(*Purpose:This code was written in order to estimate
226Ra and 222Rn acitivites before equilibrium is established*)
(*Author:This code was written by Charles A Wilson IV using Wolfram Mathematica 10.4*) (*Copyright 2017. Last Modified:4/24/2018*)
(*Notes:The Bateman equation was used for this code, utilizing the solution published by Skrable et al.1974*)
(*Notes:This approach was inpsired by a simmilar approach published by Li et al.2015*) (*Notes:This code is part of a submission
to Operational Radiation Safety by Wilson et al.2018*) (*Notes:For questions contact Charles Wilson [email protected]*) (*Notes:This code may be modified and redistributed
provided that the original authors and source are credited*) (*Inputs*)
(*Time in days after sealing sample*) t1=.208333
t3=1.041667
(*Sample Run Time in days*) t2=10/24
t4=10/24
(*activity according to HpGe Gennie 2k provided in Bq*) ActPb1 = 583.6388*.037
ActBi1 = 565.5061*.037 ActPb2 =1584.641*.037 ActBi2 = 1631.606*.037 (*Decay Constants day^-1*) λ1 = 0.693/ (1602*365.25) λ2 = 0.693/ (3.823)
λ3 = 0.693/ (26.8/60/24) λ4 = 0.693/ (19.7/60/24) (*End Inputs*)
(*Total disentigrations over th sample time*) DisentPb1 =ActPb1*t2*24*60*60
DisentBi1 =ActBi1*t2*24*60*60 DisentPb2 =ActPb2*t2*24*60*60 DisentBi2 = ActBi2*t2*24*60*60 (*Equations for 214Pb and 214Bi*) Pb1=IntegrateA0* λ3 λ1
λ2- λ1 λ2
λ3- λ1 Exp[-λ1*t] + λ1
λ1- λ2 λ2
λ3- λ2 Exp[-λ2*t] + λ1 λ1- λ3
λ2
λ2- λ3 *Exp[-λ3*t] + B0* λ3 λ2
λ3- λ2 *Exp[-λ2*t] + λ2
λ2- λ3 *Exp[-λ3*t] ,{t, t1, t1+t2}
Bi1=IntegrateA0* λ4 λ1 λ2- λ1
λ2 λ3- λ1
λ3
λ4- λ1*Exp[-λ1*t] +
+ +
λ1 λ1- λ2
λ2 λ3- λ2
λ3
λ4- λ2 *Exp[-λ2*t] + λ1 λ1- λ3
λ2 λ2- λ3
λ3
λ4- λ3 *Exp[-λ3*t] + λ1
λ1- λ4 λ2 λ2- λ4
λ3
λ3- λ4 *Exp[-λ4*t] + B0* λ4 λ2
λ3- λ2 λ3
λ4- λ2 *Exp[-λ2*t] + λ2 λ2- λ3
λ3
λ4- λ3 *Exp[-λ3*t] + λ2
λ2- λ4 λ3
λ3- λ4 *Exp[-λ4*t] ,{t, t1, t1+t2}
Pb2=IntegrateA0* λ3 λ1 λ2- λ1
λ2
λ3- λ1 Exp[-λ1*t] + λ1 λ1- λ2
λ2
λ3- λ2 Exp[-λ2*t] + λ1
λ1- λ3 λ2
λ2- λ3 *Exp[-λ3*t] + B0* λ3 λ2
λ3- λ2 *Exp[-λ2*t] + λ2
λ2- λ3 *Exp[-λ3*t] ,{t, t3, t3+t4}
Bi2=IntegrateA0* λ4 λ1 λ2- λ1
λ2 λ3- λ1
λ3
λ4- λ1*Exp[-λ1*t] + λ1
λ1- λ2 λ2 λ3- λ2
λ3
λ4- λ2 *Exp[-λ2*t] + λ1 λ1- λ3
λ2 λ2- λ3
λ3
λ4- λ3 *Exp[-λ3*t] + λ1
λ1- λ4 λ2 λ2- λ4
λ3
λ3- λ4 *Exp[-λ4*t] + B0* λ4 λ2
λ3- λ2 λ3
λ4- λ2 *Exp[-λ2*t] + λ2 λ2- λ3
λ3
λ4- λ3 *Exp[-λ3*t] + λ2
λ2- λ4 λ3
λ3- λ4 *Exp[-λ4*t] ,{t, t3, t3+t4}
(*Simplification in terms of original N of 222Rn and 226Ra*) Pb1o=CoefficientList[Pb1,{B0, A0}]
Bi1o=CoefficientList[Bi1,{B0, A0}]
Pb2o=CoefficientList[Pb2,{B0, A0}]
Bi2o=CoefficientList[Bi2,{B0, A0}]
(*Matrix Math*)
F= {{Pb1o[[1, 2]], Pb1o[[2, 1]]},{Bi1o[[1, 2]], Bi1o[[2, 1]]}, {Pb2o[[1, 2]], Pb2o[[2, 1]]},{Bi2o[[1, 2]], Bi2o[[2, 1]]}}
Nn= {{DisentPb1},{DisentBi1},{DisentPb2},{DisentBi2}};
W= {{DisentPb1, 0, 0, 0},{0, DisentBi1, 0, 0},{0, 0, DisentPb2, 0},{0, 0, 0, DisentBi2}};
R=Inverse[Transpose[F].Inverse[W].F].Transpose[F].Inverse[W].Nn;
(*Initial N of 222Rn and 226 Ra in Bq*) R[[1, 1]]
R[[2, 1]]
(*Initial activity of 222Rn and 226 Ra in Bq*) R[[1, 1]] *0.693/ (1602*365.25*24*3600) R[[2, 1]] * 0.693/ (3.823*24*3600) 2
