DIRECTORATE-GENERAL JRC
JOINT RESEARCH CENTRE
Institute for Transuranium Elements
Unit: Nuclear Fuels
Title / Virtual Nuclides:
A Formalism for the Application of the Bateman Solution to Mixtures
Author(s) / J. Magill and J. Galy
Report Nr.: / JRC-ITU-TN-2001/15 (revision 1)
Classification: / Unclassified
Type of report: / Progress report for the development of Nuclides 2000 2nd edition “Nuclides.Net”
Name / Date / Signature
The author / J. Galy / 6th June 2001
approved by the
project leader / J. Magill / 6th June 2001
released by the
Director / R. Schenkel / 6th June 2001
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Virtual Nuclides:
A Formalism for the Application of the Bateman Solution to Mixtures
J. Magill and J. Galy
Abstract:
The solution to the general equations governing first order linear processes was first given by Bateman for the case of radioactive transformations in 1910. This solution can be applied to obtain the quantities of “daughters” present at any time starting from a single parent.
In this paper, we show how the formalism developed by Bateman can be extended to multiple parent mixtures through the concept of a “virtual parent”. This virtual parent decays on timescale much shorter than any processes of interest to the required mixture. The procedure for obtaining the half-life, the daughters, the branching ratios, and the atomic weight ratio of the virtual parent for the real nuclide mixture are described in detail. Applications of this procedure are given for the radioactive decay of simple and complicated nuclide mixtures. In a second application, the concept of the virtual nuclide is applied to dosimetry and shielding calculations on nuclide mixtures.
Finally, a new module is described for use in Nuclides 2000, 2nd edition, for handling nuclide mixtures through the concept of the virtual parent.
Virtual Nuclides:
A Formalism for the Application of the Bateman Solution to Mixtures
Contents
1. General Equation for the Kinetics of Linear First Order Phenomena 5
2. Bateman Solution to the Differential Equations 5
3. Convergent and Divergent Branches 6
4. Application of the Bateman Solution to Mixtures: Virtual Parents 6
5. Properties of the Virtual Parent 7
6. A Simple Two Component Mixture 8
7. More Complicated Mixtures 11
8. An Application to Fission 13
9. Application of Virtual Nuclides to Dosimetry and Shielding 17
10. A New Module for Nuclides 2000 19
11. References 21
- General Equation for the Kinetics of Linear First Order Phenomena
A general linear first order process is shown schematically in fig.1.
where Qi are the amounts of species i present at time t, kQ1 is the total removal constant for the species Qi, kQi,Qi+1 is the partial removal constant leading to the production of species Qi+1, etc, and kQi,Qi+1=BRQi,Qi+1 × kQi,Qi+1 where BRQi,Qi+1 is the branching ratio. The source terms Si are the constant independent rates of production of the ith species.
- Bateman Solution to the Differential Equations
The differential equations governing the main “chain” shown in fig. 1 are given by
dQ1/dt = S1 – kQ1 × Q1
dQ2/dt = S2 + kQ1,Q2 × Q1 – kQ2×Q2
.
.
dQi/dt = Si + kQi-1,Qi × Qi-1 – kQi×Qi
.
dQn/dt = Sn + kQn-1,Qn × Qn-1 – kQn × Qi
The solution to these equations, first obtained by Bateman [1] and discussed by Skrable [2], can be expressed in the form:
(1)
where Qi(0) is amount of the ith species at time t=0. The kn are the total removal constant for species n (k = ln(2)/t1/2 = 0.69315/t1/2), kn,n+1 is the partial decay constant (partial removal constant) and is related to the branching ratio BRn,n+1 through the relation kn,n+1 = BRn,n+1.kn. Clearly, form equation (1) the total removal rate constants must be distinctly different, otherwise the terms in the denominator become indeterminate. For radioactive transformations, this is generally the case.
- Convergent and Divergent Branches
The solution to the differential equations given in equation (1) for the various species produced in series is shown in fig. 1. If branching occurs, as indicated in the figure, the solution (1) must be applied to all possible chains. As an example, we consider the radioactive decay of Ac225. The schematic decay is shown in fig. 2 together with the various paths by which Ac225 can decay. The breakdown into linear chains is shown in table 1. Equation (1) must be applied to each of these three chains. In the evaluation of the total quantities of any species, care is required not to count the same decay more than once.
