Magnetically Quantized Continuum Distorted Waves
D S F Crothers, D M McSherry and S F C O'Rourke
Theoretical and Computational Physics Research Division
Department of Applied Mathematics and Theoretical Physics
Queen's University Belfast, Belfast BT7 1NN
Northern Ireland
PACS: 34.50 Fa, 34.70+e
Abstract. The continuum distorted-wave eikonal initial-state (CDW-EIS) theory of Crothers and McCann (1983) used to
describe ionization in ion-atom collisions is generalised (G) to GCDW-EIS, to incorporate the azimuthal angle
dependence of each CDW in the final state wave function. This is accomplished by the analytic continuation of
hydrogenic-like wave functions from below to above threshold, using parabolic coordinates and quantum numbers
including magnetic quantum numbers, thus providing a more complete set of states. At impact energies lower than 25
keVu-1, the total ionization cross section falls off, with decreasing energy, too quickly in comparison with experimental
data. The idea behind and motivation for the GCDW-EIS model is to improve the theory with respect to experiment, by
including contributions from non-zero magnetic quantum numbers. We also therefore incidentally provide a new
derivation of the theory of continuum distorted waves for zero magnetic quantum numbers while simultaneously
generalising it. Comparison of our theoretical calculations are made with available experimental data.
the usual angles of θ and φ to be subscripted also.
This is to represent the fact that they are either targetbased or projectile-based as indicated by the subscript
T or P respectively.
In recent work, (Nesbitt et al 2000 and McGrath et
al 2000) no evidence was found by our experimental
and theoretical groups for saddle-points for collisions
of 40 keV protons with either He or H 2 . However
INTRODUCTION
The continuum distorted-wave eikonal-initial-state
(CDW-EIS) model was first introduced by Crothers
and McCann (1983) who both presented and applied it
to the single ionization of H, He + , Li 2+ and Be 3+ by
a proton. Since then the theory has been extensively
studied and extended to model the single ionization of
multielectron atoms ranging from helium to argon.
One of the benefits of the EIS wave function is that
it is normalised at all times and impact parameters.
The advantages of the EIS and the final state CDW
two-centre wave functions are that all long range
Coulomb boundary conditions are obeyed. In essence
the CDW final state is a two-centre product of two
CDWs, each of which is a zero magnetic quantum
number analytic continuation of the well known exact
hydrogenic bound state with parabolic quantum
numbers and coordinates. One of the CDWs is targetbased and one projectile-based so that most of the postcollision interaction is included.
In this paper a new derivation of continuum
distorted-wave (CDW) theory is presented. We have
also generalized the CDW final state to non-zero
magnetic quantum numbers which leads to a more
complete set of states. The more general and therefore
more complicated geometry of the system has required
for 100 keV proton collisions with H 2 CDW-EIS
calculations predict saddle-points in contradiction with
the experimentalists. This was puzzling and indeed
incongruous, since saddle-point electrons are normally
expected to be associated with lower impact energies.
We were therefore motivated to reconsider the very
basis of CDW-EIS.
DERIVATION OF THE GENERALIZED
CDW FINAL STATE
In this section we derive the generalized CDW
final state by analytic continuation of hydrogen wave
functions from below to above threshold using
parabolic coordinates and magnetic quantum numbers
thus providing a more complete set of states. In
deriving the generalized CDW final state we follow
the analysis of Schiff (1968) in his treatment of the
hydrogen atom. We consider the hydrogen-like ion to
CP680, Application of Accelerators in Research and Industry: 17th Int'l. Conference, edited by J. L. Duggan and I. L. Morgan
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142
where
consist of a two-particle system of an electron of
charge − e and an atomic nucleus of charge + Ze .
These interact via a Coulomb attractive potential of
α=
2
− Ze = − Z in atomic units.
r
r
h 2 ( n1 + n 2 + m + 1)
)
n1
∑
(αη ) =
Lm
m +n
η = r − z = r (1 − cos θ )
=
(2)
2
 4  ∂  ∂u  ∂

 ξ + η  ∂η η ∂η  + ∂ξ

 ξ ∂u   + 1 ∂ 2 u 


 ∂ξ   ηξ ∂φ 2 
2
m ! n1!
(αξ ) =
Lm
m +n
2
(3)
=
(4)
(n1 − k )!( m + k )! k!
(8)
Zη 
M  − n1 ,1 + m ,


n 
(( n 2 + m )!) 2
m !n2 !
n2
∑
k =0
2
( − 1) k ( n 2 + m )! (αξ ) k
( n 2 − k )!( m + k )! k!
.
Zξ 
M  − n 2 ,1 + m ,


n 
(10)
(
)
u n n m (η , ξ , φ ) = exp Z (η − ξ ) − imφ (ηξ )
1 2
2n
m
2
(αη )
Lm
m +n
1
(9)
to equation (8) then the analytic continuation of the
unnormalised wave function may be written as
(η , ξ , φ ) =
α (ξ + η ) 
exp −
 (ξη )


