IOP PUBLISHING
PHYSICA SCRIPTA
Phys. Scr. 77 (2008) 065005 (8pp)
doi:10.1088/0031-8949/77/06/065005
Amplitude-phase methods for analyzing
the radial Dirac equation: calculation of
scattering phase shifts
Karl-Erik Thylwe
KTH-Mechanics, Royal Institute of Technology, S-100 44 Stockholm, Sweden
E-mail: [email protected]
Received 7 February 2008
Accepted for publication 22 April 2008
Published 21 May 2008
Online at stacks.iop.org/PhysScr/77/065005
Abstract
Approaches inspired by a recent amplitude-phase method for analyzing the radial Dirac
equation are presented to calculate phase shifts. Regarding the spin- and pseudo-spin
symmetries of relativistic spectra, the coupled first-order and the decoupled second-order
differential forms of the radial Dirac equation are investigated by using a novel and the
‘classical’ amplitude-phase methods, respectively. The quasi non-relativistic limit c → +∞
of the amplitude-phase formulae is discussed for both positive and negative energies. In the
positive (E > mc2 ) low-energy region, the relativistic effects of scattering phase shifts are
discussed based on two scattering potential models. Results are compared with those of
non-relativistic calculations. In particular, the numerical results obtained from a rational
approximation of the Thomas–Fermi potential are discussed in some detail.
PACS numbers: 03.65.NK, 03.65.Pm.
described in some detail for the coupled first-order Dirac
equations [4], and a more classical amplitude-phase approach
is fully pursued for the decoupled second-order Dirac
equations. Formulae for calculating scattering phase shifts are
obtained, also in the non-relativistic limit, and the numerical
applications of the two amplitude-phase approaches are
compared.
In section 2, the different versions of the radial Dirac
equations are presented and certain relativistic (pseudo)
spin symmetries are pointed out. Section 3 introduces
the amplitude-phase decompositions for the two different
versions of the radial Dirac equations and phase-shift
formulae are derived. The amplitude-phase methods are
applied to two low-energy scattering systems in section 4,
one that has no relativistic symmetry and another that has a
so-called spin symmetry. Conclusions are given in section 5.
1. Introduction
Since its introduction in the 1930s [1], the amplitude-phase
method has frequently been discussed and improved in
studies of non-relativistic scattering and bound-state models
of atomic physics [2]. An amplitude-phase type approach for
solving coupled radial Schrödinger equations of scattering
states was recently presented in [3]. However, similar
decompositions of radial Dirac solutions into amplitudes and
phases seem to be rare, see a note in [4]. In an approximate
context, amplitudes and phases are also important ingredients
of semiclassical (WKB/phase-integral) approaches; see [5]
and further references therein. The basic idea with an
amplitude-phase decomposition is to find ‘almost constant’
(related to so-called adiabatic) quantities, and also to provide
a link to semiclassical mechanics that is formulated by
action-angle variables or using phase-integral techniques [6].
In the present work, the amplitude-phase approach
is discussed in connection with two different versions of
the radial Dirac equation containing a central Lorentz
four-vector potential with vanishing space-like components
combined with a scalar potential. The recent advances in
(pseudo) spin-symmetric Dirac systems are briefly considered
here, see [7]. A new type of amplitude-phase approach is
0031-8949/08/065005+08$30.00
2. Radial Dirac equations
The Dirac solutions for central potentials of the present type
are factorized into an angular part and a radial part (see
Ginocchio [7], Berestetskii et al [8], or Messiah [9]). In the
present analysis, the amplitude-phase approach is discussed
1
© 2008 The Royal Swedish Academy of Sciences
Printed in the UK
Phys. Scr. 77 (2008) 065005
Karl-Erik Thylwe
in connection with the radial two-spinor components F(r )
(upper) and G(r ) (lower); see [8].
where a prime (0 ) denotes differentiation with respect to r .
