Phy5645/Cross Section Relation

Consider the scattering of a particle from a real spherically symmetric potential. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d} \sigma (\theta) }{\mathrm{d} \Omega }} is the differential cross section and $\sigma$ is the total cross section, show that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma \leq \frac{4\pi}{k}\sqrt{\frac{\mathrm{d} \sigma (0) }{\mathrm{d} \Omega }}}

for a general central potential using the partial-wave expansion of the scattering amplitude and the cross section.

Solution:

The differential cross section is related to the scattering amplitude through

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d} \sigma (\theta)}{\mathrm{d} \Omega} = |f_{k}(\theta)|^2}

Since $|f|^{2}=(Ref)^{2}+(Imf)^{2}\geq (Imf)^{2}$

therefore, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d} \sigma (\theta)}{\mathrm{d} \Omega} \geq (Im f_{k}(\theta))^{2} }

On the other hand, from the optical theorem we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma =\frac{4\pi}{k} Im f_{k}(\theta)) \leq \frac{4\pi}{k}\sqrt{\frac{\mathrm{d} \sigma (0) }{\mathrm{d} \Omega }}}

For a central potential the scattering amplitude is

$f_{k}(\theta )={\frac {1}{k}}\sum _{l=0}^{\infty }(2l+1)e^{i\delta _{l}}sin\delta _{l}P_{l}(cos\theta )$

and, in terms of this, the differential cross section is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d} \sigma (\theta)}{\mathrm{d} \Omega} = \frac{1}{k^2}\sum_{l = 0}^{\infty}\sum_{l^{\prime} = 0}^{\infty}(2l + 1)(2l^{\prime} + 1) e^{i(\delta _{l}- \delta _{l^{\prime}})} sin\delta _{l}sin\delta _{l^{\prime}} P_{l} (cos \theta)P_{l^{\prime}} (cos \theta)}

The total cross section is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma = \frac{4\pi ^2}{k^2}\sum_{l = 0}^{\infty}(2l + 1) sin^2\delta _{l}}

Since $P_{l}(1)=1$ we can write

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d} \sigma (0)}{\mathrm{d} \Omega} = \frac{1}{k^2}\left [\sum_{l = 0}^{\infty}(2l + 1) e^{i\delta _{l}} sin\delta _{l} \right ]^2}

${\frac {\mathrm {d} \sigma (0)}{\mathrm {d} \Omega }}={\frac {1}{k^{2}}}\left[\sum _{l=0}^{\infty }(2l+1)sin\delta _{l}cos\delta _{l}+isin^{2}\delta _{l}\right]^{2}$

${\frac {\mathrm {d} \sigma (0)}{\mathrm {d} \Omega }}={\frac {1}{k^{2}}}\left[\sum _{l=0}^{\infty }(2l+1)sin\delta _{l}cos\delta _{l}\right]^{2}+{\frac {1}{k^{2}}}\left[\sum _{l=0}^{\infty }(2l+1)sin^{2}\delta _{l}\right]^{2}$ $\Rightarrow {\frac {\mathrm {d} \sigma (0)}{\mathrm {d} \Omega }}\geq {\frac {1}{k^{2}}}\left[\sum _{l=0}^{\infty }(2l+1)sin^{2}\delta _{l}\right]^{2}={\frac {k^{2}\sigma ^{2}}{16\pi ^{2}}}$