2
2
( − 1) k ( n1 + m )! (αη ) k
M ( a, b, z ) = exp( z ) M ( b − a, b, − z )
It is clear (Schiff 1968) that in parabolic coordinates
the unnormalised bound states of the hydrogen-like
wave function may be written as
1 2m
(7)
Now applying the Kummer relation, equation
(13.1.27) of Abramowitz and Stegun (1970)
We seek eigenfunctions u(η , ξ , φ ) of the form
un n
(6)
and
− 2Ze u = Eu where E < 0.
ξ +η
u (η , ξ , φ ) = f (η )g (ξ )Φ(φ ).
k =0
((n1 + m )!) 2
to give the resulting equation
−h
2µ
.
The above threshold analytical continuation of the
hydrogen-like atom bound states in parabolic
coordinates with parabolic quantum numbers may be
obtained by first rewriting the associated Laguerre
polynomials in terms of confluent hypergeometric
functions and secondly by then choosing appropriate
separation constants. Thus in terms of the regular
Kummer hypergeometric functions we obtain
1
φ
nh 2
n
function is Ψ ( r, t ) = u ( r ) exp − i Et . It is convenient
h
to rewrite the wave equation in terms of parabolic
coordinates where the parabolic coordinates η , ξ and
φ are defined in terms of either spherical polar
coordinates ( r , θ , φ ) or Cartesian coordinates ( x, y, z )
by the formulae
= r + z = r (1 + cos θ )
y
= tan −1
x
µZe 2
α=Z .
2
2 
2  2
(1)
− h  ∂ + ∂ + ∂  u + Vu = Eu
2 µ  ∂x 2 ∂y 2 ∂z 2 
where µ is the reduced mass and the total wave
ξ
=
In atomic units and assuming an infinitely massive
nucleus, α reduces to
We start from the basic separated Schrödinger
equation in terms of the spatial relative motion of the
two particles with separation r = ( x, y, z ) ,
(
µZe 2
(5)
((n1 + m )!) 2
1
m
(αξ ) exp( − imφ )α ( 2π ) − 2
Lm
m + n2
m ! n1!
143
m
2
− Zη 
M  m + 1 + n1 , 1 + m ,


n 
((n 2 + m )!)
2
Zξ  α m
.
M  − n 2 ,1 + m ,


n  ( 2π ) 12
m !n2 !
2
 Γ1 + iZ + m  
 
υ
2 
m
m ! Γ1 + iZ − 

υ
2 
(11)
The choice of separation constant can now be made to
model the physical behaviour of the continuum. Thus
we set Z = −iυ (Crothers 1989 equation A30) where
n
v is the impact velocity of the collision and
ξ − η = 2 z so that the continuum hydrogen-like atom
wave function may be written in the form
m
( − i)
m
.
M  − i Z +
, 1 + m , − i( υ ⋅ r + υr ) 
1
 υ

