Download Scattering Matrix Formulation of the Total Photoionization of Two

Journal of the Korean Physical Society, Vol. 56, No. 6, June 2010, pp. 1799∼1806
Scattering Matrix Formulation of the Total Photoionization
of Two-electron Atoms
Min-Ho Lee and Nark Nyul Choi∗
School of Natural Science, Kumoh National Institute of Technology, Gumi 730-701
(Received 15 April 2010, in final form 27 April 2010)
A rigorous derivation of an exact quantum-mechanical formula is presented for the total photoionization of two-electron atoms in terms of local scattering matrices that describe local dynamics
separately in the inner and the outer parts of a surface dividing the whole configuration space. The
exact formula is a correct extension of a previously known formula that misses the contribution of
the channels closed at the dividing surface. The validity of semiclassical treatments of the outer
part can be examined by evaluating the contribution of channels closed at the dividing surface.
PACS numbers: 32.80.Fb, 03.65.Sq, 34.80.Kw, 03.65.Nk
Keywords: Scattering matrix, Photoionization cross section, Green’s function, Semiclassical method, Twoelectron atom
DOI: 10.3938/jkps.56.1799
I. INTRODUCTION
Following the discovery of doubly-excited states by
Madden and Codling [1], low-energy resonance spectra
of two-electron atoms were at first explored from the
viewpoint of the configuration interaction and the quantum defect theory [2,3]. Also, it was found that low-lying
doubly-excited states could be labelled by approximate
quantum numbers that could be understood in terms
of group-theoretical quantities [4,5]. Further progress in
both experiment and numerical methods made it possible
to observe irregular fluctuations in the photoionization
spectrum beyond the region of low-lying doubly-excited
states and to examine a breakdown of the approximate
quantum numbers [6]. As the energy approached the
double-ionization threshold, the resonances belonging to
different N -manifolds became strongly mixed, and the
distribution of nearest neighbor spacings became closer
to the Wigner distribution representing a chaotic system
[7]. The numerical and the experimental results in Ref. 8
clearly show a selective breakdown of labelling individual
resonances in terms of approximate quantum numbers.
Recognizing a two-electron atom as a kind of restricted
three-body system which is well known to be nonintegrable, Richter et al. [9] showed that its classical dynamics had a mixed phase space containing regular and
chaotic regions by investigating the structures of three
invariant subspaces. Semiclassical approaches based on
their findings in the underlying classical dynamics allowed progress in qualitative understanding of the pho∗ Corresponding
Author; E-mail: [email protected]
toionization spectrum: for example, the widely different
resonance widths in the same N -manifold, the regularity
in the resonances classified as corresponding to a part
of the phase space around the collinear nucleus-electronelectron configuration, and the adiabatic coordinate systems [6,9]. However, the question “what happens to twoelectron resonances when their energy approaches the
break-up threshold?” [10], i.e., “what is the characteristic of the chaotic signal of photoionization when their
energy appoaches the double-ionization threshold?”, still
remains far from being resolved without understanding
the main mechanism of the chaos in two-electron atoms.
Recently, a series of studies on the structure of phase
space beyond the invariant subspaces revealed that the
triple collision is a main source of chaos in two-electron
atoms; i.e., the chaotic part is mainly generated by the
complex folding patterns of the stable and the unstable manifolds of the triple collision point and the double
escape point [11]. Based on their findings on the phase
space structure around the stable and the unstable manifolds of the triple collision, Byun et al. [12] predicted that
the fluctuation (i.e., resonant part) in the total photoionization cross section of two-electron atoms was linked to
a set of infinitely unstable classical orbits starting and
ending in the nonregularizable triple collision - the socalled closed triple collision orbits (CTCO) - and that
the amplitude of the fluctuation decayed algebraically
as the energy approached E = 0, the double-ionization
threshold, with an exponent determined by the triple
collision singularity. They used a semiclassical argument
based on a quantum-mechanical formula for the total
photoionization cross section in terms of local scattering
matrices. The formula was introduced earlier by Granger
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Journal of the Korean Physical Society, Vol. 56, No. 6, June 2010
and Greene [13] for a semiclassical treatment of quantum
chaos in diamagnetic atoms. However, the formula is a
kind of approximate one because the contributions from
channels closed at the surface dividing the inner and the
outer configuration spaces were not included in the formula.
