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Deflection function and differential cross section = MESO-m3c-defl

The deflection function expresses the deflection angle of a scattered particle as a function of the collision parameter. In the study of differential cross sections, the deflection function can be used for three purposes: The value of the classical differential cross section is proportional to the derivative of the deflection curve $\vert\dr b/\dr{\mit\Theta}\vert$ at the relevant angle, while the semiclassical interference structure needs the calculation of phase shifts by integrating over the deflection cube. Moreover, the deflection curve is a handy visual link at the adjustment of the potential parameters to fit the measured and calculated differential cross sections.

Consider two colliding particles interacting via a spherically symmetric potential function V(r) and suppose that the relative motion of the particles can be described by classical mechanics. Fig. MESO-m3c-defl-F1 shows the scattering trajectory in the centre of mass system near the scattering centre M in the case of a central-force field, consisting of attractive forces at large distances and short-range repulsive forces. The particles approach each other with an impact parameter b and a relative kinetic energy at infinity E.
\end{figure} Figure MESO-m3c-defl-F1: Scattering of a particle by a symmetric central-force field with centre at M.

Using polar coordinates $r,\varphi$ with the origin at M, the scattering angle ${\mit\Theta }$ can be deduced by applying the conservation theorems for energy and angular momentum.
[unfold details of the derivation]

This leads to the classical deflection-angle formula:

Theta =\pi-2b\int\limits_{R_0}^\infty
\frac{\dr r}{r^2[1-V(r)/E-b^2/r^2]^{1/2}}
where the distance of closest approach R0 is the outermost zero point of the square root term:
By substituting the potential functions in the classical scattering function, the deflection curve can be calculated numerically. For small angle scattering, the impact parameter approximation is often used, in which it is assumed that $V(r)/E \ll 1$.

We consider the deflection functions for ion-pair formation (in a centre of mass system), which can be described via the crossing of ionic and covalent potential surfaces [The crossing of potential surfaces is clarified in a segment of a mesoscopic <i>Theoretical methods</i> module (link type: `detailed in/clarified in/project'; target: MESO-m3c-mod#crossing potentials)]).

An asymmetric collision process needs a summation over the different contributions to the total scattering angle. Incoming on the covalent curve and outgoing on the ionic one, the scattering angle is given by:

$\displaystyle {\mit\Theta}=\pi$ - $\displaystyle b\int_{R_\crm}^\infty\frac{\dr
R}{R^2 (1-U_{\cov}(R)/E_i - b^2/R^2)^{1/2}}$  
  - $\displaystyle b\int_{R_\crm}^\infty\frac{\dr
R}{R^2(1-U_{\ion}(R)/E_i - b^2/R^2)^{1/2}}$  
  - $\displaystyle 2b\int_{R_0}^{R_\crm}\frac{\dr R}{R^2(1-U(R)/E_i - b^2/R^2)^{1/2}},$
where Ucov(R) is the covalent potential function, Uion(R) the ionic potential function
and U(R) in the third term is the ionic or covalent potential function, depending on the scattering path considered inside the crossing radius Rc [Why the third term depends on the path is explained in the mesoscopic module on the model (link type: `elaborated in/explained in/project'; target: MESO-m3c-mod#trajectories)]. R0 means the distance of closest approach. The exclusive use of the initial parameters b and Ei is permitted by an appropriate choice of the zero point of the potential energy at the entrance channel of the collision.

Experimentally, only the absolute value of the classical deflection function $\theta = \vert\Theta\vert$ is meaningful, because the deflection angles $\Theta$ and $- \Theta$ resulting from repulsive resp. attractive scattering on different sides of the target cannot be distinguished.

Typical shape of the deflection function

Fig. MESO-m3c-defl-F2 shows the typical shape of the deflection function for ion-pair formation.

\end{figure} Figure MESO-m3c-defl-F2: Typical deflection curves for chemi-ionization scattering (CM system). The two curves due to ionic and covalent scattering are connected $b\approx R_\crm$. Because of the several interference features, the ionic curve is split up into b, c and e branches, the covalent curve into a and d branches.

The deflection function shows several peculiar features. Because there are two trajectories leading to ionization, the deflection functions consist of two parts connected at $b\approx R_\mathrm{c}$, resulting in a closed deflection function. Of course there is no large-b deflection curve because the classical turning point has to be smaller than or equal to Rc to have the possibility of ionization. In the case in which the classical turning point is equal to Rc, both trajectories merge leading to the connection point of the two branches of the deflection function.