One may introduce the diagonal matrix coefficient
Q2 = − D2 + U2 + D0


κ(κ+1)
[E−Vv ]2 −[mc2 +Vs ]2
−
0
2
2
(h̄c)
r
,
=
κ(κ−1)
[E−Vv ]2 −[mc2 +Vs ]2
0
−
2
2
(h̄c)
r
2.1. First-order radial Dirac equations
The two coupled radial Dirac equations for central potentials
are, according to [7, 8, 10], given by
(2.8)
κ
(mc2 + Vs ) + E − Vv
dF
=− F +
G,
dr
r
h̄c
so that the equations can be written as
(2.1)
00
9 + Q2 9 = U0 9,
dG
κ
(mc2 + Vs ) − E + Vv
F + G.
=
h̄c
dr
r
with the coupling given by
The nonzero integer parameter κ in (2.1) is here related to the
orbital angular momentum quantum number ` and is given by
κ = −(` + 1) 6 −1 for ` = 0, 1, 2 . . ., and κ = ` > 1 for
` = 1, 2 . . .. One observes that the roles of the Dirac
components F and G are interchanged by the simultaneous
substitutions κ → −κ, E → −E and Vv → −Vv . To proceed,
let the equations (2.1) be written in matrix form with separated
diagonal and off-diagonal coefficient functions, i.e.
d9(r )
= (D + U) 9(r ),
dr
U =
F
9=
G
U=
mc2 +Vs −(E−Vv )
h̄c
Vv0 −Vs0
h̄c
0
V 0 −V 0
0
(2.3)
.
(2.4)
=
One thus finds
0
− E−Vvv+mcs2 +Vs
0
V 0 +V 0
v
s
− E−Vv −mc
2 −V
s
00
ψ + R ψ = 0,
where
U
=
0
h̄c
mc2 +Vs +E−Vv
h̄c
mc2 +Vs −(E−Vv )
0
.
(2.10)
with
!
.
!
.
(2.12)
As a final exact transformation, the first-order derivatives
in the diagonal equation (2.11) can be removed by putting
F
9=
G
(E − Vv + mc2 + Vs )1/2
0
f
=
.
0
(E − Vv − mc2 − Vs )1/2
g
(2.13)
!
0
−1
UU
The original equation (2.2) implies the following useful
identity:
"
#
−1 d9
9 =U
− D9 ,
(2.5)
dr
−1
!
with
and the off-diagonal matrix U is defined as
mc2 +Vs +E−Vv
h̄c
Vv0 +Vs0
h̄c
−
In equation (2.9), the coupling in the term containing U0 can
partly be removed for two relativistic symmetries [7]: the socalled spin symmetry is characterized by d(Vv − Vs )/dr = 0;
and the so-called pseudo-spin symmetry is characterized by
d(Vv + Vs )/dr = 0. Hence, the upper/lower Dirac components
become one-sided decoupled for Vv − Vs = 0 and for
Vv + Vs = 0.
In the general case, the coupling between components
is removed by substituting the identity (2.5) into the right
hand side of (2.9), thus yielding the completely decoupled
equations
00
0 9 − U0 U−1 9 + Q2 + U0 U−1 D 9 = 0,
(2.11)
(2.2)
is a column solution, D is a diagonal matrix
κ
−r 0
D=
κ ,
0
r
0
0
where
0
(2.9)
(2.6)
Relation (2.5) can be used to express any of the solution
components entirely in terms of the other one, which will be
used in section 2.2
Rf =
2.2. Second-order decoupled radial Dirac equations
By differentiating equation (2.2) and using the same equation
to eliminate the first derivative of 9 one obtains
00
9 = D2 + U2 + D0 9 + U0 9,
(2.7)
2
f
ψ=
,
g
Rf
0
R=
,
0 Rg
(2.14)
(2.15)
(2.16)
(E − Vv )2 − (mc2 + Vs )2 κ(κ + 1)
−
(h̄c)2
r2
0
0
Vv − Vs
κ
Vv00 − Vs00
+
−
mc2 + Vs + E − Vv r
2 mc2 + Vs + E − Vv
2
3
Vv0 − Vs0
−
(2.17)
4 mc2 + Vs + E − Vv
Phys. Scr. 77 (2008) 065005
Karl-Erik Thylwe
and
Rg =
with − Ẽ − = E + mc2 :
(E − Vv )2 − (mc2 + Vs )2 κ(κ − 1)
−
(h̄c)2
r2
Vv0 + Vs0
κ
Vv00 + Vs00
+
+
mc2 + Vs − (E − Vv ) r 2 mc2 + Vs − (E − Vv )
2
3
Vv0 + Vs0
−
.