/ Ac225 / Ac225 / Ac225¯ / ¯ / ¯
Fr221 / Fr221 / Fr221
¯ / ¯ / ¯
At217 / At217 / At217
¯ / ¯ / ¯
Bi213 / Bi213 / Rn217
¯ / ¯ / ¯
Po213 / Tl209 / Po213
¯ / ¯ / ¯
Pb209 / Pb209 / Pb209
¯ / ¯ / ¯
Bi209
¯
Stable / Bi209
¯
Stable / Bi209
¯
Stable
Figure 2. Schematic decay of Ac225. The colours used indicate the type of decay (yellow: alpha emission, blue: beta emission, black: stable). / Table 1. The three “linear chains” for the decay of Ac225 giving the various paths by which the nuclide can decay.
- Application of the Bateman Solution to Mixtures: Virtual Parents
In practice, it often arises that one has not a single parent which can serially transform, but a mixture of parents. In the case of radionuclides, for example, one may be interested in an inventory of nuclides such as in spent fuel. In the algorithm for the solution to equation (1), one could, of course, just sum over the set of parents in the initial configuration. However, already existing algorithms for solving equations (1), may be rather complicated due to the fact that various subsidiary quantities (such as the gamma emission rates, isotopic powers, dose rates etc.) are also evaluated. To extend such algorithms to cover multiple parents is problematic.
A much simpler idea is to define a virtual parent which decays into the required mixture. Consider a mixture containing the components P1, Q1 and R1 at time zero as shown in fig.3. For simplicity, the source terms (corresponding to Si in fig.1) are omitted. The components P1, Q1 and R1 are completely independent of one another and decay serially to their daughters Pi, Qi and Ri. The components P1, Q1 and R1 present in the initial mixture can be considered to be “daughters” of a virtual parent V also shown schematically in fig. 3. If one sets the half-life of V to be much smaller than the times of interest in the evolution of P, Q, R, then the parent V decays essentially instantaneously to P1, Q1 and R1 which then can decay further to their respective daughters. In the real mixtures, there will be specified amounts of P1, Q1 and R1 expressed in terms of the numbers of atoms, masses, activities, etc. These quantities are used to set the branching ratios of the parent V as described in more detail below.
Figure 3. Schematic representation of the decay of a mixture of component P1, Q1 and R1. The components of the original mixture can be considered as daughters of the virtual nuclide V.5. Properties of the Virtual Parent
In one deals only with the numbers N of atoms, the properties of the virtual nuclide V are fully prescribed (for decay calculations) by specification of the half-life tV, the daughters (e.g. P1, Q1 and R1), the branching ratios (BRVP1, BRVQ1, BRVR1).
If calculations, however, are to be made based on masses and activities, the virtual parent must be given an atomic weight ratio AWRV, and a “conversion” half-life conT. The atomic weight ratio is required to obtain the number of atoms from a given mass; whereas the conversion half-life is required to obtain the number of atoms from a given activity.
Half-life of virtual nuclide, tV: clearly, the half-life of the virtual nuclide must be very much less that the half-lives of the daughters. A typical half-life might be 1 ps (10-12 s) such that after a period of approximately ten half-lives only the first generation daughters are present. Once the half-life has been set, the decay constant kV of the virtual nuclide is given by
kV = ln2/tV(s)
where tV(s) is the half-life of the V in seconds.
Daughters of the virtual nuclide: the daughter products of the virtual nuclide V are simply the components on the nuclide mixture we wish to treat. If the mixture contains the nuclides P1, Q1 and R1 at time zero, then these are the daughters of V.
Branching Ratios: In the original mixture, e.g. P1, Q1 and R1, one usually has to deal with the masses or activities of P1, Q1 and R1 rather than the number of atoms. However, the branching ratios are defined in terms of the number of atoms. As a first step, therefore, the masses or activities have to be converted to the number of atoms through the relations:
Once the numbers of atoms are known, the branching ratios BR are then given by:
BRVP1 = NP1/NV, BRVQ1 = NQ1/NV, BRVR1 = NR1/NV
where NV is the number of atoms of the virtual parent i.e. NV = NP + NQ+ NR.
