2.3 The exact partial wave method

In the exact partial wave method we include the exact numerical atomic potential in the unperturbed Hamiltonian. In this case the interaction (for bremsstrahlung from a neutral atom) is not with the Coulomb potential (as in the Elwert-Haug case) but with the screened Coulomb potential, so no form factor approximation is needed. Furthermore, no approximation in $(\kappa/Z\alpha)^2$ is made, but rather the full partial wave solutions are obtained numerically to an order in $\kappa$ needed for convergence of the matrix element. The Dirac equation is separated into angular and (coupled) radial equations. We obtain numerical solutions of specified energy E of the radial Dirac equation, labeled by the quantum numbers $\kappa$ and m. The quantum number $\kappa=\pm 1,\pm 2,\pm 3$..., corresponds to eigenvalues of the operator $-(\vec\sigma\cdot\vec L + 1)$; m is the eigenvalue of J_z, where $\vec L$ is the orbital angular momentum operator, and J_z is the z component of the total angular momentum operator. A linear combination of eigenstates $\psi_{\kappa m}$ is taken which obeys the appropriate boundary conditions (plane wave plus incoming or outgoing spherical wave). The photon interaction ``operator'', $\vec\alpha\cdot{\vec\epsilon}^{\,*}e^{-i\vec k\cdot\vec r}$, is expanded into a multipole series [12]. We may represent the three fold summation over incident and outgoing electron partial waves and photon angular momenta in the matrix element symbolically as $\sum_{\cal K}{\cal M}_{\cal K}$, where ${\cal K}(L,\kappa_1,\kappa_2)$ represents the photon angular momenta and parity and the electron angular momentum quantum numbers. Thus we write

$$M_{fi} = \sum_{\cal K} {\cal M}_{\cal K},$$ (3)

and the expression for the polarized cross section is

$$d^3\!\sigma^{\rm pol} \propto \left\vert \sum_{\cal K} {\cal M}_{\cal K}\right\vert^2.$$ (4)

The matrix elements in this summation involve radial integrals of the form

$$\begin{eqnarray*}S_1 & = & \int_0^\infty g_{\kappa_1}j_l(kr)f_{\kappa_2}\,dr,\\ S_2 & = & \int_0^\infty g_{\kappa_2}j_l(kr)f_{\kappa_1}\,dr, \end{eqnarray*}$$

where $g_\kappa$ and $f_\kappa$ are the small and large components of the initial and final spinors and j_l(kr) are the spherical Bessel functions. Such integrals over rapidly oscillating functions are evaluated using the well tested computer program of Tseng [13]. These integrals depend only on p₁, p₂, kand Z. The summation in Eq. (3) to obtain the total matrix element is performed numerically before squaring to obtain the cross section. A more detailed description of our formalism can be found in [4].