Hybrid Theory of Electron-Hydrogenic Systems Elastic Scattering

Abstract

Accurate electron-hydrogen and electron-hydrogenic cross sections are required to interpret various experiments. Scattering from such simple systems allows us to test various scattering theories. The incident electron produces a distortion of the target which results into a long-range polarization potential in the Schrödinger equation for the scattering function. There are also short-range correlations. In this article, we show how to take into account these two effects at the same time variationally. The polarization is considered to take place whether the electron is outside the target-electron orbit or inside it. Phase shifts have been calculated for the scattering from hydrogen atom, He⁺ and Li⁺⁺ positive ions. The S-wave phase shifts calculated here have lower bounds and they increase as the number of terms in the correlation function is increased. It is shown that only short expansions are needed to obtain accurate results. The results are compared with the previous calculations. Resonance parameters for the resonances in He and Li⁺ have been calculated by calculating phase shifts in the resonance regions. The results are compared with the previous calculations and the merit of the present approach compared to the previous approach is pointed out.

Appendix

We will briefly describe the non-iterative method for solving integrodifferential equations [23]. Consider the equation for the scattering function u(r) given by

$$ \biggl[ \frac{d^{2}}{dr^{2}} + V(r) + k^{2} \biggr]u(r) = g(r)\int_{ 0}^{ \infty} f(x)u(x)dx. $$

(A.1)

Let

$$ u(r)=u_0(r) + Cu_1(r). $$

(A.2)

The constant C represents the definite integral in Eq. (A.1).

Substituting u(r) in Eq. (A.1), we get two equations

$$ \biggl[ \frac{d^{2}}{dr^{2}} + V(r) + k^{2} \biggr]u_{0}(r) = 0, $$

(A.3)

$$ \biggl[ \frac{d^{2}}{dr^{2}} + V(r) + k^{2} \biggr]u_{1}(r) = g(r). $$

(A.4)

These two equations can be solved easily. The substitution of (A.2) in C gives

$$ C = \int_{0}^{\infty} f(x)u_{0}(x)dx + C \int_{0}^{\infty} f(x)u_{1}(x)dx = I_{0} + CI_{1}. $$

(A.5)

Having calculated u ₀ and u ₁, I ₀ and I ₁ can be calculated. From the above equation, we can now solve for C. We get

$$ C = \frac{I_{0}}{(1 - I_{1})}. $$

(A.6)

Equation (A.1) can now be written as

$$ \biggl[ \frac{d^{2}}{dr^{2}} + V(r) + k^{2} \biggr]u(r) = Cg(r). $$

(A.7)

Now the right-hand side is known and this equation can be solved for u(r). This method can be generalized to any number of definite integrals.