2
( 2π ) 2
Setting
(( n 2 + m )!)
m !n2 !
2
m
m
M ( m + 1 + n1 , 1 + m , iυη )
( − i)
1
( 2π ) 2
(12)
.
M
1
n1 = −1 − 1 m , n 2 = iZ − 1 m .
υ 2
2
m
)
)
(15)
2
[
]
Dυ− ( r; Z ) = m D−+υ ( r; Z ) *
D υ± ( r; Z ) =
(
(16)
) (
exp ± imφ + πv M ± iv + 1 m , 1 + m , ± iυr − iυ ⋅ r
2
2
(13)
M
Thus the analytic continuation of the hydrogen-like
wavefunction from below to above threshold is given
by
(υr − υ ⋅ r )
m
2
m
,1 + m , − i( υ ⋅ r + υr )
2
m
2
m
following:
)
(12 m ,1 + m , m iυr − iυ ⋅ r )(υr + υ ⋅ r ) m / 2
(υr − υ ⋅ r ) m / 2
m
Γ(1 +
= exp( iυz − imφ )(υr + υ ⋅ r ) 2
1
2
m m iv )Γ(1 +
2
Γ (1 + m )
( m2 ) m
M ( ,1 + m , i(υr − υ ⋅ r ) )
m
2
m !Γ (− )
2
Γ
(
M − iv +
(
Γ1+
( m2 ,1 + m , i(υr − υ ⋅ r ))ΓΓ(1 +(1 + +miv) )
2
implies a suitable choice of n1 and n 2 is the
m
2
m
2
m
2
where we have written the generalized continuum
distorted wave (GCDW) having dropped the spatial
exponent. Now we have
no longer integers as they were for the case of bound
states and since n = iZ = n + n + 1 + m then this
υ
)
m
Here the first M function models the physical
behaviour of the outgoing wave and the second M
function simulates the physical characteristics of the
ingoing wave. In the continuum since n1 and n 2 are
(η , ξ , φ )
1 2m
(
Dυ− (r; Z ) = exp π ν − imφ (υr + υ ⋅ r )
2
(υr − υ ⋅ r )
M ( − n 2 , 1 + m , − iυξ )
un n
and
1 2
m
m ! n1!
r =∞,
at
Crothers and Dubé (1993), generalized to include
magnetic-quantization dependence we obtain
exp( iυz − imφ )(υr + υ ⋅ r ) 2 (υr − υ ⋅ r ) 2
(( n1 + m )!)
renormalizing
representing the Coulomb distorted wave function,
u n n m (η, ξ , φ ) in the continuum in the notation of
u n n m (η , ξ , φ ) =
12
2
ν=Z,
υ
(14)
2
It may also be noted that in
m
1
2
m)
.
(17)
Dυ+ ( r; Z ) , the first M
function is outgoing whereas the second is ingoing
and vice versa for
single-centred
144
m
Dυ− ( r; Z ) . This is in essence a
non-zero
magnetically
quantized
generalized continuum distorted wave theory. It is
quite clear we have a radially coherent set of final
states since they all have the same radial asymptotic
behaviour.
This may be compared with the wave treatment of
Crothers et al (2002). Here R is the position vector
of P relative to T where P denotes the projectile
and T denotes the target. The position vectors of the
electron relative to the target and projectile
respectively are defined as rT and rP . The electron
APPLICATION OF THE
GENERALIZED CDW FINAL STATE
TO THE GENERALIZED (G) TWOCENTRE CDW IONIZATION STATE
WITHIN AN IMPACT PARAMETER
TREATMENT
(
velocity of the projectile is given by υ with k and
p the momentum of the ejected electron relative to the
target and projectile respectively. The ingoing waves
We now apply the formalism adopted in section 2 to
address the problem of ionization in ion-atom
collisions.
We derive a generalized two-centre
continuum distorted wave function which correctly
models the physical representation of the ejected
electron which travels in the presence of two Coulomb
potentials namely that due to the target and that to the
projectile. Thus this section generalizes the theory in
section 2 where we confined our discussion to a single
Coulomb potential and hence the wave function
derived in equation (17) was a single centred
wavefunction. In the present work we assume the
independent electron model which simplifies the
description of the ionization process to a one electron
system. Our approach is within the semiclassical
straight-line impact parameter ρ time dependent ( t )
formalism of Crothers and McCann (1983). As in
section 2 we generalize the notation of Crothers and
Dubé (1993) equation (132) in which m P = 0 = mT , to
for χ k− in equation (18) are obtained by using the
definition stated in equation (17).
written explicitly as
mT
mT

mT
 υ

exp π T − imT φT (krT + k ⋅ rT ) 2 Γ1 +

2
2



( krT − k ⋅ rT )
mP

Dk±′ (rP , Z P ) exp ±

iZ P Z T
υ
mT
m

M  T ,1 + mT , i( krT − k ⋅ rT ) 
 2


mT
Γ 1 +
+ iυ T

2

2
2
mT
2






mT
M  − iυ T +
,1 + mT ,−i(k ⋅ rT + krT )


2


2
distorted wave states of energy k thus
3
These may be
Dk− (rT ; Z T ) =
obtain ingoing and outgoing generalized continuum
χ k± = ( 2π ) − 2 exp(ik ⋅ rT )E k ,± 1 υ ( r, t )
)
translation factor is exp −1 iυ ⋅ r , where r is the
2
electron position vector relative to the midpoint of the
nuclei. The nuclear charge of the target is denoted by
Z T and that of the projectile by Z P . The impact
Dk± (rT ; Z T )
Γ 2 (1 + mT
)