(2.18)
4 mc2 + Vs − (E − Vv )
Rf =
For understanding the non-relativistic (Schrödinger) limit
(c → +∞) for which the positive scattering energies E > mc2
are approaching mc2 , it may be instructive to introduce Ẽ + =
E − mc2 in the coefficients R f and Rg . One thus obtains the
equivalent expressions
Rf =
Rg =
2m +
κ(κ + 1)
( Ẽ − Vv − Vs ) −
r2
h̄ 2
Vv0 − Vs0
κ
Vv00 − Vs00
+
−
2mc2 + Ẽ + − Vv + Vs r
2[2mc2 + Ẽ + − Vv + Vs ]
2
( Ẽ + − Vv )2 − Vs2 3
Vv0 − Vs0
+
−
(h̄c)2
4 2mc2 + Ẽ + − Vv + Vs
(2.19)
−
( Ẽ − Vv )
(h̄c)2
+
+
2
− Vs2
.
2m −
κ(κ − 1)
( Ẽ + Vv − Vs ) −
r2
h̄ 2
0
0
Vv + Vs
Vv00 + Vs00
κ
+
+
2mc2 + Ẽ − + Vv + Vs r 2(2mc2 + Ẽ − + Vv + Vs )
2
( Ẽ − + Vv )2 − Vs2 3
Vv0 + Vs0
+
−
.
(h̄c)2
4 2mc2 + Ẽ − + Vv + Vs
(2.22)
For ‘negative’ scattering energies E < −mc2 ( Ẽ − > 0) it is
F that tends to zero as c → +∞ and the lower component G
(or g) becomes the physically important one. The coefficient
Rg simplifies again for the pseudo-spin symmetry (d(Vv +
Vs )/dr = 0) and becomes invariant with respect to the
substitution κ → −κ + 1. Similarly, R f simplifies again for
the spin symmetry (d(Vv − Vs )/dr = 0).
2m +
κ(κ − 1)
( Ẽ − Vv − Vs ) −
2
r2
h̄
2
Vv0 + Vs0
Vv0 + Vs0
κ 3
−
−
4 Ẽ + − Vv − Vs
Ẽ + − Vv − Vs r
Vv00 + Vs00
2( Ẽ + − Vv − Vs )
(2.21)
and
and
Rg =
2m −
κ(κ + 1)
( Ẽ + Vv − Vs ) −
2
r2
h̄
Vv00 − Vs00
Vv0 − Vs0
κ
+
−
Ẽ − + Vv − Vs r 2[ Ẽ − + Vv − Vs ]
2
3
Vv0 − Vs0
( Ẽ − + Vv )2 − Vs2
−
+
4 Ẽ − + Vv − Vs
(h̄c)2
2.3. Scattering boundary conditions
At the origin r = 0 the physically relevant Dirac solution is
assumed to be bounded or integrable in some sense; see [11].
The boundary condition for this solution as r → +∞ defines
the scattering phase shifts for each partial wave [8]. From
the first-order differential equations (2.1) together with (2.13),
(2.14) and (2.17) one can thus write for the present type of
central short-range potentials
(2.20)
Obviously the components f and g behave quite differently as
c → +∞. In this limit, the component G (unlike g) vanishes
and it is the parameter function R f and the component f
that become of primary interest. One easily identifies the
terms corresponding to the Schrödinger limit, where Ẽ + then
corresponds to the non-relativistic Schrödinger energy. Note,
however, that the terms in the second and third lines in
(2.19) cannot be rigorously neglected for singular potentials
at sufficiently small radial distances.
