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| '''Calculation of Gamow Factor for Alpha decay of Nuclei'''
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| [[Image:Gamow.jpg|thumb|500px]]
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| Since the <math>\alpha</math>-decay happens in the nulcie then we can assume that an <math>\alpha</math>-decay is formed in the nucleus just before its emission (although <math>\alpha</math> particle doesnot exist in the nucleus).
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| Inside the nucleus the particle will experience nuclear interaction which mostly attractive and outside the nucleus the inetraction would be coulombic(replusive).
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| Since the mathematical form of the nuclear interaction is not known we can model it by a square well type potential for the present purpose.
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| Outside the range of the nuclear interaction would be coulombic. So the coulomb interaction is,
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| <math>V_{coul} = \frac{1}{4\pi\epsilon_{0}}\frac{2z_{1}e^{2}}{r}</math>
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| where <math>Z_{1}</math> is the atomic number of the rest of the nucleus(after decay).
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| From the WKB apporximation we know that at the turning point, <math>E= V(x)= V_{coul} = \frac{1}{4\pi\epsilon_{0}}\frac{2z_{1}e^{2}}{R_{c}}</math> | | From the WKB apporximation we know that at the turning point, <math>E= V(x)= V_{coul} = \frac{1}{4\pi\epsilon_{0}}\frac{2z_{1}e^{2}}{R_{c}}</math> |
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From the WKB apporximation we know that at the turning point, 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 E= V(x)= V_{coul} = \frac{1}{4\pi\epsilon_{0}}\frac{2z_{1}e^{2}}{R_{c}}}
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 R_{c} = \frac{1}{4\pi\epsilon_{0}}\frac{2z_{1}e^{2}}{E}}
Now the Transition probabilty
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 T\cong \Theta ^{2}}
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where 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 \Theta = e^{-\int_{b}^{a}q(x)dx}}
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 q(x)= \frac{1}{\hbar}\sqrt{2m\left(V(x)-E\right)}}
In the present problem 
and 
Now,
![{\displaystyle =\left({\frac {2m}{\hbar ^{2}}}\right)^{\frac {1}{2}}\left({\frac {2z_{1}e^{2}}{4\pi \epsilon _{0}}}\right)^{\frac {1}{2}}\int _{R}^{R_{c}}\left[{\frac {1}{r}}-{\frac {1}{R_{c}}}\right]^{\frac {1}{2}}dr}](https://wikimedia.org/api/rest_v1/media/math/render/svg/704c867fbfd68503d98e2352a56e3dc0856925ed)
let, ![{\displaystyle I=\int _{R}^{R_{c}}\left[{\frac {1}{r}}-{\frac {1}{R_{c}}}\right]^{\frac {1}{2}}dr}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b09dbdd1fdfee1a3afae0789b0d6929db2f56438)
Put,
and
![{\displaystyle I=R_{c}^{\frac {1}{2}}\left[\theta -sin\theta cos\theta \right]_{0}^{cos^{-1}{\sqrt {\frac {R}{R_{c}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aa79c198a1ee43e754aac6919586bef605162a1)
![{\displaystyle I=R_{c}^{\frac {1}{2}}\left[cos^{-1}{\sqrt {\frac {R}{R_{c}}}}-sin\left(cos^{-1}{\sqrt {\frac {R}{R_{c}}}}\right)cos\left(cos^{-1}{\sqrt {\frac {R}{R_{c}}}}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f3e9d4836a3cccf4afec6f2113a890089721da1)
![{\displaystyle I=R_{c}^{\frac {1}{2}}\left[cos^{-1}{\sqrt {\frac {R}{R_{c}}}}-{\sqrt {\frac {R}{R_{c}}}}{\sqrt {1-{\frac {R}{R_{c}}}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd8ea106e695fc1a0431d281b0751adffd8b22b)
![{\displaystyle I=R_{c}^{\frac {1}{2}}\left[cos^{-1}{\sqrt {\frac {R}{R_{c}}}}-{\sqrt {{\frac {R}{R_{c}}}-\left({\frac {R}{R_{c}}}\right)^{2}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bee83f3d745631af80ff96e0e49ddbdc1b021b4)
Let us consider 
Then we have
where 
Setting, charge of 
particle = 2= 
(in general)
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 \int q(x)dx = \left ( \frac{2Mz_{1}z_{2}e^{2}R_{c}}{\hbar^{2}4\pi\epsilon_0} \right )^{\frac{1}{2}}\left [\frac{\pi}{2} - 2\left(\frac{R}{R_{c}}\right)^{\frac{1}{2}} \right ]}
Now 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 T\cong e^{-2\int q(x)dx} = exp\left [ -\frac{\pi z_{1}z_{2}e^{2}}{\hbar 4\pi\epsilon_0} \left (\frac{2M}{e} \right )^{2} + \frac{4}{\hbar} \left ( \frac{2z_{1}z_{2}e^{2}MR}{4\pi\epsilon_0} \right )^{\frac{1}{2}}\right ]}
Now putting 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 E= \frac{1}{2}mv^{2}}
, veloctiy of the particle
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 T\cong exp\left ( \frac{-2\pi z_{1}z_{2}e^{2}}{4\pi\epsilon_0\hbar v} \right )exp \left ( \frac{32z_{1}z_{2}e^{2}MR}{4\pi\epsilon_0\hbar^{2} } \right )^{\frac{1}{2}}}
The 1st exponential term is known as the Gamow factor. The Gamow factor determines the dependence of the probability on the speed (or energy) of the alpha particle.
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