Single and Triple Differential Cross Sections for DoublePhotoionization of H- Page: 3 of 12
This article is part of the collection entitled: Office of Scientific & Technical Information Technical Reports and was provided to UNT Digital Library by the UNT Libraries Government Documents Department.
Extracted Text
The following text was automatically extracted from the image on this page using optical character recognition software:
3
R(r)
eRo Re(r)
(a) ECS contour
Tm(r)
[y6(p
appo)- 6(p- po) p'Z(rir),
(6)
8 - where the arrows above the partial derivatives indicate
the direction to which they operate and the delta func-
0 5 10 15 20 '2 ' 30 tions enforce the evaluation of the amplitude along the
hypersphere defined by p r1 + r. The hyperangle
a is defined by tana r2/ri, while Q1 and Q2 are the
spherical polar angular coordinates of electrons 1 and 2,
(b) Outgoing Waves under ECS respectively.
The triple differential cross section describing the an-
I: (Color online) Exterior complex scaling in a single gular distributions of both ejected electrons and the en-
dimension (a) ECS contour in the complex plane ergy sharing between them is given bydemonstrating the rotation of the radial coordinate beyond
Ro into the upper-half complex plane by angle 0. (b) Real
(solid line) and imaginary (dashed line) components of an out-
going wave solution for a model problem [47] along the ECS
contour. The oscillatory nature of the outgoing waves is ex-
ponentially damped by the coordinate transformation beyond
Ro. Inside of Ro the outgoing wave is the physical solution.
The distorted wave "testing functions" 4--(ki,ri) in
Eq. 3 are momentum-normalized atomic Coulomb func-
tions with charges Z equal to the charge of the nuclear
potential in V of Eq. 4, which is Z 1 in the case
of H-. With that choice of effective nuclear charge on
the testing functions, the finite-volume amplitude inte-
gral above projects out single-ionization contamination
from the double-ionization channel by orthogonality of
the Coulomb functions to the residual bound one-electron
atom [34]. This choice for Z is in contrast to the usual
"Peterkop condition" [32],1
k1 k2'which is formally adopted to eliminate an overall volume-
dependent phase. This phase, however, has been shown
to have no effect on the calculated cross sections [51]. It
must be stressed that the final state in Eq. 3 is not givend33d
dEi dQi dQ242
__k1k2 f(k1, k2) 2
wc(7)
C. Partial Wave Decomposition of t, and the
Double Photoionization Amplitude
Following the prescription for practical calculation of
the double photoionization amplitude utilized for both
atomic helium [25] and molecular hydrogen [29], we seek
to decompose the full scattered wave into angular com-
ponents on a radial grid in order to implement exterior
complex scaling. Thus, the scattered wave function that
solves Eq. 2 is expanded as
sc Vim,12m2 (r1, r2)Ylim,(Pi)Y2m2 (r2),
lim1 12m2 r1r2
(8)
where unlike our earlier helium treatment, we have not
explicitly partitioned the sum into direct and exchange
components, but have instead summed over angular con-
figurations (i. e., lm-pairs) of the individual electrons.
This sum, of course, is over lm-pair configurations that
give an overall L 1, M 0 state required by pho-
toabsorbtion selection rules. The two-dimensional ra-
dial function Vl1mi,12m2(r, r2) multiplying the productby a product of Coulomb functions but is contained in
the outgoing wave 4'; the Coulomb functions serve to
extract the double ionization amplitude from all other en-
ergetically allowed processes (e. g. single ionization chan-
nels) contained in the exact solution.
The six-dimensional finite-volume integral of Equa-
tion 3 also leads to a further computational simplifica-
tion by application of Green's theorem, thus allowing the
amplitude to be computed as a surface integral. This al-
lows the amplitude to be evaluated by considering only
the asymptotic form of the scattered wave. The five-
dimensional surface integral evaluated in hyperspherical
coordinates is given by
f f/2 p sin2 acos2
f(ki, k2) dQ1 dQ2 dp da 2
02
<b)-"(ki, rl)*<D(-)(k2, r2)*- I I
/ % I I -
TI , I , I j 1 I 1
1 , 1 j I i I I 1
-I I j I I -
Ii i , I Il I 1_
K , 1J 1J J I10.8
0.4
0
-0.4-0.
FIG.
rad(ia nZ Z 1 1
k1 k2 k1 k2
Upcoming Pages
Here’s what’s next.
Search Inside
This article can be searched. Note: Results may vary based on the legibility of text within the document.
Tools / Downloads
Get a copy of this page or view the extracted text.
Citing and Sharing
Basic information for referencing this web page. We also provide extended guidance on usage rights, references, copying or embedding.
Reference the current page of this Article.
Yip, Frank L.; Horner, Daniel A.; McCurdy, C. William & Rescigno,Thomas N. Single and Triple Differential Cross Sections for DoublePhotoionization of H-, article, February 15, 2007; United States. (https://digital.library.unt.edu/ark:/67531/metadc896689/m1/3/: accessed April 20, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.