For the spin symmetric case (d(Vv − Vs )/dr = 0) it is
seen that the coefficient R f (see equations (2.17) and (2.19))
simplifies considerably. In particular, the equation for the
radial component f becomes invariant with respect to the
substitution κ → −κ − 1. The coefficient Rg , however, does
not simplify here. In fact, it may contain singular terms for
r > 0 if the potentials are repulsive and Ẽ + > 0. For the
pseudo-spin symmetric case (d(Vv + Vs )/dr = 0) it is Rg that
simplifies and it is R f that may contain singular terms. The
radial component g becomes invariant with respect to the
substitution κ → −κ + 1. Note that the symmetries imply an
unspecified constant in the potential sum Vv + Vs that causes a
shift in the scattering energy range.
For the sake of completeness, one should also take a
closer look at the equations for the negative-energy scattering
states in the limit c → +∞. In the same way as above, one
derives the following equivalent expressions for R f and Rg
F ∼ (E + mc2 )1/2 f
∼ (E + mc2 )1/2 sin kr − π `/2 + δ`,κ ,
G ∼ (E − mc2 )1/2 g
∼ (E − mc2 )1/2 cos kr − π `/2 + δ`,κ ,
r → +∞,
(2.23)
r → +∞,
where, according to [8], δ`,κ is the scattering phase shift
parameterized by ` and the Dirac parameter κ, and k is the
asymptotic wavenumber given by
k=
E 2 − m 2 c4
h̄ 2 c2
1/2
.
(2.24)
3. Amplitude-phase decompositions
The preceding section introduced several equivalent forms
of the radial Dirac equation. The second-order decoupled
equation (2.14) containing no first-order derivatives are of the
form that has been analyzed by amplitude-phase methods in
several earlier articles [2]. Recently, a novel amplitude-phase
method [3] was developed for analyzing also second-order
coupled equations of the type (2.9). With these earlier results
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Phys. Scr. 77 (2008) 065005
Karl-Erik Thylwe
in mind, it is not surprising that one can approach the
original first-order coupled Dirac equations (2.1) by a similar
amplitude-phase ansatz, see a recent note [4]. This section
concludes with a discussion of the quasi non-relativistic
(Schrödinger) limit c → +∞.
The non-normalized, real, Dirac components are from (2.13),
(3.2) and (3.7) given by
3.1. Second-order decoupled radial Dirac equations
containing no first-order derivatives
Possible branch points in the factors [E − Vv ± (mc2 + Vs )]1/2
in (3.9) are avoided with the potential models and energies
discussed in the present work. These factors can be taken
merely as indicators of the relative sizes of the Dirac
components. Hence, scattering with positive energies is
dominated by F in the non-relativistic limit, and scattering
with negative energies (E < −mc2 ) is dominated by G in the
non-relativistic limit.
F = k 1/2 (E − Vv + mc2 + Vs )1/2 u f sin φf ,
G = k 1/2 (E − Vv − mc2 − Vs )1/2 u g sin φg .
Since f - and g-solutions satisfy decoupled second-order
ordinary differential equations of the standard single-channel
type, one can write the basic real-valued amplitude-phase
solutions as (see [2])
f b = u f sin φ f ,
φ 0f = u −2
f ,
gb = u g sin φg ,
φg0
=
u −2
g ,
with φ f (r ) → 0,
with φg (r ) → 0,
r → +0,
r → +0,
(3.1)
3.2. First-order coupled radial Dirac equations
The amplitude-phase method has very recently been discussed
in the case of coupled second-order ordinary differential
equations of the Schrödinger type [3]. Being consistent with
this approach, the second-order version of the radial Dirac
equation, equation (2.9), is solved by a solution ansatz
(see also [4]):
9 = u exp(iφ),
(3.10)
uF
where u =
, and φ is a real scalar phase (the same for
uG
both amplitude components) satisfying
or, alternatively, as
f (±) = u f exp ±iφ f , φ 0f = u −2
f , with φ f (r ) → 0,
r → +0,
(±)
g
= u g exp ±iφg , φg0 = u −2
g , with φg (r ) → 0,
r → +0,
(3.2)
for the more general exponential (Jost-type) solution
components. The amplitude functions u f and u g for any of the
solutions above, obtained from the decoupled equation (2.14),
has to satisfy the Milne-type equations given by [1]
u 00f + R f u f = u −3
f ,
φ 0 = u −2 ,
(3.3)
−1/4
(+∞) = k −1/2 ,
u 0f (+∞) = 0,
∗
Note that u f and u g introduced in the previous subsection in
(3.1) and (3.2) are real-valued functions unlike the complex
valued amplitude functions u F and u G introduced in the
present subsection. The real phase φ in this approach is thus a
‘common’ phase for the amplitude components. While φ is an
accumulated real phase, it combines in this section with the
two ‘local’ phases and two absolute values of the complex
amplitude components, so that one has a total of five real
quantities (instead of four) that defines a fundamental set
of solutions. The extra quantity has to be accompanied by
an auxiliary equation, which in this case is the fundamental
phase-amplitude relation (3.11). The success of introducing
the extra quantity φ will ultimately rely on the numerical
behaviors of the amplitude components.