(23)
where
(
)

ln υR m υ t 

2
(18)
υT =
ZT
(24)
k
where
and
k′ = p = k − υ
(19)
R = rT − rP = ρ + υt
(20)
r = 1 (rT + rP )
2
(21)
mP
Dp− (r P ; Z P ) =
m


Γ  1 + P + iυ P 
2


Γ 2 (1 + m P )
 υ

exp π P − im P φ P ( prP + p ⋅ r P )
 2

2


E 1 ( r, t ) = exp ± 1 iυ ⋅ r − 1 iυ 2 t − i k t  . (22)
k ,± υ
2
8
2 

2
145
mP
2
m 

Γ 1 + P 
2 

( prP − p ⋅ rP )
m

M  P ,1 + m P , i( prP − p ⋅ rP ) 
 2

mP
2
m


M  − iυ P + P ,1 + m P ,−i(p ⋅ r P + prP ) 
2


1


M  − iυ T +
m T ,1 + m T , − ikr T − i k ⋅ r T 
2


1

M  mT ,1 + mT , ikrT − ik ⋅ rT  (krT + k ⋅ rT )
2


(25)
where
(krT
υP =
ZP
p
.
− k ⋅ rT )
( prP − p ⋅ r P )
between ( rT , k ) and ( υ, k ) , φ P appearing in equation
(25) is the angle between ( rP , p ) and ( υ, p ) . It should
be pointed out that, although p = k − υ, p and k are
skew vectors. Also the uniform two-centre nature of
Γ 2 (1 + mT
)

mP
Γ 1 +

2





)
mP
( prP + p ⋅ r P )
2
( r, t ) = exp(ip ⋅ rP + iρ ⋅ k )E
p, 12 v
mP
2
 mP

M
,1 + m P , i ( prP − p ⋅ r P )
 2





mP
M  − iυ P +
,1 + m P ,−i(p ⋅ r P + prP )


2


χ k± may be confirmed by noting that
k,− 12 v
2
2
(26)
This generalizes the treatment of Crothers and Dubé
(1993). It should be noted that the target and projectile
continuum waves contain magnetic quantum numbers
mT and m P respectively. In eq (23) φT is the angle
exp(ik ⋅ rT )E
(
Γ 1 + 12 mT + iυ T
mT
mT


mP
Γ 1 +
+ iυ P 


2


( r, t )
Γ 2 (1 + m P
(27)
)
(29)
.
Consideration of the analytic behaviour of the
confluent hypergeometric functions for large
arguments allows the asymptotic form to be extracted
so that
The coupling between the P and T components is
minimal at finite separation and in any case dies out
before infinity where degeneracy occurs. This is due
to the exp( −imT φT − im Pφ P ) term.
Moreover
mP
summation of any cross sections over degeneracies is
required. If we let mT = m P = 0 , then equation (22)
Dp− (rP ; Z P ) ≅ exp( − im Pφ P )( prP + p ⋅ rP )iv p .
p >>1
reduces down to
(28)
Hence the final Two-Centre GCDW state for a threebody continuum Coulomb wave function may be
expressed as a sum of products of two-body Coulomb
waves with their corresponding magnetic quantum
numbers and the individual wave function is given by

1
2
1
8
1
2
3

 πυ
 1

Γ1 + mT  exp − im p φ p exp(− imT φT ) exp T
 2

 2
(
)
 πυ
exp P
 2
2
8
2
)
 πυ 
exp T  M ( − iυ T ,1,−ikrT − ik ⋅ rT )Γ(1 + iυ T )
 2 
 πυ
exp P
 2

χ k− = (2π )− 2 exp ik ⋅ rT − iυ ⋅ r − iυ 2 t − ik 2 t 
3
(
χ k− ≈ ( 2π ) − 2 exp ik ⋅ rT − 1 iυ ⋅ r − 1 iυ 2 t − 1 ik 2 t


 M ( − iυ P ,1,−iprP − ip ⋅ rP )Γ(1 + iυ P )

(30)



which agrees with equation (17) of Crothers and
McCann (1983) as derived from their equations (7)
and (16) representing the final state of their CDW
Approximation. This also implies that if we take
m P = 0 = mT in the generalized form of the CDW-



EIS (GCDW-EIS) it reduces down to the usual CDW-
146
EIS approximation. This provides one good test for
our computer programs.
ACKNOWLEDGEMENTS
One of us (DMcS) acknowledges support by a DENI
(now DEL) Distinction Award.
CONCLUSIONS
REFERENCES
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with outgoing and one with ingoing waves, which
reflects reality and includes an azimuthal phase factor
with magnetic-quantum-number dependence. It is
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