Decay rate always wins over diffusion

The sources establish that the behavior of a radioactive material in a substance is governed by a reaction-diffusion equation, $\partial_t n - D \nabla^2 n = -\lambda n$, where radioactive decay is incorporated as a negative source term or sink ($\kappa = -\lambda n$), reflecting the inherent decrease in the number density ($n$) of radioactive nuclei without production. This sink is defined by the decay constant ($\lambda$), which is related to the half-life ($\tau$) by $\lambda=1 / \tau$. Physically, the diffusivity ($D$) and the decay rate ($\lambda$) exert independent and accelerated control over the system; a higher diffusivity speeds up the spatial redistribution and peak flattening while conserving mass, whereas a higher decay rate dictates the system's temporal lifespan by acting as an accelerated mass sink, causing the concentration profile to shrink exponentially downwards. Crucially, in the combined scenario, the decay mechanism's rapid destruction of the material ensures that the overall mass vanishes before extensive diffusion can occur, confirming that the decay rate ultimately dominates the total existence time of the material within the domain.

📎IllustraDemo

A derivative illustration based on our specific text and creative direction

🏗️Structural clarification of Condensed Notes

block-beta
columns 5
CC["Criss-Cross"]:5

%% Condensed Notes

CN["Condensed Notes"]:5
RF["Relevant File"]:5
NV["Narrated Video"]:4 VO["Voice-over"] 
PA("Plotting & Analysis")AA("Animation & Analysis")KT("Summary & Interpretation") ID("Illustration & Demo") PO("Polyptych")

%% Proof and Derivation

PD["Proof and Derivation"]:5
AF("Derivation Sheet"):5
NV2["Narrated Video"]:4 VO2["Voice-over"]
PA2("Plotting & Analysis")AA2("Animation & Analysis")KT2("Summary & Interpretation") ID2("Illustration & Demo") PO2("Polyptych")

classDef color_1 fill:#8e562f,stroke:#8e562f,color:#fff
class CC color_1

classDef color_2 fill:#14626e,stroke:#14626e,color:#fff
class CN color_2
class RF color_2

classDef color_3 fill:#1e81b0,stroke:#1e81b0,color:#fff
class NV color_3
class PA color_3
class AA color_3
class KT color_3
class ID color_3

classDef color_4 fill:#47a291,stroke:#47a291,color:#fff
class VO color_4
class PO color_4

%% Proof and Derivation

classDef color_5 fill:#307834,stroke:#307834,color:#307834
class PD color_5
class AF color_5

classDef color_6 fill:#38b01e,stroke:#38b01e,color:#38b01e
class NV2 color_6
class PA2 color_6
class AA2 color_6
class KT2 color_6
class ID2 color_6

classDef color_7 fill:#47a291,stroke:#47a291,color:#47a291
class VO2 color_7
class PO2 color_7

🗒️Downloadable Files - Recursive updates

Asset

Copyright Notice

All content and images on this page are the property of Sayako Dean, unless otherwise stated. They are protected by United States and international copyright laws. Any unauthorized use, reproduction, or distribution is strictly prohibited. For permission requests, please contact [email protected]

</aside>