To proceed, on substitution into equation (2.2), the
amplitude u in (3.10) turns out to satisfy the first-order
differential equation (cf [4])
(3.5)
f = k 1/2 f b ,
(3.7)
g=k
(3.11)
9 = u ∗ exp(−iφ).
u g (+∞) = Rg−1/4 (+∞) = k −1/2 , u 0g (+∞) = 0.
(3.6)
These amplitude solutions are expected to be almost constant
from r = +∞ until the approach of the transition points
(where Rg, f = 0) and the inner region containing the origin.
The asymptotic boundary conditions (3.5) and (3.6) above for,
respectively, u f and u g are obviously the same.
Finally, the physical solutions satisfying the
real-parameter problem with boundary conditions (2.23)
can be represented by
1/2
with u 2 = u • u ∗ = |u F |2 + |u G |2 .
An independent solution of the real Dirac equation (2.9) is
obviously given by the complex conjugate of (3.10), i.e.
u 00g + Rg u g = u −3
(3.4)
g .
Often, for slowly varying central scattering potentials such
that R f and Rg are negative (a classically forbidden region)
as r → +0, it is known to be computationally advantageous
for calculating u f and u g to use the boundary conditions at
infinity, i.e.
u f (+∞) = R f
(3.9)
gb ,
u 0 = (D + U − i φ 0 1) u,
(3.12)
where 1 is a unit matrix. All basic amplitude-phase equations
(3.3), (3.4) and (3.12) are nonlinear equations, but among
them (3.12) appears to have the simplest coefficients.
The central idea with the amplitude-phase analysis is
to find (and to make use of) ‘almost constant’ solutions
of the present type of nonlinear amplitude equations. It is
and the phase shifts are determined from either of the
expressions below; see (2.23) and (3.1):
δ`,κ = lim φ f − kr + π`/2
r →+∞
= lim φg − kr + π(` − 1)/2.
(3.8)
r →+∞
4
Phys. Scr. 77 (2008) 065005
Karl-Erik Thylwe
important to note here that the method is still exact even
if arbitrary solutions are used for the amplitude quantities.
However, computational advantages with the method are only
expected for special, slowly varying amplitude solutions.
Such ‘almost constant’ solutions must be determined from
special boundary conditions. Thus, since the derivatives of
the parameter functions D and U become zero as r → +∞
one may actually find ‘almost constant’ solutions from a
boundary condition there. Hence, assuming stationary (exact
constant) amplitude functions in the limit r → +∞ one
obtains boundary conditions satisfying
(3.13)
U(+∞) − i φ 0 (+∞)1 u(+∞) = 0.
3.3. Non-relativistic limit
The relativistic energy is assumed to be positive here. The
second-order decoupled amplitude equation (3.3) with the
parameter function R f (r ) written in the same way as (2.19)
provides a direct link between the ‘upper’ radial Dirac
component and the radial solution of the Schrödinger theory.
One just needs to consider the equation and the boundary
conditions for the amplitude u f (r ), and to make the following
substitutions:
!1/2
2m Ẽ +
,
(3.21)
k → k̃ =
h̄ 2
Using equations (2.4) and (2.24) and assuming φ 0 (+∞) > 0
the eigenvalue problem (3.13) implies the boundary
conditions
φ 0 (+∞) = k,
(3.14)
with k defined by (2.24), and
(E + mc2 )1/2
−1/2
,
u(+∞) = (2Ek)
i(E − mc2 )1/2
κ → −(` + 1),
R f (r ) → R̃ +f (r ) =
(3.15)
where the factor (2Ek)
is determined from the relation
(3.11). With these boundary conditions one can thus integrate
(3.12) to obtain the special amplitude-phase solution 9
in (3.10). The physical Dirac solution satisfying (2.1)
and the boundary conditions (2.23) is then found to be
some real-valued linear combination of the amplitude-phase
solution 9 and its complex conjugate, or, equivalently the real
or imaginary part of 9 given by (3.10).