In this paper, we will present a detailed derivation of
an exact quantum-mechanical formula for the total photoionization cross section of two-electron atoms in terms
of local scattering matrices. For that purpose, we introduce a surface that divides the whole configuration
space into inner and outer parts by applying the quantum surface-of-section technique [14–17]. The paper is
organized as follows: The basic ingredients, such as hyperspherical coordinates and local scattering matrices,
are introduced in Sec. II.. A detailed derivation of the formulas for the photoionization cross section is presented
in Sec. III.. In Sec. IV., discussions are given with numerical estimates of the contribution of closed channels.
1. Hyperspherical Coordinates and Adiabatic
Channels
The Scrödinger equation for a two-electron atom in
the infinite-nucleus-mass approximation is
·
¸
1
1
Z
Z
1
− 521 − 522 − −
+
− E Ψ̃(r1 , r2 ) = 0,
2
2
r1
r2
r12
(1)
where r1 and r2 are the position vectors of the two electrons, r12 is the inter-electron distance, and Z is the
charge of the nulceus. Atomic units (~ = e = me =
4π²0 = 1) are used throughout. The hyperspherical coordinates {R, Ω = (α, r̂1 , r̂2 )} with
µ ¶
q
ri
r2
R = r12 + r22 , α = tan−1
, r̂i =
(2)
r1
ri
are well known to be suited for describing the electronelectron correlations, and the Schrödinger equation in
these coordinates is written as [3,18]
¶
µ
1 ∂2
+ HR Ψ = EΨ
(3)
−
2 ∂R2
with Ψ(R, Ω) = R5/2 sin(α) cos(α) Ψ̃(r1 , r2 ), where Ψ̃
denotes a solution of Eq. (1) in Cartesian coordinates.
The Hamiltonian HR with
1 Λ2
1
+ V (Ω)
2
2R
R
which is the angular momentum operator in 6 dimensions [3, 5]. Here, li denotes the 3d angular momentum operator for electron i, and V (Ω)/R is the threebody Coulomb potential written in hyperspherical coordinates:
V (Ω) = −
Z
Z
1
. (5)
−
+√
cos α sin α
1 − r̂1 · r̂2 sin 2α
The adiabatic Hamiltonian in Eq. (4) acts on five angle
variables Ω alone and has a discrete spectrum; that is,
HR Φn (Ω; R) = Un (R) Φn (Ω; R)
(6)
with adiabatic channel functions Φn and adiabatic potentials Un depending parametrically on R. The set of adiabatic channel functions at a given R will be used in the
following as a complete basis set for the 5-dimensional
hypersurface at R.
II. BASIC INGREDIENTS FOR THE
SCATTERING APPROACH TO
PHOTOIONIZATION
HR =
depends parametrically on the hyperradius R with the
‘grand angular momentum’ operator Λ defined as
¶
µ
1
l21
l22
∂2
+
− ,
Λ2 = − 2 +
∂α
cos2 α sin2 α
4
(4)
2. Local Scattering Matrix
For quantization of multidimensional bound systems,
Bogomoly [14] introduced semiclassical Poincaré mapping as an analogue of the surface-of-section reduction of
classical dynamics. Smilansky and coworkers [15,16] developed a scattering approach leading to a construction
of exact quantum Poincaré mapping for two-dimensional
billiards. The scattering approach was further generalized to almost arbitrarily bound Hamiltonian systems by
Prosen [17]. Some of the techniques developed for quantum Poincaré mapping are quite suitable for the present
purpose although photoionization is apparently far from
quantization of bound systems. We will summarize them
here with a slight adaption for the photoionization of
two-electron atoms.
Let φi be the initial bound state. A hyperradius R0
is chosen such that the surface Σ0 defined as R = R0
encloses the support of the initial state φi . The surface
naturally leads to a partition of the whole configuration
into physically distinct regions. Specifically, the dynamics in the inner region is insensitive to the total energy
of the system and produces a smooth background part
in the photoionization spectrum while the outer region is
responsible for the complex resonance structure near the
double-ionization threshold [12]. The dynamics in each
region can be separately treated by introducing two independent inner (I) and outer (O) scattering systems.
