Abstract
Full Text
B. A. Zon, N. L. Manakov, L. P. Rapoport
A Semiphenomenological Green’s Function of the Optical Electron in a Complex Atom
(Presented by Academician Ya. B. Zel’dovich, 19 II 1969)
The most widely used method for calculating oscillator strengths and photoionization cross sections of complex atoms is the semiempirical method, usually called the quantum-defect method (QDM) ((^{1,2})). The method is based on the fact that the main contribution to the dipole matrix elements associated with transitions of the optical electron comes from the regions of large (r), where the effective one-particle potential (V(r)) may, with good accuracy, be regarded as Coulombic ((^{1})). In this region, as the radial wave function of the electron one may take a Coulomb function with a principal quantum number determined from the experimentally observed spectrum of the atom:
[
E_{nl}=-\frac{Z^{2}}{\nu_{nl}^{2}}Ry,
]
where (\nu_{nl}=n-\mu_l(E_{nl})) is the effective principal quantum number, (n=1,2,\ldots); (\mu_l) is the quantum defect; (Z) is the charge of the residual ion. The radial wave function chosen in this way is incorrect in the region of the atomic core; however, as has already been noted, this region is also immaterial for problems connected with electromagnetic transitions in atoms.
In the present note the QDM approximation is used to construct the Green’s function of the optical electron of an atom, which makes it possible to calculate multiphoton atomic processes (coherent and combination scattering, multiphoton ionization, etc.).
The problem reduces to solving the differential equation for the Green’s function (G_E(\mathbf r,\mathbf r')) in the region where (V(r)=-\alpha Z/r). After separation of the angular parts, the solution of the equations for the radial Green’s function (g_l(E;r,r')), corresponding to the (l)-th orbital angular momentum*,
[
\left{\frac{1}{2m}\left[\frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}\frac{d}{dr}\right)-\frac{l(l+1)}{r^{2}}\right]+\frac{\alpha Z}{r}+E\right}g_l(E;r,r')
=\frac{\delta(r-r')}{rr'},
\tag{1}
]
satisfying the asymptotic condition ((^{3}))
[
g_l(E;r,r')
\underset{r,r'\to\infty}{\longrightarrow}
-\frac{2m}{rr'k}
\exp\left{ i\left[kr_{>}-\xi\ln 2kr_{>}-l\frac{\pi}{2}+\sigma_l+\delta_l\right]\right}
]
[
{}\times
\sin\left[kr_{<}-\xi\ln 2kr_{<}-l\frac{\pi}{2}+\sigma_l+\delta_l\right],
]
can be represented in the region under consideration in the form
[
g_l(E;r,r')=
\frac{ima\xi}{rr'}e^{\pi\xi}
\left[
(-1)^{l+1}e^{2i(\sigma_l+\delta_l)}
W_{-i\xi,l+1/2}(-2ikr_{>})
W_{-i\xi,l+1/2}(-2ikr')
+\right.
]
[
\left.
{}+
W_{i\xi,l+1/2}(2ikr_{<})
W_{-i\xi,l+1/2}(-2ikr_{>})
\right],
\tag{2}
]
where (W) is the Whittaker function; (k=i/a\nu); (\xi=-1/ak); (\nu=Z/\sqrt{-ERy}); (a) is the Bohr radius; (r_{>(<)}) is the larger (smaller) of the quantities (r) and (r'); (\sigma_l=\arg\Gamma(l+1-\nu)); (\delta_l) is the phase of the (l)-th partial wave associated with the deviation of (V(r)) from a pure Coulomb field.
* The system of units used in the work is one in which (c=\hbar=1,\ e^{2}=\alpha\simeq 1/137).
Formula (2) defines the Green’s function in the region (E>0), if it is possible in some way to find the phases (\delta_l). In the case (\delta_l=0), (2) goes over into the expression for the Green’s function of the hydrogen atom ((^4)). For the analytic continuation of the phases into the region (E<0), we shall use the quantum-defect method proposed in ((^2,^5)) and based on knowledge of the experimental spectrum of the atom. Let us note that in ((^6)) the equivalence is shown of the quantum-defect method and the effective-radius approximation, well known in scattering theory ((^7)). The indicated method gives, for (\delta_l) at (E<0), the relation
[
\operatorname{ctg}\delta_l(k^2)\to (1-e^{2\pi i\nu})
\left[\operatorname{ctg}\pi\mu_l(E)+i\frac{1}{1-e^{-2\pi i\nu}}\right]
]
((\mu_l(E)) is the quantum defect), using which we obtain
[
g_l(E;r,r')=
\frac{ma\nu}{rr'}e^{i\pi\nu}
\left[
\pi e^{i\pi\mu_l}
\left{
\sin\pi(\nu+\mu_l)\Gamma(\nu-l)\Gamma(l+1+\nu)
\right}^{-1}
\times
\right.
]
[
\left.
{}\times
W_{\nu,l+1/2}!\left(\frac{2r}{a\nu}\right)
W_{\nu,l+1/2}!\left(\frac{2r'}{a\nu}\right)
-
W_{-\nu,l+1/2}!\left(-\frac{2r_<}{a\nu}\right)
W_{\nu,l+1/2}!\left(\frac{2r_>}{a\nu}\right)
\right];
\tag{3}
]
(\mu_l(E)) is interpolated from the experimentally observed spectrum of the atom.
It is easy to verify that the Green’s function (3) has poles at the points
[
E_{nl}=-\frac{Z^2}{(n-\mu_l)^2}Ry,
]
where (n) is an integer, corresponding to the real spectrum of the atom, and the residues at these poles are equal to
[
\left[
arr'\nu^2
\left(1+\frac{\partial\mu_l}{\partial\nu}\right)
\Gamma(\nu-l)\Gamma(l+1+\nu)
\right]^{-1}
W_{\nu,l+1/2}!\left(\frac{2r}{a\nu}\right)
W_{\nu,l+1/2}!\left(\frac{2r'}{a\nu}\right),
]
which coincides with the product of the eigenfunctions for the state with effective principal quantum number (\nu) and orbital angular momentum (l).
The Green’s function defined by expressions (2) and (3) makes it possible to calculate the amplitudes of multiphoton transitions in a complex atom, using the previously proposed method for calculating radial integrals for the hydrogen atom ((^4,^8)).
Voronezh State University
Received
7 II 1969
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