A derivation of a phase shift formula using the first-order
coupled amplitude-phase method may now be achieved by
using just one of the component solutions, say F(r ). Hence,
let the complex amplitude component u F (r ) be written as
U→
− Ẽ
+
−Vv −Vs
h̄c
2mc2
h̄c
0
!
.
`+1
h̄
0
0
uF−
− iφ u F .
uG =
2mc
r
(3.24)
(3.25)
This result suggests that u G vanishes as c → +∞, while
the term (2mc/h̄) u G appearing in (3.12) will still be finite.
Therefore, the contribution of u G in the phase φ 0 in (3.11)
can be neglected, but otherwise u G remains significant in
the differential equation (3.12). In fact, u G as well as c turn
out to become auxiliary quantities in the calculation of the
component amplitude u F . It is then convenient to introduce
a new auxiliary function that also eliminates c from the
differential equations, for example:
(3.16)
The constant A is not important here, but the phase integral
φ(r ) is to be specified so that F(r ) satisfies the boundary
condition at the origin. Assuming [2, 11]
0 6 r < +∞,
0
One element of this matrix is singular as c → +∞, indicating
that special care is required. Hence, from the amplitude
equations (3.12) one obtains for not too small values of r , but
sufficiently large that the central potentials have vanished:
so that one can write, from (3.10), a real representation of the
‘upper’ spinor component F(r ) as
F(r ) = A|u F (r )| sin φ(r ) + arg u F (r ) .
(3.17)
|u F (r )| 6= 0,
2m +
`(` + 1)
( Ẽ − Vv − Vs ) −
. (3.23)
2
r2
h̄
For the coupled first-order amplitude-phase solutions (3.12)
this means, provided r is not too small, that
−1/2
u F (r ) = |u F (r )|ei arg u F (r ) ,
(3.22)
ũ G = h̄cu G ,
(3.26)
so that (3.12) transforms to the non-relativistic equations
(3.18)
(3.19)
! 2m
−iφ 0 + `+1
uF
u 0F
2
r
h̄
=
. (3.27)
ũ 0G
ũ
−( Ẽ + − Vv − Vs ) −iφ 0 − `+1
G
r
Furthermore, in the present problem one has
arg u F (+∞) = 0, so that the phase shift defined in (2.23) is
obtained from (3.17) by the formula
The transformation means that the boundary condition (3.14)
becomes
φ 0 (+∞) → k̃,
(3.28)
δ`,κ = lim (φ(r ) − kr ) + π`/2.
and that the ‘stationary’ amplitude solutions satisfying (3.27)
as r → +∞ become
1
u F (+∞)
−1/2
= k̃
.
(3.29)
h̄ 2
ũ G (+∞)
i 2m
k̃
it is thus required that
φ(0) = − arg u F (0).
r →+∞
(3.20)
Note that the phase φ(r ) defined here is obtained by
numerically integrating φ 0 along with the vector amplitude u
using the boundary conditions (3.14), (3.15) and (3.19).
5
Phys. Scr. 77 (2008) 065005
Karl-Erik Thylwe
Computations are performed using atomic units (m =
e = h̄ = 1) and taking the speed of light approximately as
c = α −1 ≈ 137, α being the fine structure constant. The
relevant numerical integrations use r = rmax ≈ 5 × 103 and
r = rmin ≈ 10−8 and the numerical tolerance is 10−8 . The
MatLab integrator ‘ode23’ for non-stiff ODE equations
appears to be more accurate than the faster routine ‘ode15s’
suitable for stiff ODEs. The methods (AP1) and (AP2) are
similar as concerns computational speeds and accuracies.
