2.1 The soft photon limit

In the limit that the photon energy k goes to zero with one electron energy fixed (the soft photon limit), R_l1,l2(E₁,E₂) is known to diverge as 1/k² [4]. We wish to determine the sign of R_l1,l2 in this limit. To obtain the leading divergent term in R_l1,l2, following Lassetre [10], we first partition the radial integral in Eq. (4) into contributions from two regions,

$$\begin{eqnarray*}\int_0^\infty & = & \int_0^{r_c} + \int_{r_c}^\infty \\ & = & I_0 + I_d, \end{eqnarray*}$$

where r_c is chosen to be sufficiently large that the wavefunctions will have reached, at least approximately [11], their asymptotic form given by Eq. (6). If we analytically evaluate the integral I_d using (6) and retain only the dominant term in the soft photon limit, we obtain

$$I_d \rightarrow N_{l_1} N_{l_2} {4\sqrt{E}\over\pi} {\sin\Delta_{l_1,l_2}(E)\over k^2}\mbox{ as }k\rightarrow 0,$$ (9)

where we have taken $E_1\approx E_2=E$. Here $\Delta_{l_1,l_2}(E) = \delta_{l_<} - \delta_{l_>} + \sigma_{l_<} - \sigma_{l_>}$, where l_;SPMlt; (l_;SPMgt;) is the lesser (greater) of l₁ and l₂. In the soft photon limit, I_d diverges as 1/k². Since the integrand is well behaved in the region from 0 to r_c, the integral I₀ does not give a divergent contribution (for finite r_c) even in the soft photon limit.

Thus we obtain (retaining only I_d) [12]

$$R_{l_1,l_2}(E_1,E_2) \rightarrow N_{l_1} N_{l_2} {4\sqrt{E}\... ...a_{l_1,l_2}(E)\over k^2} \mbox{\quad as\quad} k\rightarrow 0.$$ (10)

We see that in the soft photon limit R_l1,l2(E₁,E₂) is singular with a sign that depends upon the phase shift difference $\Delta_{l_1,l_2}(E)$ and on the signs of N_l1 and N_l2. Whenever $\Delta_{l_1,l_2}(E)$ crosses $n\pi$, for integers n, as a function of E the matrix element in the soft photon limit passes through zero and changes sign. Examples of such crossings can be found in realistic elastic scattering phase shift calculations [13,14,15]. Of course, it may be possible for $\Delta_{l_1,l_2}(E)$ to become equal to $n\pi$ without crossing it. In such a case, the soft photon matrix element would have a zero but would not change sign. We do not address such phenomena explicitly here and we will be concerned with and refer to ``zero crossings'' throughout this paper.

In the pure Coulomb case $\Delta_{l_1,l_2}(E)$ can be written analytically, using the relation [16]

$$\arg\Gamma(z+1) = \arg\Gamma(z) + \tan^{-1}{y\over x},$$ (11)

where z=x+iy and x and y are real and the range of the arctangent is taken to be $-\pi/2$ to $\pi/2$. Using this relation and Eq. (3) we obtain

 

$$\Delta_{l_1,l_2}^{\rm coul}(E) -\tan^{-1}{\eta_{\rm ion}\over l_< + 1},$$ =

$$-\sin^{-1}\left\{{\eta_{\rm ion}\over l_< + 1} \left[1+{\eta_{\rm ion}^2\over (l_< + 1)^2}\right]^{-1/2}\right\},$$ = (12)

with the range of the arcsine function taken to be $-\pi/2$ to $\pi/2$. For an attractive Coulomb potential ( $\eta_{\rm ion} < 0$), we have $0<\Delta_{l_1,l_2}^{\rm coul}(E) < \pi/2$. The Coulomb normalization is given by Eq. (8) and is always positive. Thus, in an attractive pure Coulomb potential, the matrix element in the soft photon limit R_l1,l2(E,E) never changes sign. In the pure Coulomb case we obtain the soft photon result [17]

$$R_{l_1,l_2}^{\rm coul} = - {4\sqrt{E}\over\pi}N_{l_1}N_{l_2}... ...1} \left[1 + {\eta_{\rm ion}^2\over (l_<+1)^2}\right]^{-1/2},$$ (13)

which is always positive for an attractive Coulomb potential ( $\eta_{\rm ion} < 0$), so that there are no soft photon zero crossings in this case. We note that in a pure Coulomb potential it has been shown [18] that, more generally, the matrix element R_l1,l2 has no zeros, is always positive and is a monotonically decreasing function as one goes away from the singularity of the soft photon limit.

