July, 6, 2007 Investigations of Geometry Influence on the Fission Fragments Behaviour

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1. Theoretical Calculations

Relativistic charged particles lose energy in matter primarily by ionization. The mean rate of energy loss is given by the Bethe_Bloch equation,

- [math]\frac{dE}{dx}[/math]=K[math]z^2\frac{Z}{A}[/math][math]\frac{1}{\beta^2}[/math][[math]\frac{1}{2}\ln\frac{2m_ec^2\beta^2\gamma^2T_{max}}{I^2}[/math] - [math]\beta^2[/math] - [math]\frac{\delta}{2}[/math]]
[math]\frac{K}{A}[/math] = 0.307075 MeV [math]g^{-1} cm^2[/math]
[math]T_{max}[/math]=61 keV
For example, for incident particle Ce-140 and target U-238, we have following results for energy loss
Density of Uranium is 19.1 [math]\frac{g}{cm^3}[/math] [1]

- [math]\frac{dE}{dx}[/math]=[math]0.3071*58^2*19.1[/math] [math]{\frac{92}{140}} {\frac{1}{0.033^2}} \frac{1}{2}\ln[\frac{2*0.51*0.033^2*0.061}{883.2^2*10^{-12}}][/math] = 2.66 [math]\frac{MeV}{nm}[/math]
The half length [math]X_{\frac{1}{2}}[/math] is the distance needed to reduce the intensity in half

[math]X_{\frac{1}{2}}[/math] = [math]\frac{61}{2*2.66*10^3}[/math] nm = 0.0115 nm

Time t=[math]\frac{0.0115*10^{-9}}{0.033*3*10^8}[/math] = [math]1.16*10^{-9}[/math] nsec

2. Simulation

(i) Simulated half-length for totally ionized unexited Ce-140 (A=140, Z=58, Q=58, E'=0) having energy 61 keV propagated thru the U-238 target:

Ce attenuation U 2.jpg


[math]X_{\frac{1}{2}} = 0.33 nm = 3.3 A [/math]

Time needed to the Ce-140 ion to pass half-length:

Time = [math]3.3*10^{-17} s[/math]

(ii) Energy spectrum of Ce-140 ions (E = 61 keV). Number of incident ions was 20,000:

Energy sp Ce.jpg