2. Zeros in free-free matrix elements

We begin by writing the non-relativistic dipole bremsstrahlung matrix element for a spin-less electron

$$M_{fi}^{\rm nrd} = \int \psi_2^* \left({\vec\epsilon}^{\,*}\cdot\vec p\right)\psi_1 d^3r,$$ (1)

where $\psi_2$ and $\psi_1$ are continuum solutions of the full three dimensional Schrödinger equation representing the initial and final states of the electron respectively, $\epsilon$ is the photon polarization vector and $\vec p$ is the momentum operator (as discussed, for example, in [4]). The partial wave decomposed form of this matrix element is written [4],

 

$${M^{\rm nrd}_{fi}} (ik/p_1p_2)\sum_m(\hat\epsilon)^*_m \sum_{l_1,m_1,l_2,m_2} \left[... ...i^{l_2}e^{i(\delta_{l_1}+\sigma_{l_1})} e^{i(\delta_{l_2}+\sigma_{l_2})}\right.$$ =

$$\quad\times Y_{l_1,m_1}^*(\hat p_1) Y_{l_2,m_2}(\hat p_2) R_{l_1,l_2}(E_1,E_2)$$

$$\quad\left.\times\sqrt{3(2l_1+1)\over 4\pi(2l_2+1)} \left<l_11;00\vert l_11;l_20\right> \left<l_11;m_1m\vert l_11;l_2m_2\right>\right],$$ (2)

where the ionic Coulombic phase shift

$$\sigma_l = \arg \Gamma(l+1+i\eta_{\rm ion}),$$ (3)

$\delta_l$ is the (energy dependent) short range phase shift, $\eta_{\rm ion} = -Z_{ion}/p$ is the Coulomb parameter corresponding to the ionic charge $Z_{\rm ion}$( $\eta_{\rm ion}=\sigma_l\equiv0$ for a neutral atom), $\vec p_{1,2}$ are the electron momenta (with magnitudes p_1,2) and the unit vectors $\hat p_{1,2} = \vec p_{1,2}/p_{1,2}$), $\hat \epsilon$ is a unit vector in the direction of the photon polarization, and Y_LM are the spherical harmonics. [In all of our equations we use atomic units ( $m=e=\hbar=1$, m the mass of the electron, e the charge of the electron, and $\hbar$Planck's constant divided by $2\pi$).] In the summation in Eq. (2), m represents the z component of the angular momentum of the photon, $l_1,\,m_1$ ($l_2,\,m_2$) represent the angular momentum and its z component for the initial (final) electron. We have used the notation of [5] for the Clebsch-Gordan coefficients $\left<LL';MM'\vert LL';JM_J\right>$ and the notation for a spherical vector $(\vec \epsilon)_m$ derived from a Cartesian vector $\vec \epsilon$ (and the corresponding unit vector $\hat \epsilon$)

$$\begin{eqnarray*}(\vec \epsilon)_m = \left\vert\vec \epsilon\right\vert Y_{1m}(\hat \epsilon). \end{eqnarray*}$$

The bremsstrahlung cross section can then be obtained as

$$\begin{eqnarray*}{d^3\!\sigma\over dkd\Omega_2d\Omega_1} = {4\pi^{2}\alpha k} \left\vert M_{fi}^{\rm nrd} \right\vert^2, \end{eqnarray*}$$

where $\alpha=e^2/\hbar c$ is the fine structure constant.

The radial matrix element is written in terms of the radial wave functions as

$$R_{l_1,l_2}(E_1,E_2) = \int dr\, \phi_{l_2}(E_2;r) \, r \, \phi_{l_1}(E_1;r).$$ (4)

We note this radial matrix element is the same as would appear in the partial wave decomposition of the matrix element for absorption of a photon by an electron scattering from an atom or ion - inverse bremsstrahlung. Thus the considerations which follow for this matrix element also apply to the process of inverse bremsstrahlung.

The radial wavefunctions $r^{-1}\phi_l(E;r)$ which enter the radial matrix element R_l1,l2(E₁,E₂) are defined as the real-valued solutions of the radial Schrödinger equation

$${d^2\over dr^2}\phi_l(E;r) - \left[ {l(l+1)\over r^2} + 2V(r) - 2E\right]\phi_l(E;r) = 0,$$ (5)

with the asymptotic forms

 

$$\phi_l(E;r) \rightarrow N_l\left({2\over\pi\sqrt{E}}\right)^{1/2} \sin(pr-\eta_{\rm ion}\ln 2pr-l\pi/2+\delta_l+\sigma_l) \quad\mbox{as }r\rightarrow\infty,$$ (6)

$$\phi_l(E;r) \rightarrow r^{l+1} \quad\mbox{as }r\rightarrow 0.$$ (7)

The normalization constant N_l is chosen so that, as above, the coefficient of r^l+1 near the origin is unity. In the Coulomb case,

 

$$N^{\rm coul}_l = {\Gamma(2l+2)\over 2^le^{-\pi\eta/2}\left\vert\Gamma(l+1+i\eta)\right\vert}.$$ (8)

We require that the phase shifts $\delta_l$ and the normalization N_l be continuous functions of energy [6], as are the Coulomb normalization and phase. For most potentials the normalization constant, so defined, will not change sign as a function of energy, since an energy for which N_l=0 would correspond to a bound state in the continuum [6]. Note that, since the photon carries one unit of angular momentum (through the vector $\hat \epsilon$), the Clebsch-Gordan coefficients in Eq. (2) are zero unless $l_2=l_1\pm 1$ and m₂=m₁+m, as expected in the dipole approximation.

In the following subsections we consider the radial matrix element, Eq. (4), for dipole transitions, as a function of the energies E₁ and E₂. This matrix element is a continuous function of both E₁ and E₂ (except when E₁=E₂), if (as above) we suitably choose the definition of the normalizations of the wavefunctions $\phi_l(E;r)$ [7,8]. Note we have used wavefunctions ($\phi_l$) normalized so that the coefficient of r^l+1 in the expansion of $\phi_l(E;r)$ about r=0 is unity. Since this normalization condition is independent of E, Poincaré's theorem [6] applies and the $\phi_l$ are analytic in E. In this way we can be sure that R_l1,l2(E₁,E₂), as defined above, is analytic except when E₁=E₂ and that for E real and E>0 the two limiting cases we will consider are limiting cases for R_l1,l2 of the same continuous matrix element [9,8].