Differential cross section

Due to this composite character of the deflection function there are up to four impact parameters leading to scattering over the same angle for small-angle scattering. Even at large-angle scattering there are always two impact parameters for one angle. By simple addition of the contributions of both trajectories, the classical chemi-ionization differential cross section is given by [The derivation of this formula is given in a macroscopic module (link type: `explained in/elaborated in/project/wider range'; target: MACRO-m3c-diff]

$\displaystyle I(\theta)=\frac{1}{\sin \, \theta}\sum_{i=1,2,\ldots}\
P_{b_{i}}(1-P_{b_{i}})b_{i}\bigg\vert\frac{\dr b_{i}}{\dr {{\mit\Theta}
The same impact parameters are important in the semiclassical treatment [More on the semiclassical treatment in Ford (link type: (elaborated in/external'; target: Ford] extended with the stationary-phase approximation [Somewhere, more details and more context concerning this approximation should be given (link type: `elaborated in'; target: stationary phase)]. Then the scattering amplitude $f(\theta)$ is built up again either by four or by two contributions:
leading to the differential cross section
\begin{displaymath}I(\theta) = \big\vert f(\theta)\big\vert^2.
For simplicity we shall split up the deflection curve into five branches a, b, c, d and e, where a refers to that part of the deflection curve corresponding to attractive covalent scattering, b to the outer attractive ionic branch, c to the inner attractive ionic branch, d to the repulsive covalent branch and e to the repulsive ionic branch as indicated in Fig. MESO-m3c-defl-F2. Then the possible contributions $f_i(\theta)$ to $f(\theta)$ are given in lowest-order stationary-phase approximation by
$\displaystyle f_\ar(\theta)$ = $\displaystyle \bigg[ \Big\vert\frac{\dr b_\ar}{\dr{\mit\Theta}}\Big\vert b_\ar
P_{b_\ar}(1-P_{b_\ar})/\sin\theta \bigg]^{1/2}$
    $\displaystyle \times\exp\big[ \ir(2\eta_{b_\ar}+kb_\ar\theta-\pi) \big]\equiv
A\er^{\ir\alpha},{\rm }$
$\displaystyle f_\br(\theta)$ = $\displaystyle \bigg[ \Big\vert\frac{\dr b_\br}{\dr{\mit\Theta}}\Big\vert b_\br
P_{b_\br}(1-P_{b_\br})/\sin\theta \bigg]^{1/2}$  
    $\displaystyle \times\exp\bigg[ \ir\Big(2\eta_{b_\br}+kb_\br\theta-\frac{1}{2}\pi\Big) \bigg]\equiv
B\er^{\ir\beta},{\rm }$
$\displaystyle f_\crm(\theta)$ = $\displaystyle \bigg[ \Big\vert\frac{\dr b_\crm}{\dr{\mit\Theta}}\Big\vert b_\crm
P_{b_\crm}(1-P_{b_\crm})/\sin\theta \bigg]^{1/2}$  
    $\displaystyle \times\exp\big[ \ir(2\eta_{b_\crm}+kb_\crm\theta-\pi) \big]\equiv
C\er^{\ir\gamma},{\rm }$
$\displaystyle f_\dr(\theta)$ = $\displaystyle \bigg[ \Big\vert\frac{\dr b_\dr}{\dr{\mit\Theta}}\Big\vert b_\dr
P_{b_\dr}(1-P_{b_\dr})/\sin\theta \bigg]^{1/2}$  
    $\displaystyle \times\exp\big[ \ir(2\eta_{b_\dr}+kb_\dr\theta-\frac{1}{2}\pi) \big]\equiv
D\er^{\ir\delta},{\rm }$
$\displaystyle f_\er(\theta)$ = $\displaystyle \bigg[ \Big\vert\frac{\dr b_\er}{\dr{\mit\Theta}}\Big\vert b_\er
P_{b_\er}(1-P_{b_\er})/\sin\theta \bigg]^{1/2}$  
    $\displaystyle \times\exp\bigg[ \ir\Big(2\eta_{b_\er}+kb_\er\theta-\frac{1}{2}\pi\Big) \bigg]\equiv
where $\eta_\br$ indicates the phase shift and k is given by $k =(2\mu
E_\ir)^{1/2}$. For a certain scattering angle $\theta$ where four contributions form the cross section (for instance the contributions from the branches a, c, d and e), the differential cross section is given by:
$\displaystyle I(\theta)$ = $\displaystyle \big\vert A\er^{\ir\alpha}+C\er^{\ir\gamma}+D\er^{\ir\delta}+
  = $\displaystyle A^2+C^2+D^2+E^2+2AC\cos(\alpha-\gamma)$  
    $\displaystyle +2AD\cos(\alpha-\delta)+2AE\cos(\alpha-\varepsilon)$  
    $\displaystyle +2CD\cos(\gamma-\delta)+2CE\cos(\gamma-\vare)$  
    $\displaystyle +2DE\cos(\delta-\vare).$