the
both
parameter
+ H(d) by the former
where
given
approximation
it is pointed
the probability 2s state
for
methods
nl =
yield approximately
2s,
holds
impact
The
smaller
impact
the impulse
while
capture
transition
than
those
(within
is less than
parameter,
for small
29.
approximation
into the 2p state
of the
this only
Is,
between
for the electron
are always
out that in the impulse of capture
all values
approximation
formulation,
approximation
+ Hf,
of validity) Kramers
in an impact
Brinkman-Kramers
H+ + H(ls)
latter. Furthermore, into
the
that
in the
parameters.
proof the
its range
for capture BrinkmanNevertheless
the same ratio for the total 2s and 2p cross sections.
1. Irttrodz~tion. Predictions of rates of electron from neutral atoms H++X+H(nz)+X+
capture
by fast protons (1)
are often made in the first order Brinkman-Kramers approximation (henceforth referred to as the OBK approximationl)) in order to avoid the great mathematical complexity of higher order approximations. Again, interpretation of the dominance of particular final orbital angular momenta I in experiments on capturez) may be aided by impact parameter arguments based on the OBK approximation. In this paper we compare some such predictions with those of a second order (“impulse”) approximation for the processes H+ + H(ls)
-+ H(nZ) + Hf
where nz = Is, 2s, 2$. 2. Transition
probability.
(4 (3)
The cross section for process (2) may be written +I
in the usual notationz) -
1164
-
TRANSITION
PROBABILITIES
FOR ELECTRON
1165
CAPTURE
where
Rif = <$f Iv,l y;>
(5)
and vf
=
vl2
+
v23,
(6)
The term Vi2 does not contribute to zero order in (m/M), except perhaps at relativistic energies, so that we may write & Expanding
= <+f
iv231
uli+>.
(7)
!Pi in terms of the resolvent 1
Goz =
Et - K -
where K is the kinetic energy operator,
Rif =
(#f
Vl3
+
(8)
is
we have to second order 1
Iv231[ 1 +
K
_
v13
+
is
‘13-j b)
(9)
where the first term in (9) is the Brinkman-Kramers approximation, and both terms constitute the impulse approximation, when Et = E, for the mth member of the complete set implied by the summation (9). Turning now from a wave treatment to an impact parameter treatment, we write m
QI~-~z(~~) = 2s b(pM2 0
where a(p)nl and velocity functions,
p dpnai
(10)
is the (complex) transition amplitude at impact parameter p vi for capture into the &state. If we expand Rtf in Legendre
c
Z=O
(2 + 1)Cd%)~z(cos0)
and assume a classical relation between angular momentum meter
(11)
and impact para-
1 = ktp then replacing
the summation
by an integration 44
and
(12) it is easily shown that
= a(p),l,
a, (13)
where p = kr kind.
kf, and JO(X) is the zero order Bessel function
of the first
1166 J. P. COLEMAN, M. R. C. MCDOWELL, We have used (13) to compute
J. VAN DEN BOS AND F. J. DE HEER
a(p),2 in the impulse approximation
the matrix elements calculated by Coleman and McDowells); (10) provided a check on the numerical work. The OBK transition ties are readily calculated initially by M ay 4) : &)7&z =
analytically
the
following
equation probabiliresult
used
cc.00
(2:)s viss
e ik?&(kz,
=
from
from
--oo
k,, c( -
iv,) !&(kz,
k,, c( + *vi) dk, dk,
(14)
-cm
where a = AEdf/vz and G-(x, y, z) is the Fourier transform with respect to Y = (x, y, z) of the function @and qiis the Fourier transform of -Q/r. Expressions for a(p),l, nl = Is, 2s, 2p,, 2p, in terms of modified Bessel functions of the second kind are given in the appendix. This result was obtained independently by one of us (J. v. d. B.). 3. Results and discussion. In fig. la and 1b we show the quantity p ia( as a function of p for the symmetric resonance case at 25 keV and 100 keV,
Fig. Fig. 1. a) Transition
probabilities I) III)
Fig.
1. b). Transition
OBK Born
probabilities I) III)
OBK Born
1.
b
for H+ + H( 1s) j H( 1s) + H+. II) Impulse IV) for
E = 25 keV.
McCarroll
H+ + H( 1s) --f H(ls) II)
Impulse
IV)
McCarroll.
+
Ii-‘.