The I system is formed by the potential on the inner
side of Σ0 , and its constant continuation on the outer
Scattering Matrix Formulation of the Total Photoionization of Two-electron · · · – Min-Ho Lee and Nark Nyul Choi
H O , which is given by
T Nn
r mn
s mn
Σ0
H
RMN
t nN
side (cf. Fig. 1 in Ref. 17), i.e., its Hamiltonian H I is
defined as
½
1 ∂2
HR for R < R0
I
H =−
+
(7)
2
HR0 for R > R0 .
2 ∂R
The O system is similarly defined by the Hamiltonian
½
p
E −p
Um (R0 )
iκm = i Um (R0 ) − E
1 ∂2
=−
+
2 ∂R2
½
HR0 for R < R0
HR for R > R0 .
ψnI(+) (R, Ω) =
(8)
∞
X
Φm (Ω; R0 )
√
2πkm
m=1
h
× e−ikm (R−R0 ) δmn
i
+e+ikm (R−R0 ) smn ,
(9)
where Φm (Ω; R0 ) is the adiabatic channel functions,
Eq. (6), with adiabatic potential Um (R0 ) at R = R0 ,
the channel wavenumber km is defined by
for open channels, i.e., E ≥ Um (R0 )
for closed channels, i.e., E < Um (R0 ),
and s is an inner-region scattering matrix, which contains
all the information about the correlated two-electron dynamics near the nucleus.
For the O scattering system, there are two kinds of
ψnO(+) (R, Ω)
O
In the region R ≥ R0 , the solutions of the I scattering
system with outgoing boundary conditions can be written in the form
Fig. 1. (Color online) Schematic diagram for the dividing
surface Σ0 and the local scattering matrices s, r, t, R, and T .
km =
-1801-
(10)
incident wave, one coming from the left (i.e., from R <
R0 ) or one from the right (i.e., from R = ∞). The
asymptotic form of the solutions with outgoing boundary
conditions is written as
£
¤ √
½ P∞
)
Φm (Ω; R0 ) e+ikm (R−R0p
δmn + e−ikm (R−R0 ) rmn / 2πkm for R ≤ R0
m=1
= P∞
0
+ikN
R
0
TN n / 2πkN
for R → ∞
N =1 ΦN (Ω; ∞) e
(11)
for incident waves from the left and
( P∞
O(+)
ψN (R, Ω)
=
√
e−ikm (R−R0 ) tmN / 2πkm
m=1 Φm (Ω; R0 ) h
i p
P∞
0
0
−ikM
R
+ikM
R
0
Φ
(Ω;
∞)
e
δ
+
e
R
/ 2πkM
M
M
N
M
N
M =1
for incident waves from the right. Here, ΦM (Ω; ∞) and
0
kM
are, respectively, the adiabatic channel functions and
the wavenumbers at R = ∞, and r, t, R, and T are outerregion scattering matrices containing all the information
on the energy-sensitive two-electron dynamics outside
the core region. A schematic diagram is presented in
for R ≤ R0
for R → ∞
(12)
Fig. 1 for a pictorial representation of the dividing surface and the local scattering matrices.
As in Ref. 16, by using the reality of the adiabatic
potentials, one can find a set of relations in the matrix
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Journal of the Korean Physical Society, Vol. 56, No. 6, June 2010
s:
soo s∗oo = 1,
sco s∗oo = is∗co ,
(13)
(14)
soo s∗oc
= −isoc ,
i (s∗cc − scc ) = sco s∗oc ,
(15)
(16)
where the indices o and c in the submatrices stand for
open and closed channels. The vanishing divergence of
the probability current leads to another set of relations
[16]:
¡ †¢
s oo soo = 1,
(17)
¡ †¢
¡ ¢
s oo soc = i s† oc ,
¢
¡¡ ¢
¡ ¢
i s† cc − scc = s† co soc .