The first-order radial Dirac equation (3.12) do not
decouple as r → +∞. One needs to calculate small and large
Dirac components in the whole range of r . Furthermore,
the amplitudes are complex valued, unlike those of equation
(3.3). When starting the integration at a finite distance, say at
r = rmax , one may try to modify the boundary conditions
(3.15) by replacing the exact values at infinity with
approximate ones (for example an adiabatic solution that
neglects derivatives of the solution itself) at r = rmax . The
first-order Dirac equations then reduce to an approximate
eigenvalue problem at r = rmax , i.e.
!
κ
E+mc2
−iφ 0 (rmax ) − rmax
u F (rmax )
h̄c
2
κ
u G (rmax )
−iφ 0 (rmax ) + rmax
− E−mc
h̄c
0
(4.2)
≈
.
0
4. Application
In an early presentation [4], the new amplitude-phase
approach applicable to the first-order coupled Dirac equations
gave accurate numerical S-matrix results for the four-vector
Coulomb potential with zero space components. For Coulomb
potentials, one can easily find exact close-form analytic
results relevant for scattering and also bound-state energies.
However, in the present applications no exact results are yet
available. Instead, one may use the consistency between the
new and the classical amplitude-phase approach to estimate
the accuracy of the results here. The reliability of the
classical amplitude-phase method is well documented even
for calculations with complex energies and complex orbital
angular momenta [2, 3].
4.1. Relativistic rational-fraction Thomas–Fermi
potential models
Two relativistic potential models that reduce to the same
non-relativistic Schrödinger potential, i.e. Vv (r ) + Vs (r ), are
considered for positive scattering energies (E > mc2 ); see
equation (3.23). The non-relativistic model is a particular
so-called rational approximation of the Thomas–Fermi (RTF)
potential that has been discussed recently in the context
of low-energy scattering and complex-angular momentum
(Regge-pole) resonances (see [12, 13]):
−Z e2
VRTF (r ) =
,
r (1 + a Z 1/3r ) (1 + bZ 2/3r 2 )
a = 0.0126,
One finds the solutions of the components of u(rmax ) by a
standard eigenvalue technique. A slowly varying (adiabatic)
3
solution that neglects terms of the order O(1/rmax
) for
kr max κ is given by
(4.1)
b = 1.5874.
1/2
κ2
φ 0 (rmax ) ≈ k 2 − 2
,
rmax
Here, Z (> 0) is the nuclear charge number, e > 0 is the
Coulomb charge unit, and a, b are parameters provided by
Mzesane et al [12] given in atomic units.
(4.3)
with k defined by (2.24), and

1/2 
κ
2 1/2
0
(E
+
mc
)
φ
(r
)
+
i
max
u F (rmax )


rmax
=N
1/2  .
u G (rmax )
κ
2 1/2
0
i(E − mc )
φ (rmax ) − i rmax
4.1.1. Non-symmetric RTF model. The first potential model
consists of a four-vector time component Vv (r ) ≡ VRTF (r )
with Vs (r ) ≡ 0. None of the relativistic symmetries are
relevant here. Hence, the phase shifts δ`,κ are expected
to be different for different values of ` and κ unless
these parameters are sufficiently large, in which case the
radial wave components mainly stay outside of the potential
region.
(4.4)
The normalization factor N is determined from (3.11) and is
now given by
N = (2φ 0 k E)−1/2 .
(4.5)
4.1.2. Spin-symmetric RTF model. The other model has the
relativistic spin symmetry Vv (r ) ≡ Vs (r ) ≡ 21 VRTF (r ) defined
with the same parameters as in (4.1); see [14]. One expects
from the relativistic equations that the phase shifts will satisfy
δ`,κ = δ`,−κ−1 as in the non-relativistic theory.
