Law of decay

Exponential decrease of a quantity from the initial value N - z. B. the number of radioactive atomic nuclei in a given substance sample - with time t .

The law of decay is the name commonly used in physics for the equation that describes an exponential decrease in quantities over time. In nuclear physics , the law of decay indicates the number of atomic nuclei in a radioactive substance sample that have not yet decayed at a given point in time . This number is ${\displaystyle N}$ ${\displaystyle t}$

${\displaystyle N(t)=N_{0}\cdot \mathrm {e} ^{-\lambda t}}$,

where is the number of nuclei present at the beginning ( ) and the decay constant of the nuclide in question . ${\displaystyle N_{0}}$ ${\displaystyle t=0}$ ${\displaystyle \lambda }$

Derivation

If one looks at a radioactive preparation with initially atomic nuclei and the activity , the following applies to the number of nuclei that have not yet decayed during the period : ${\displaystyle N_{0}}$ ${\displaystyle A}$ ${\displaystyle N}$ ${\displaystyle t}$

${\displaystyle {\begin{aligned}A&=-{\frac {\mathrm {d} N}{\mathrm {d} t}}\qquad {\text{mit }}A=\lambda \cdot N\\-\lambda \cdot N&={\frac {\mathrm {d} N}{\mathrm {d} t}}\\-\lambda \cdot \mathrm {d} t&={\frac {1}{N}}\cdot \mathrm {d} N\\\int _{0}^{t}-\lambda \cdot \mathrm {d} t'&=\int _{N_{0}}^{N}{\frac {1}{N'}}\cdot \mathrm {d} N'\\-\lambda t-(-\lambda \cdot 0)&=\ln(N)-\ln(N_{0})\\-\lambda t&=\ln \left({\frac {N}{N_{0}}}\right)\\\mathrm {e} ^{-\lambda t}&={\frac {N}{N_{0}}}\\N(t)&=N_{0}\cdot \mathrm {e} ^{-\lambda t}\end{aligned}}}$

So after the time there are still some starting cores left. ${\displaystyle t}$ ${\displaystyle N_{0}}$ ${\displaystyle N(t)}$

Average lifespan

The decay constant ( lambda ) is the reciprocal of the mean life , i.e. the time after which the number of atoms has decreased by the factor . ( Tau ) differs from the half-life only by the constant factor : ${\displaystyle \lambda }$ ${\displaystyle \tau =1/\lambda }$ ${\displaystyle \mathrm {e} =2{,}71828\dotso }$ ${\displaystyle \tau }$ ${\displaystyle T_{1/2}}$ ${\displaystyle \ln 2}$

${\displaystyle T_{1/2}={\frac {\ln 2}{\lambda }}=\tau \cdot \ln 2\approx 0{,}693\cdot \tau }$

This results in the following form for the law of decay:

${\displaystyle N(t)=N_{0}\cdot e^{-{\frac {\ln(2)}{T_{1/2}}}t}}$

Web links

• Java animation of the law of decay