(18)
(19)
One can easily find the symmetry s = sT by comparing
both sets of relations above. Similarly, one can find two
sets of relations in the outer-region scattering matrices:

∗
∗
∗
∗
∗
+ iToc
+ Too roc
Roo Too
+ Too roo
Roo Toc
+ Too t∗oo − 1
Roo Roo
∗
∗
∗
∗
∗
=0

roo t∗oo + too Roo
roo roo
+ too Too
−1
roo roc
+ too Toc
+ iroc
∗
∗
∗
∗
∗
∗
∗
∗
∗
rco too + tco Roo − itco rco roo + tco Too − irco rco roc + tco Toc + i(rcc − rcc )
(20)

†
†
†
Roo
Roo + t†oo too − 1
Roo
Too + t†oo roo
Roo
Toc + t†oo roc − it†co
†
†
†
†
†
†
†
=0

roc − irco
Toc + roo
roo − 1
Too
Too + roo
too
Too
Roo + roo
Too
†
†
†
†
†
†
†
Toc
Roo + roc
too + itco Toc
Too + roc
roo + irco Toc
Toc + roc
roc − i(rcc
− rcc ).
(21)

and

The symmetries in the outer-region scattering matrices:
(22)
mula for the total photoionization cross section. We start
with the well-known Fermi’s golden rule for the partial
photoionization cross section [20]
can be derived by using the conservation of generalized
fluxes [19] and the above relations in Eqs. (20) and (21)
.
σN (E) = (4π 2 /c)(E − Ei )| < ΨN |Dφi > |2 , (23)
rmn = rnm , TN n = tnN , RM N = RN M ,
III. PHOTOIONIZATION CROSS SECTION
Equipped with local scattering matrices and their symmetries, we will derive an exact quantum-mechanical for-
(−)
ΨN (R, Ω)
→
X
M ∈open
r
(−)
where Ei is energy of the initial state φi , c is the velocity
of light, N is the channel index at the asymptotic limit
(−)
R → ∞, and ΨN is the energy-normalized final state
with the ingoing boundary conditions. By definition, the
(−)
wavefunction ΨN (R, Ω) is written at R → ∞ as
´
0
0
1
1 ³ +ikM
R
∗
ΦM (Ω; ∞) p 0
e
δM N + e−ikM R SM
N ,
2π
kM
where S is the ‘global’ scattering matrix, and open means
the channels open at R = ∞. The exponential functions
(24)
in Eq. (24) should be replaced by Coulomb functions,
but we will continue to use exponential functions as we
Scattering Matrix Formulation of the Total Photoionization of Two-electron · · · – Min-Ho Lee and Nark Nyul Choi
did in Eqs. (11) and (12) because the final result of the
derivation does not depend on this. It is noted that the
S-matix in Eq. (24) is unitary and symmetric:
S † S = SS † = 1;
S = ST .
(25)
-1803-
viding surface Σ0 can be expanded in the functions
Φn (Ω; R0 ) exp(±ikn y), where y = R − R0 . Thus, one
(−)
can write the wavefunction ΨN (R, Ω) in that interval
in the following form:
From the previous section, one can see that any
wavefunction in an infinitesimal interval around the di-
r
(−)
ΨN (R, Ω)
=
1
2π
Ã
X
¢
1 ¡ −ikn y −
Φn (Ω; R0 ) √
e
anN + e+ikn y a+
nN
kn
n∈open
X
+
n∈closed
!
¢
1 −iπ/4 ¡ +κn y −
e
bnN + e−κn y b+
,
Φn (Ω; R0 ) √ e
nN
κn
where y = R − R0 .
Now, we want to express the coefficients a+ , a− , b+ ,
and b− in Eq. (26) in terms of the local scattering matrices representing the inner (R < R0 ) and the outer
(R > R0 ) region dynamics, which are sketched in Fig. 1.
From Eqs. (9), (11), (12), (24), and (26), one can obtain
the following equations:
 −  
 + 
anN
rnn0 rnm0 tnM 0
an0 N
 b−  =  rmn0 rmm0 tmM 0   b+ 0  (27)
mN
mN
∗
TM n0 TM m0 RM M 0
δM N
SM
0N
and
µ
a+
n0 N
b+
m0 N
¶
µ
=
sn0 n sn0 m
sm0 n sm0 m
¶µ
a−
nN
b−
mN
X
one can see from Eqs. (25), (27), and (28) that
A− = (1 − rs)−1 tS ∗ ,
£
¤−1
S ∗ = R + T s(1 − rs)−1 t
;
S = R + T s(1 − rs)−1 t.