In a neutral or partially ionized atom, the detailed behavior of the phase shifts depends upon the potential under consideration. In general we can require [6] that $\delta_l\rightarrow 0$ as $E\rightarrow \infty$; $\sigma_l$ has the same property. Thus we have

$$\Delta_{l_1,l_2}(E) \rightarrow 0\quad\mbox{as } E\rightarrow\infty.$$ (14)

For neutral atoms, invoking Levinson's theorem [19,6], we can require that the short range phase shifts, taken to be continuous in energy [6], go to $n_l\pi$ as $E\rightarrow 0$, where n_lis the number of bound states with angular momentum l (with the exception of potentials with a virtual l=0 bound state at threshold, sometime called a ``half bound state'', in which case we define n₀ as the number of actual l=0 bound states plus ${1\over 2}$). Therefore

$$\Delta_{l_1,l_2}(E) \rightarrow \left(n_{l_<} - n_{l_>}\right)\pi\quad\mbox{as } E\rightarrow 0.$$ (15)

For neutral atom potentials which do not support an l=0 virtual bound state at threshold $R_{{l_1,l_2}}(E,E)\rightarrow 0$ as $E\rightarrow 0$. For such potentials, since $\Delta_{l_1,l_2}(E)$ is a continuous function of E, if $\left\vert \Delta_{l_1,l_2}(0)\right\vert > \pi$ there will be at least $\left\vert n_{l_<}-n_{l_>}\right\vert - 1$ additional zero crossings in R_l1,l2(E,E) for E > 0 (and a zero in the limit $E\rightarrow \infty$), assuming there are no bound states in the continuum. For neutral atom potentials which do support an l=0virtual bound state at E=0, the number of zero crossings for $E\ge 0$, in addition to the zero at infinite energy, is $\left\vert n_{l_<} - n_{l_>}-1/2\right\vert$ when $\left\vert \Delta_{l_1,l_2}\right\vert \ge 3\pi/2$. For such cases the matrix element is non-zero at E₁=E₂=0. We note that these conditions for soft photon zero crossings would force a zero in the soft photon radial matrix element. They are sufficient, but not necessary, conditions for the existence of zero crossings in the soft photon limit. In Section 3 we will give an example of zeros in the soft photon limit, corresponding to Ramsauer-Townsend minima in elastic scattering, which are not required by the arguments above.

For positive ions, we can use a generalization of Levinson's theorem [20,21,22] which gives the zero energy phase shifts in terms of zero energy quantum defects $\mu_l(0)$. We make the replacements

$$\begin{eqnarray*}n_{l_>} & \rightarrow & \mu_{l_>}(0),\\ n_{l_<} & \rightarrow & \mu_{l_<}(0), \end{eqnarray*}$$

and use the result for the difference of zero energy Coulomb phase shifts, $\Delta_{l_1,l_2}^{\rm coul}(0) = \pi/2$, from Eq. (12), to obtain instead of (15)

$$\Delta_{l_1,l_2}^{\rm ion}(0) = \left(\mu_{l_<}(0) - \mu_{l_>}(0)\right)\pi + \pi/2.$$ (16)

If $\left\vert\Delta_{l_1,l_2}^{\rm ion}(0) \right\vert > \pi$ then soft photon zeros (in addition to the zero for $E\rightarrow \infty$) for E > 0are required in the ionic cases. Again, this condition is sufficient but not necessary. There will be a soft photon zero at E=0 in the ionic case only if $\Delta_{l_1,l_2}^{\rm ion}(0){\rm mod\,}\pi = 0$. In a study of photoionization, Yang [23] gives quantum defect differences at threshold as a function of Z for all Z, showing that, for the potentials considered in his work, this condition is satisfied for many atoms. For example, at Z=30 Yang finds that $\mu_0 - \mu_1 \approx 0.6$ giving $\Delta_{l_1,l_2}^{\rm ion}(0) \approx 1.1\pi$.

In this subsection we have shown that the sign of the free-free radial matrix element in the soft photon limit can be obtained from a knowledge of the elastic scattering phase shifts. We have also demonstrated that Levinson's theorem or its generalization can be applied to deduce the required existence of zero crossings in the soft photon radial matrix elements in some cases. We will show in the next subsections that sign changes in the soft photon matrix element are indications of the existence of zero crossings in the general bremsstrahlung radial matrix element R_l1,l2(E₁,E₂), even away from the soft photon limit.