With the improved boundary conditions above one may
integrate (3.12) together with the ‘reduced’ integrand φ 0 − k
towards the origin. By proper care of the reverse direction of
integration one obtains by this procedure:
Z
rmax
4.2. Numerical aspects and results
(φ 0 − k)dr = φ(rmax ) − φ(rmin ) + k(rmin − rmax ).
rmin
(4.6)
Here φ(rmin ) ≈ −argu F (rmin ) and one finally calculates the
phase shift in (3.20) from the expression
Z rmax
(φ 0 − k) dr − krmin − arg u F (rmin )
δ`,κ ≈
For numerical comparisons, the ‘new’ amplitude-phase
method based on the first-order differential equation (3.12),
denoted (AP1), is compared with the classical method that
uses the second-order differential equation (3.3), presently
denoted (AP2). The latter equation has the familiar form that
is known to provide reliable and accurate numerical results
with the classical amplitude-phase method [2].
rmin
+ π `/2 + γ (rmax ),
6
(4.7)
Phys. Scr. 77 (2008) 065005
Karl-Erik Thylwe
5. Conclusions
Table 1. Phase shift results for the RTF potential. Here (D) denotes
the consistent relativistic amplitude-phase results for the
non-symmetric potential model, (NR) the consistent
amplitude-phase result in the non-relativistic Schrödinger limit, and
(D-SS) denotes the consistent relativistic amplitude-phase result for
the spin symmetric potential model. Ẽ + = 10−5 × mc2 au and
Z = 36 (krypton).
`
κ
δ`,κ (D)
δ`,κ (NR)
δ`,κ (D − SS)
0
1
1
2
2
3
3
−1
1
−2
2
−3
3
−4
10.781398
7.240633
7.240633
3.336945
3.336945
0.083086
0.083086
10.781432
7.240663
7.240663
3.336946
3.336946
0.083086
0.083086
10
10
10
−11
10.883515
7.308504
7.263108
3.337161
3.337020
0.083087
0.083086
···
0.005931
0.005931
···
0.005931
0.005931
0.005931
0.005931
Two equivalent versions of the radial Dirac equation with
a central (time-like) four-vector potential together with
a scalar potential are discussed as starting points for
calculating accurate numerical scattering phase shifts. The
‘classical’ amplitude-phase method (AP2) using a single
(one-component) second-order ordinary differential equation
is found to be stable and accurate. Some limitations in this
approach for more general potentials that are singular at the
origin are already recognized in the literature [11]. A new
amplitude-phase method (AP1) using the coupled first-order
radial Dirac equations (2.1) is discussed in some detail in
this study, see also a recent note [4]. In this approach, the
phase can also be defined to be real. The stationary condition
at r = +∞ for the initial amplitude functions is generalized
in this work by adiabatic considerations to a more useful
condition at large finite distances.
The ‘new’ amplitude-phase method (AP1) turns out to be
significantly simpler than the ‘classical’ method (AP2) [6]. No
lengthy transformations involving derivatives of the potential
are needed. Furthermore, it is shown in section 3.3 that
this new method applies also to the non-relativistic radial
Schrödinger equation. Numerical calculations in section 4
show that relativistic corrections may be physically significant
for the first few partial waves also at quite low energies
(see [12]). It also turns out that the spin symmetric RTF
potential model provides phase shifts that agree better with
those in the non-relativistic limit than the asymmetric RTF
model does.
Table 2. Same as for table 1 but with Z = 54 (xenon).
`
κ
δ`,κ (D)
δ`,κ (NR)
δ`,κ (D − SS)
0
1
1
2
2
3
3
−1
1
−2
2
−3
3
−4
12.757974
9.822773
9.822773
3.372859
3.372859
0.092530
0.092530
12.758008
9.822795
9.822795
3.372861
3.372861
0.092531
0.092531
10
10
10
−11
12.959700
9.937276
9.865572
3.373650
3.373160
0.092532
0.092532
···
0.006357
0.006357
···
0.006357
0.006357
0.006357
0.006357
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(4.8)
that can be solved analytically. In this study
γ (rmax ) ≈ −
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+
+
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2 krmax 24 k 3 rmax
80 k 5 rmax
is used.
The results in table 1 correspond to the nuclear charge
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methods) for estimating the relativistic effects. It is obvious
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energies where also resonances are expected to occur.
Finally, in tables 1 and 2 the phase shifts δ`,κ (D − SS) are
the agreeing results of methods (AP1) and (AP2) for the spin
symmetric potential model.
7
Phys. Scr. 77 (2008) 065005
Karl-Erik Thylwe
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8