(30)
(31)
Thus, remembering Eq. (9), we can express the wave(−)
function ΨN (R, α) restricted in the inner region (R ≤
R0 ) as
¶
.
(28)
(−)
ΨN (R, α) =
The summation convention was used here and will be
used hereafter. Using a notation
µ ±¶
a
A± =
.
(29)
b±
σtot (E) =
(26)
X
ψnI(+) A−
nN .
(32)
n∈(open+closed)
Putting this expression into Eq. (23), we write
σN (E)
N
= (4π 2 /c)(E − Ei )
X
£
¤
£
¤
I(+)
< Dφi |ψnI(+) > (1 − rs)−1 tS ∗ nN × S T t† (1 − s† r† )−1 N n0 < ψn0 |Dφi >
n,n0 ,N
2
= (4π /c)(E − Ei ) d† (1 − rs)−1 t t† (1 − s† r† )−1 d,
where d is the atomic dipole vector
dn =< ψnI(+) |Dφi > .
(34)
(33)
Note that the summation over channel indices at Σ0 includes not only open channels but also closed channels.
In the above equation, Eq. (33), the total photoionization was expressed in terms of local scattering matrices,
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Journal of the Korean Physical Society, Vol. 56, No. 6, June 2010
but still it is not a desirable formula because the appearance of the matrix t prevents it from being easily
regarded as a generalization of the simple formula [12,
13] that neglects contributions of channels closed at Σ0 .
In order to avoid long expressions, hereafter we will
use a short notation for matrices such that
µ
¶
Aoo Boc
Aoo + Boc + Cco + Dcc ≡
.
(35)
Cco Dcc
It can be easily seen that, for open channels m,
< R, Ω|ψoI(+) (s† )om >
(
r
X
1
1
=
Φn (Ω; R0 ) √
2π n∈open
kn
¤
£ ikn y
−ikn y †
× e
δnm + e
(s )nm
X
+
n∈closed
Now, we introduce
for y = R − R0 ≥ 0.
∆ ≡ d† (1 − rs)−1 tt† (1 − (rs)† )−1 d − d† 1oo d, (36)
where 1 is the identity matrix. Using Eqs. (22) and (20),
one can show that
†
= tT
(37)
= 1oo − roo (r† )oo − roo (r† )oc − rco (r† )oo
−rco (r† )oc + i[(r† )co − roc ] + i[(r† )cc − rcc ]. (38)
Thus, ∆ is written as
∆ = d† [(1 − rs)−1 1oo (1 − (rs)† )−1 − 1oo ]d
+d† (1 − rs)−1 {−roo (r† )oo − roo (r† )oc
−rco (r† )oo − rco (r† )oc + i[(r† )co − roc ]
+i[(r† )cc − rcc ]}(1 − (rs)† )−1 d.
)
(44)
([s1oo − i1cc ]d)m for m ∈ closed is written as
(s1oo − i1cc d)m = smo do − idm
= smo < ψoI(+) |Dφi >
I(+)
−i < ψm
|Dφi >
∗
tt
eiπ/4
Φn (Ω; R0 ) √ e−κn y (s† )nm
κn
I(+)
= < ψoI(+) (s† )om + iψm
|Dφi >. (45)
It can be also easily seen that, for closed channels m,
I(+)
< R, Ω|ψoI(+) (s† )om + iψm
>
(
r
X
eiπ/4
1
Φn (Ω; R0 ) √
=
2π
κn
n∈closed
£ κn y
¤
× e δnm + e−κn y (s† )nm
X
1
+
Φn (Ω; R0 ) √ e−ikn y (s† )nm
kn
n∈open
(39)
It can be also shown that
for y = R − R0 ≥ 0.
)
(46)
I(−)
(1 − rs)−1 1oo (1 − (rs)† )−1 − 1oo
= 1oo (rs)† (1 − (rs)† )−1
+ (1 − rs)−1 rs1oo
+ (1 − rs)−1 rs1oo s† r† (1 − (rs)† )−1 (40)
One may want to introduce ψn
and
In fact, ψn
defined above is nothing but the solution of the I scattering system with ingoing boundary
conditions. By virtue of sT = s, it can be shown from
Eqs. (44) and (46) that
³
I(−)
ψo
I(−)
´
³
=
ψc
I(+)
ψo
such that
¶
´µ †
(s )oo (s† )oc
I(+)
.
