Phy5645/Cross Section Relation: Difference between revisions
(New page: Consider the scattering of a particle from a real spherically symmetric potential. If <math>\frac{\mathrm{d} \sigma (\theta) }{\mathrm{d} \Omega }</math> is the differential cross section ...) |
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The differential cross section is related to the scattering amplitude through | The differential cross section is related to the scattering amplitude through | ||
<math>\frac{\mathrm{d} \sigma (\theta)}{\mathrm{d} \Omega} = |f_{k}(\theta)|^2</math> | <math>\frac{\mathrm{d} \sigma (\theta)}{\mathrm{d} \Omega} = |f_{k}(\theta)|^2</math> | ||
Since <math>| f |^2 = (Re f )^2 + (Im f )^2 \geq (Im f )^2</math> | Since <math>| f |^2 = (Re f )^2 + (Im f )^2 \geq (Im f )^2</math> | ||
therefore, \frac{\mathrm{d} \sigma (\theta)}{\mathrm{d} \Omega} \geq (Im f_{k}(\theta))^{2} | therefore, \frac{\mathrm{d} \sigma (\theta)}{\mathrm{d} \Omega} \geq (Im f_{k}(\theta))^{2} | ||
On the other hand, from the optical theorem we have | On the other hand, from the optical theorem we have | ||
<math> \sigma =\frac{4\pi}{k} Im f_{k}(\theta)) \leq \frac{4\pi}{k}\sqrt{\frac{\mathrm{d} \sigma (0) }{\mathrm{d} \Omega }}</math> | <math> \sigma =\frac{4\pi}{k} Im f_{k}(\theta)) \leq \frac{4\pi}{k}\sqrt{\frac{\mathrm{d} \sigma (0) }{\mathrm{d} \Omega }}</math> | ||
For a central potential the scattering amplitude is | For a central potential the scattering amplitude is | ||
<math>f_k(\theta) = \frac{1}{k}\sum_{l = 0}^{\infty}(2l + 1) e^{i\delta _{l}} sin\delta _{l} P_{l} (cos \theta)</math> | <math>f_k(\theta) = \frac{1}{k}\sum_{l = 0}^{\infty}(2l + 1) e^{i\delta _{l}} sin\delta _{l} P_{l} (cos \theta)</math> | ||
and, in terms of this, the differential cross section is | and, in terms of this, the differential cross section is | ||
<math>\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)</math> | <math>\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)</math> | ||
The total cross section is | The total cross section is | ||
<math>\sigma = \frac{4\pi ^2}{k^2}\sum_{l = 0}^{\infty}(2l + 1) sin^2\delta _{l}</math> | <math>\sigma = \frac{4\pi ^2}{k^2}\sum_{l = 0}^{\infty}(2l + 1) sin^2\delta _{l}</math> | ||
Since <math> P_{l | Since <math> P_{l}(1)= 1</math> we can write | ||
<math>\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</math> | <math>\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</math> | ||
Revision as of 18:42, 7 December 2009
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 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} 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 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 | f |^2 = (Re f )^2 + (Im f )^2 \geq (Im f )^2}
therefore, \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
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 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 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 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}
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) sin\delta _{l}cos\delta _{l} + isin^2\delta _{l} \right ]^2}
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) 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} 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 \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}}}