2.2 The high energy limit

We now want to determine the sign of the matrix element R_l1,l2(E₁,E₂) in the limit $E_1 \rightarrow \infty$. (For convenience of notation we will discuss only the case where E₂ is fixed and E₁ varies but the same arguments apply when reversing the roles of the energies.) For large E₁, $\phi(E_1;r)$ will oscillate for $p_1 r \gg 1$. Since the other terms in the integrand of (4) are slowly varying on this scale, the important contributions to the integral occur when r is of the order of 1/p₁, since for larger distances the integrand oscillates rapidly. The radial matrix element for fixed l in this high energy limit will be determined in this small r region. Since a realistic atomic potential would have nuclear Coulombic behavior at this length scale, it is sufficient to use wave functions $\phi_l(E;r)$ for a particle in a Coulomb potential with nuclear charge Z. That is, we can use the reduced Coulomb functions $\phi_l(E;r) = F_l(\eta;pr)$ (normalized to r^l+1 near the origin), where $\eta = -Z/p$ is the (nuclear) Coulomb parameter. It has previously been demonstrated [24] that the dipole Coulomb free-free matrix element is positive everywhere (and thus in the high energy limit considered here). It is nevertheless useful to obtain an expression for the matrix element in this high energy limit. We may expand the wavefunction for the slow electron, $\phi_{l_2}(E_2;r)$, about r = 0, keeping terms up to order r^l₂+2 (the first two terms):

$$\phi_{l_2}(E_2;r) = r^{l_2+1} \left[1+B_{l_2}(\eta_2) r\right],$$ (17)

where

$$B_{l_2}(\eta_2) = {p_2 \eta_2 \over l_2 +1}.$$ (18)

Inserting this expansion in Eq. (4), and using the Coulomb function for $\phi_l(E;r)$,we obtain

$$R_{l_1,l_2}(E_1,E_2) \rightarrow \int_0^\infty \left[r^{l_2... ...ta_1; p_1 r) \, dr \mbox{\quad as\quad} E_1\rightarrow \infty.$$ (19)

The first two terms in the expansion (17) are needed in (19) since they contribute to the same order in the small parameter $\eta_2$ (while further terms contribute pair-wise in higher orders in $\eta_2$ or p₂/p₁[3,25]). The reduced Coulomb function F_l can be expressed in terms of the confluent hypergeometric function M(a,b,z), [26]

$$\begin{eqnarray*}F_l(\eta;z) = z^{l+1}e^{-iz}M(l+1-i\eta,2l+2,2iz). \end{eqnarray*}$$

Eq. (19) can be evaluated in a straightforward manner by using an integral representation for the confluent hypergeometric function given in [27]. Landau and Lifshitz [28] give

 

$$J_{\alpha\gamma}^\nu \int_0^\infty e^{-\lambda z}z^\nu F(\alpha,\gamma,kz)\, dz$$ = (20)

$$\Gamma(\nu+1)\lambda^{-\nu-1}F(\alpha,\nu+1,\gamma,k/\lambda),$$ = (21)

convergent if $\Re\nu > -1$ and $\Re\lambda > \left\vert\Re k\right\vert$. To satisfy these constraints, as is conventional in defining the bremsstrahlung matrix element, we introduce the exponential $e^{-\lambda r}$ into the integrand of Eq. (20) and take the limit $\lambda\rightarrow 0^+$ after integrating, obtaining

$$\rightarrow p_1^{-l_2-3}p_2^{l_2+1}\Gamma(l_2+l_1+4)$$ R_l1,l2(E₁,E₂)

$$\times \left[ i^{-l_2-l_1-4}F(l_1+1-i\eta_1,l_2+l_1+4,2l_1+2;2+0i)\right.$$

$$\left. (l_2+l_1+5)i^{-l_2-l_1-5}B_{l_2}(\eta_2){p_2\over p_1}\right.$$ +

$$\left.\quad\times F(l_1+1-i\eta_1,l_2+l_1+5,2l_1+2;2+0i) \strut\right],$$ (22)

where F(a,b,c;z) is the hypergeometric function and $\Gamma(z)$ is the gamma function. Since l₁ and l₂ are both integers, these hypergeometric functions can be evaluated analytically [29]. The additional constraint $l_2=l_1\pm 1$simplifies this algebraically tedious calculation. Retaining only the leading term in $\eta_2$ we obtain

$$R_{l_1,l_1+1}(E_1,E_2) \rightarrow -2\eta_1e^{-\eta_1\pi}p_1^... ...}p_2^{l_1+2} \Gamma(2l_1+2) \mbox{ as } E_1\rightarrow\infty,$$ (23)

for upward transitions and

$$R_{l_1,l_1-1}(E_1,E_2) \rightarrow -4\eta_1e^{-\eta_1\pi}p_1^... ...} {\Gamma(2l_1+2)\over l_1} \mbox{ as } E_1\rightarrow\infty,$$ (24)

for downward transitions. Clearly

$$R_{l_1,l_2}(E_1,E_2) \rightarrow 0^+ \qquad\mbox{as }E_1\rightarrow\infty,$$ (25)

for both upward and downward transitions. That is, the matrix element R_l1,l2(E₁,E₂) is positive in the limit that $E_1 \rightarrow \infty$.