ψc
0
i
(47)
I(−)
(rs1oo s† r† )nm =
+
=
+
rno (r† )om + irnc [(s† )cc − scc ](r† )cm
irnc (s† )co (r† )om − irno soc (r† )cm
rno (r† )om
irnc ((rs)† )cm − i(rs)nc (r† )cm
(41)
for any n, m ∈ (open + closed). Substituting Eqs. (40)
and (41) into Eq. (39), we can see that
∆ = d† 1oo (rs)† (1 − (rs)† )−1 d + d† (1 − rs)−1 rs1oo d
¡
¢ ¡
¢
+ i d† [1 − rs]−1 r c [(rs)† − 1][1 − (rs)† ]−1 d c
¡
¢ ¡
¢
+ −i d† [1 − rs]−1 [rs − 1] c r† [1 − (rs)† ]−1 d c
©
ª
= 2Re d† (1 − rs)−1 r[s1oo − i1cc ]d .
(42)
One can write (s1oo d)m for m ∈ open as
(s1oo d)m = smo do = smo < ψoI(+) |Dφi >
= < ψoI(+) (s† )om |Dφi > .
(43)
< R, Ω|ψ I(−) > = < R, Ω|ψ I(+) >∗ .
(48)
Then, Eq. (42) can be rewritten as
h
∆ = 2 Re < Dφi |ψ I(+) >
i
(1 − rs)−1 r < ψ I(−) |Dφi >
(49)
i
(1 − rs)−1 r < ψ I(+) |Dφi >∗ .
(50)
h
= 2 Re < Dφi |ψ I(+) >
Combining Eqs. (33) and (36) with Eq. (50), we obtain
the final formula for the total photoionization cross section such as
¡
¢
σtot = (4π 2 /c)(E −Ei ) d†o do + 2 Re[d† (1 − rs)−1 rd∗ ] .
Uµ(R) [Atomic Units]
0.5
0
I3
I2
-0.5
-1
-1.5
-2
I1
-2.5
-3
0
5
10
15
20
25
30
35
40
Cross Section x 102 [atomic units]
1
Cross Section x 102 [atomic units]
Scattering Matrix Formulation of the Total Photoionization of Two-electron · · · – Min-Ho Lee and Nark Nyul Choi
3.655
18
(51)
It can be easily shown from Eq. (47) that
d∗o = < ψoI(−) |Dφi > = < ψoI(+) (s† )oo |Dφi >
= soo < ψoI(+) |Dφi > = soo do .
(52)
Thus, if one neglects contributions from channels closed
at R = R0 , d† (1 − rs)−1 rd∗ reduces to d† (1 − rs)−1 rsd
, and the exact formula, Eq. (51) reduces to the formula
presented in Refs. 12 and 13 as mentioned in Sec. I..
IV. DISCUSSION
We are interested in the characteristics of the fluctuations in the total photoionization cross section, especially when approaching the three-body breakup threshold, i.e., in the the limit E → 0 from below the threshold.
The surface Σ0 enclosing the support of the initial state
naturally leads to a partition of the configuration space
into physically distinct regions. In particular, quantum
contributions to the total photoionization cross section
from the inner region are insensitive to the total energy
E. The overlapping resonance structure of the cross section near the threshold E = 0 section is attributed to
contibutions from the outer region [12].
In this paper, a rigorous derivation of an exact quantum formula was presented for the total photoionization
of two-electron atoms in terms of local scattering matrices that separately describe the dynamics of the inner
and the outer parts. One of the important merits of this
formulation is the possibility of application of semiclassical methods to the treatment of the outer part. The
operator r, the only energy-sensitive term in Eq. (51),
18.4
18.6
18.8
19
Neff
3.650
3.645
3.640
15
16
17
Neff
R [Atomic Units]
Fig. 2. (Color online) Adiabatic potentials for even (red
line) and odd (blue dotted) parities for the one-dimensional
eZe model. The dividing surface is chosen to be at R = R0 =
20, where a vertical line is drawn for convenience in counting
the open channels.