E = 100 keV.
in the OBK and impulse approximations together with the so-called “Born” approximation of Jackson and Schiffs), and the results of a two-state eigenfunction expansion calculation of M c C ar r 0116). It is immediately
TRANSITION PROBABILITIES FOR ELECTRON CAPTURE
1167
10’
1
‘\
10'
;
// I/
c
,P
L_
li'
\
\
ZP
._--__
\
-..
._
\
‘1
‘\ \
10
\
\
2P
,,I
10
Y\
\ \
\ 1
\
\
25
'.\
i_
0
c \\
\
\
10
2
\\ '\\\
',2s
$
“_
_m_?L_, ;
i
4
5
10
Fig. 3. Fig. 2. Relative importance of 2.s and 2fi at E = 25 keV (OBK) and E = 22.7 keV (Impulse) (Is OBK shown for comparison) Impulse -.-.-.OBK Fig. 3. Comparison of 2s, 2p at 50 keV (OBK), 56.2 keV (Impulse) Impulse -.-.-.OBK
-
p(a,)
Fig. 4. Comparison of 2s, 29 at 225 keV (OBK), 185 keV (Impulse) Impulse -.-.-.OBK
clear that the OBK approximation grossly overestimates at all significant impact parameters in this energy range. We now compare in figs. 2,3 and 4 the predictions of the OBK and impulse approximations for capture into the 2s and 2p states at energies in the range
1168
J. P. COLEMAN,
M. R. C. MCDOWELL,
J. VAN
20 to 225 keV. At the lowest energies doubtful
validity)
BOS AND
F. J. DE HEER
(where both approximations
they agree in suggesting P lG)zpl2<
DEN
that
P lG)as)12,
P < PO =
(15)
1
but p
l43)2pl2
>
are of
p l~(p)zsl”,
p >
po =
1,
(16)
where 14f)2P12
=
I+4
2p2i2
+
la(f)2pzi2.
However, at higher energies, while this behaviour OBK approximation with somewhat larger values proximation predicts f
14f)2P12
<
f
(17)
is maintained by the of po, the impulse ap-
14f)2s12.
(18)
Assuming the relationship of eq. (12), simple physical arguments suggest that u(p)sp > u(p)ss for large p and u(p)zp < u(p)zs for small p, as obtained in the OBK approximation, in contrast to the impulse approximation at high energies. Equation (18) implies that at high energies Qrs-ss > Qrs_sp. This behaviour is also predicted by the OBK-approximation, because the significant contribution to the total cross sections comes from the low p-region. The OBK and impulse approximations are also in close agreement in their predictions of the values for the ratio R spof total 2s and 2~ cross sectionss). The fact that Qrs--ns becomes larger than Qrspnp for electron capture at higher energies has been affirmed by experiments of De Heer E.u.~) (He+ incident on He) and of Jaecks E.U. 7) (H+ incident on noble gases). Acknowledgements. One of us (J. P. C.) was supported by a grant from the Science Research Council (United Kingdom). The work of two of us (J. v. d. B. and F. J. d. H) was done as part of the research program of the Dutch Foundation Fundamenteel Onderzoek der Materie (F.O.M.) and was made possible by financial support of the organisation Giver Wetenschappelijk
Onderzoek
(Z.W.O.).
APPENDIX
OBK
expressions
for a (p),z using equation 2
2i u(p)ls
=
y
a(p)zs
=
-
df)zp,
(14)
(1 ;
272
82)
f2 [ 1 +p2
2vi
i1/2 = T
6
~
KZ(fdl
+
K&Z/l
p2)
+ 82) -
f3
1+p2
K2(fdl
+
P2)
1 48
f3 (1 + P2)’
Kdfdl
+ 82)-j
TRANSITION
PROBABILITIES
4fd2p.= where
FOR
ELECTRON
1169
CAPTURE
K3Wl + 82)
(B- 4 P3
g
(1
+
p2)*
b = a + frvg.
Received
5-1 l-65
REFERENCES 1) Mapleton, 2)
De Heer,
3)
R. A., Proc.
Phys. Sm. 85 (1965)
F. J., Wolterbeek
Muller,
1109.
L. and Geballe,
R., Physica
4)
Coleman, J. P. and McDowell, M. R. C., Proc. Phys. Sot. 88 (1965) May, R., Nuclear Fus. 4 (1964) 207, equations 20 and 21.
5)
Jackson,
6)
McCarroll,
7) Jaecks,
J. D. and Schiff, R., Proc. D., Van
Zijl,
Roy.
H., Phys.
31
(1965)
1097.
Rev. 89 (1953) 359.
Sot. A 264 (1961) 547.
B. and Geballe,
R., Phys.
Rev. 137 (1965) A 361.
1745.