Atomic Weight Ratio of the virtual nuclide, AWRV: the atomic weight ratio of the parent virtual nuclide is required to obtain the number of atoms NV from the mass MV. Since the mass of the virtual nuclide MV = MP + MQ + MR, it follows that the atomic weight ratio is given by:
AWRV = MV / (NV ×Mn),
Conversion halflife of the virtual nuclide, conT: the conversion half-life of the virtual nuclide is required to obtain the number of atoms NV from the activity AV. using the relation A = kN = ln2×N/t1/2(s). To avoid confusion with the half-life of the virtual nuclide used in decay calculations, we must introduce the conversion half-life defined by A = kN = ln2×N/conT(s). Since the activity of the virtual nuclide AV = AP + AQ + AR, it follows that the conversion half-life is given by (this is discussed in more detail in section 9 on Unit Conversion)
:
conT = (NV /AV ) × ln2
- A Simple Two Component Mixture
As an example consider a mixture of the two nuclides 232U and 60Co with initial masses 0.4 g and 0.6 g respectively. The atomic weight ratios are 230.044 and 59.4189 respectively.
Half-life of virtual nuclide, tV: assume a half-life of 1ps (this is orders of magnitude less then hte half-lives of 232U and 60Co. It follows that the decay constant of the virtual nuclides V is given by:
kV = ln2/(10-12 s) = 6.93x10+11 s
Daughters of the virtual nuclide: the daughters of V are 232U and 60Co.
Branching Ratios: since the initial quantities of the daughters are specified in grams, we must first calculate the corresponding numbers of atoms. Hence
N232U = M232U/(AWR232U × Mn) = (0.4 g) / (230.044 × 1.6749286x10-24 g) = 1.03813x1021 atoms
N60Co = M60Co/(AWR60Co × Mn) = (0.6 g) / (59.4189 × 1.6749286x10-24 g) = 6.02879x1021 atoms
Hence, NV = N232U + N60Co = 7.06692x1021 atoms, and AWRV = 84.4838
It follows that the branching ratios
BRV,232U = N232U/NV = 1.03813x1021 / 7.06692x1021 = 0.14690
BRV,60Co = N60Co/NV = 6.02879x1021 / 7.06692x1021 = 0.85310
From the branching ratios and the total decay constant, the partial decay constants are given by
kV,232U = BRV,232U × kV = (0.14690 × 6.93x10+11 s) = 1.018x10+11 s
kV,60Co = BRV,60Co × kV = (0.85310 × 6.93x10+11 s) = 5.912x10+11 s
The following equations can now be solved:
dNV/dt = -kVNV Þ NV(t) = NV(0) exp(-kVt)
dN232U/dt = kV,232UNV Þ N232U(t) = (kV,232U/kV) × NV(0) [1-exp(-kVt)]
dN60Co/dt = -kV,60CoNV Þ N60Co(t) = (kV,60Co/kV) × NV(0) [1-exp(-kVt)]
or in terms of mass
MV(t) = MV(0)×exp(-kVt), M232U(t) = M232U(¥)×[1-exp(-kVt)], M60Co(t) = M60Co(¥)×[1-exp(-kVt)]
The results of the calculations are shown in fig. 5.One can extend the decay equations to take into account the decay of the direct daughters i.e.:
One can as well choose a different half-life for the virtual nuclide as long as it is much lower than the half-lives of the daughters. Figure 6 shows results obtained with a virtual nuclide half-life of 1s.
Figure 5. Decay of the virtual parent V to give the mixture 232U+60Co
Figure 6. Decay of the mixture 232U+60Co
- More Complicated Mixtures
A more complicated mixture is an inventory of nuclides (transuranics) in spent nuclear fuel. The compositions of this light water reactor fuel at a burnup of 40 GWd/ton U is given in table 2.
Nuclide / Mass (g) per ton spent fuelNp236 / 5.3E-04
Np237 / 6.5E+02
Pu238 / 2.3E+02
Pu239 / 5.9E+03
Pu240 / 2.6E+03
Pu241 / 6.8E+02
Pu242 / 6.0E+02
Pu244 / 4.2E-02
Am241 / 7.7E+02
Am242m / 2.5E+00
Am243 / 1.4E+02
Cm242 / 5.9E-03
Cm243 / 4.3E-01
Cm244 / 3.1E+01
Cm245 / 2.3E+00
Cm246 / 3.2E-01
Cm247 / 3.7E-03
Cm248 / 2.4E-04
One can apply the same method described previously to obtain the properties of the virtual parent. The half-life of the virtual nuclide is assumed to be 1 ps. The daughters of the virtual nuclide are then the different nuclides of the spent fuel, with the corresponding branching ratios and decay constants given in table 3.