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textbook:nrctextbook:chapter15 [2025-04-28 15:53]
Merja Herzig 		textbook:nrctextbook:chapter15 [2025-05-05 11:10] (current)
Merja Herzig
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 When [[textbook:nrctextbook:chapter2#nucleus|nucleus]] B is produced by irradiating a nucleus A in a nuclear reaction, the equation for the growth rate of the resulting nucleus is: When [[textbook:nrctextbook:chapter2#nucleus|nucleus]] B is produced by irradiating a nucleus A in a nuclear reaction, the equation for the growth rate of the resulting nucleus is:
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 +{{anchor:eq_1511}}
 $$\frac{dN_B}{dt} = \sigma \times \phi \times N_A$$ ;;# $$\frac{dN_B}{dt} = \sigma \times \phi \times N_A$$ ;;#
 [XV.XI] [XV.XI]
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 When producing [[textbook:nrctextbook:chapter4|radionuclides]], the [[textbook:nrctextbook:chapter6#activity|activity]] of the nuclide is of more interest than the number of nuclei. Since $A = \lambda \times N_B$, we can replace $N_B$ with $\frac{A}{\lambda}$ in the formula XV.XIII to get: When producing [[textbook:nrctextbook:chapter4|radionuclides]], the [[textbook:nrctextbook:chapter6#activity|activity]] of the nuclide is of more interest than the number of nuclei. Since $A = \lambda \times N_B$, we can replace $N_B$ with $\frac{A}{\lambda}$ in the formula XV.XIII to get:
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 +{{anchor:eq_1514}}
  
 $$A_B = \sigma \times \phi \times N_A \left(1 - e^{-\lambda t}\right)$$  ;;# $$A_B = \sigma \times \phi \times N_A \left(1 - e^{-\lambda t}\right)$$  ;;#
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 The mass m is used instead of the [[textbook:nrctextbook:chapter15#target_nucleus|target nuclei]] number $N_A$ and the half-life t½ is used instead of the decay constant $\lambda$, in which case the formula becomes The mass m is used instead of the [[textbook:nrctextbook:chapter15#target_nucleus|target nuclei]] number $N_A$ and the half-life t½ is used instead of the decay constant $\lambda$, in which case the formula becomes
 ### ###
 +{{anchor:eq_1515}}
 $$A_B = \frac{m \times I \times 6.023 \times 10^{23} \times \sigma \times \phi}{M} \times \left(1 - e^{-\ln 2 \times \frac{t}{t_{1/2}}}\right)$$ ;;# $$A_B = \frac{m \times I \times 6.023 \times 10^{23} \times \sigma \times \phi}{M} \times \left(1 - e^{-\ln 2 \times \frac{t}{t_{1/2}}}\right)$$ ;;#
 [XV.XV] [XV.XV]
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 where $m$ is the mass of the target element, $I$ the target nuclide’s [[textbook:nrctextbook:chapter2#isotope|isotopic abundance]] of the element, 6.023·10<sup>23</sup> is the [[textbook:nrctextbook:chapter3#avogadro_number|Avogadro´s number]], and $M$ is the molar mass of the element. where $m$ is the mass of the target element, $I$ the target nuclide’s [[textbook:nrctextbook:chapter2#isotope|isotopic abundance]] of the element, 6.023·10<sup>23</sup> is the [[textbook:nrctextbook:chapter3#avogadro_number|Avogadro´s number]], and $M$ is the molar mass of the element.
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 +{{anchor:figure_152}}
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 Figure XV.2. shows the relative amount of nuclide produced in the target as a function of irradiation time. Time here is the irradiation time divided by the nuclide’s half-life, i.e. it is the number of half-lives. As seen, 50% of the maximum obtainable activity (saturation activity) is produced during one half-life, 75% during two half-lives, and about 99% during ten. Figure XV.2. shows the relative amount of nuclide produced in the target as a function of irradiation time. Time here is the irradiation time divided by the nuclide’s half-life, i.e. it is the number of half-lives. As seen, 50% of the maximum obtainable activity (saturation activity) is produced during one half-life, 75% during two half-lives, and about 99% during ten.
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 +
 {{:textbook:nrctextbook:relative_amount_of_radionuclide_in_target_as_function_of_irradiation_time_fig_15_2.png?400 |}} {{:textbook:nrctextbook:relative_amount_of_radionuclide_in_target_as_function_of_irradiation_time_fig_15_2.png?400 |}}
 Figure XV.2. The relative amount of a radionuclide in the target as a function of irradiation time up to ten half-lives of the product nuclide and the decay of the product nuclide after irradiation. Figure XV.2. The relative amount of a radionuclide in the target as a function of irradiation time up to ten half-lives of the product nuclide and the decay of the product nuclide after irradiation.
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 +{{anchor:eq_1516}}
 $$A_B = \frac{m \times I \times 6.023 \times 10^{23} \times \sigma \times \phi}{M} \times \left(1 - 2^{-\frac{t}{t_{1/2}}}\right) \times 2^{-\frac{t*}{t_{1/2}}}$$ ;;# $$A_B = \frac{m \times I \times 6.023 \times 10^{23} \times \sigma \times \phi}{M} \times \left(1 - 2^{-\frac{t}{t_{1/2}}}\right) \times 2^{-\frac{t*}{t_{1/2}}}$$ ;;#
 [XV.XVI] [XV.XVI]
textbook/nrctextbook/chapter15.1745848439.txt.gz · Last modified: 2025-04-28 15:53 by Merja Herzig

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