18.2
-1805-
18
19
20
Fig. 3. (Color online) Total photoionization cross section
σtot (E) of the lowest odd-parity
p state of the collinear He atom
as a function of Nef f ≡
2/|E| for the energy region of
[I15 , I20 ]. The lower curve (red line) is the result from the
exact formula (51) while the upper one (blue line) is from an
approximate formula that contains only the contribution of
open channels at the dividing surface R = R0 . The inset is a
magnification for the energy interval [I18 , I19 ].
can be written in a semiclassical approximation [14–17]
as
r(Ω, Ω0 , E) ≈ (2πi)−
f −1
2
X
−1/2 iSj −iπνj /2
| det M12 |j
e
,
j
(53)
where the sum is taken over all classical paths j with
fixed energy E starting at points Ω0 on the surface of section Σ0 (with momentum pointing outward) and ending
at Ω again on Σ0 without crossing the surface of section
in between. Sj (Ω, Ω0 ; E) is the action of that path, νj is
the Maslov index, and f denotes the number of degrees
of freedom (with f = 4 for two-electron atoms in three
dimensions and fixed angular momentum). Furthermore,
µ
det M12 = det
∂2S
∂Ω∂Ω0
¶−1
µ
= det
∂Ω
∂p0Ω
¶
, (54)
where pΩ are the momentum variables conjugated to Ω.
The (f − 1) × (f − 1) matrix M12 forms a 2(f − 1)dim. sub-matrix of the stability matrix describing the
linearized Poincaré map on Σ0 for fixed energy and angular momentum.
Employing the approximation in Eq. (53), Byun et al.
[12] suggested a scaling law for the energy dependence of
the amplitude of the fluctuating part of the cross section.
However, their suggestion was based on the approximate
formula derived by Granger and Greene [13], in which
only the channels open at the dividing surface were considered. Now, based on the exact formula in Eq. (51), we
can examine the validity of semiclassical approximations
by explicitly evaluating contributions from the channels
-1806-
Journal of the Korean Physical Society, Vol. 56, No. 6, June 2010
closed at the dividing surface. For this purpose, it is
helpful to compare numerical results for the total cross
section obtained by using the exact formula in Eq. (51)
and the approximate one in which the local scattering
matrices d, r, and s are replaced by doo , roo , and soo ,
respectively. A numerical study of the local scattering
matrices for the full three-body quantum problem is still
out of reach for energies E > IN with N ≈ 15 [12].
Here, IN is the N-th threshold energy of single-electron
ionization, i.e., IN = −Z 2 /2N 2 . We, therefore, choose a
model system, namely, the eZe collinear He [9].
The adiabatic Hamiltonian HR in Eq. (6) is invariant
under the exchange of two electrons. Thus, the adiabatic
channel functions have definite parities with respect to
the exchange operation. The potentials Uµ (R) for the
collinear eZe are shown in Fig. 2. We consider the photoionization of the lowest odd-parity state. Then, the
final state has even parity. Choosing the dividing surface at R = R0 = 20, we can see that there are 5 evenparity channels open at the surface. It is noticeable that
the number of open channels at the dividing surface is
much smaller than that at R = ∞. All the local scattering matrices can be accurately obtained by using a
variant method of the generalized log-derivative method
[21]. Fig. 3 shows the contribution of the channels that
are open at the dividing surface to the total photoionization cross section of the lowest odd-parity state of the
collinear He atom, as well as the exact cross section. It is
easily seen from this figure that the channels closed at the
dividing surface make an insignificant contribution to the
fluctuating part of the cross section while the magnitude
of the background part changes by 0.1%. Thus, considering that the dynamics in the classically-alloweded
region is described by open channels, we can conclude
that semiclassical methods, which employ classical trajectories starting outward from the dividing surface and
returning to it, can be safely applied to studies of the
fluctuating part of the photoionization cross section [12,
21].
ACKNOWLEDGMENTS
This work was supported by the Kumoh National Institute of Technology under contract number 2007-104068. The authors are grateful to G. Tanner for fruitful
discussions on the derivation of the exact formula.
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