Single and Triple Differential Cross Sections for DoublePhotoionization of H- Page: 4 of 12
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of spherical harmonics is represented in a product ba-
sis of 1D finite element-discrete variable representation
(FEM-DVR) functions, similar to the approach used in
molecular hydrogen DPI referenced above. The FEM-
DVR radial basis is an attractive choice because of the
computational efficiency gained as well as the natural
complementarity for implementing exterior complex scal-
The Coulomb functions in Equation 3 are similarly ex-
panded in partial waves,
1/2 (k , r) ( 2)1 t j ' (r)Y m ( r) r (1 ),
where 01 (r) is a radial Coulomb function with asymp-
0()(r) -> sin(kr + (Z/k) ln 2kr - l/2 + np(k)), (10)
as r -> o and i represents the Coulomb phase
'1(k) argf(l + 1- iZ/k), (11)
with Z equal to the nuclear charge in the one-body po-
tentials of Eq. 4, here Z 1 for H-.
By substituting the partial wave expansions of the full
scattered solution T+ (Eq. 8) and the product of testing
functions 4)z ) (k, r) (Eq. 9) into the expression for the
full amplitude (Eq. 6) and integrating over the angular
coordinates dQ1dQ2 of both electrons, we arrive at an
expression connecting the partial waves of T+ with the
product of Coulomb waves,
f (ki, k2) C 2 (l1+12),2'/1 (ki)+2'q' (k2)
,1 l 2,m2
X TI 1,12,mi,m2 (ki,k2)Yimi (k1)Y2m2 (k2) ,
due to the orthogonality of the spherical harmonics. The
sum is once again constrained to include only angular
momentum pairs for which M mi + m2 0. Using
the standard theory of rearrangement scattering, com-
bined with a two-potential formalism, we can express the
partial wave amplitudes Y1,12,m1,m2(kI1, k2) appearing in
Eq. 12 as :
-T11,12,mi,m2 (ki, k2) = X
K Qki (r1)2,k2 (r2) E h- h2 Vlimi,12m2(rl, r2))
k1dk2 J dridr 1)4 (ri)q2 (r2) x
(E -h- h2) m1,12m2((I r2),)
As before, the radial volume integral can be simplified by
application of Green's theorem,
(9(c)11,i (r)9$k2 (r2)IE -h h2 Vimi,2m2 (ri, r2))
1[9 ki(r1 )12k2 (r2) lim1,l2m2(r 1r2)
-limi,12m2(ri, r2) k '1(ri)9Km2 (r2) , dcx
where po defines the hypershere where the partial wave
amplitudes are calculated, usually just inside of the ECS
turning point Ro.
D. Cross Section Evaluation from Reduced
The partial wave amplitudes evaluated using Eqs. 13
and 15 are then returned to Eq. 12 to construct the full
double photoionization amplitude f(kl, k2). The TDCS
can then be calcuated by Eq. 7.
The single differential cross section (SDCS), describ-
ing the energy sharing between both ejected electrons, is
given by integrating the TDCS over all angles dQ1dQ2 of
electrons 1 and 2. Because of the orthonormality of the
spherical harmonics, cross terms between reduced ampli-
tudes for different angular configurations disappear, thus
the SDCS is simply given by
do, 49r2k2 (2)2
dE1 LJC k2
'11,12,m1,m2(ki, k2) 12
The total cross section for double photionization is
then given by integrating the SDCS over the energy shar-
aE dz dE ,
although the SDCS is sometimes defined to give the to-
tal cross section by integration over half energy range.
Because the SDCS is symmetric about E/2, this simply
redefines the SDCS as
dE1 2 dE1
thus also making the total DPI cross section
where hi and h2 are one-electron radial Hamiltonia
1 d2 1(1+1)
2 d,2 + 2r2
For consistency with our published SDCS results for he-
lium, we have adopted the convention of Equation 19 in
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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. (digital.library.unt.edu/ark:/67531/metadc896689/m1/4/: accessed January 17, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.