# Help:Decay Engine

Level: Intermediate

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## Background Theory

Radioactive decay is a random process. It is not possible to predict when a particular nucleus in a sample will decay. One can, however, evaluate the probability that a nucleus will decay in a time interval. The basic relation governing radioactive decay, first identified by Rutherford, is

$\frac{dQ}{dt} = -kQ$

where the left hand side gives the rate of decay of the atoms Q in terms of the number of atoms Q and a decay "constant" k which is the probability per unit time that a nucleus will decay. The minus sign indicates that Q is decreasing with time. The solution to this equation is

$Q = Q(0) exp(-kt)$

where Q(0) is the number of nuclei at time t=0. From this basic relation various useful quantities such as half-life, mean life and activity or intensity of the radioactive transformation can be derived. These are described in more detail in chapter Nuclider Explorer (sections Data Sheets and Fact Sheets).

H. Bateman

It is very often the case that the daughter product of a nuclear decay is also radioactive. In such cases one speaks of radioactive decay "chains". As an example, consider the decay chain Q1→ Q2 → Q3 →….. in which the starting or "parent" nuclide Q1 decay to the "daughter" Q2. This daughter in turn is radioactive and decay to Q3. More generally each nuclide in the decay chain Qi can "branch" to more than one daughter. In addition, there may be an external source term Sii for the production of Qi (apart from the decay of the parent).

The situation for successive radioactive decay is shown schematically in fig. 1. This general process of radioactive decay was first investigated systematically by Bateman (Proc. Camb. Phil. Soc. 16 (1910) 423. See also Skrable, K., et al., Health Physics, 27, 1974, 155-157).

Figure 1. Successive radioactive decay with branching and source terms.

Bateman's orignal paper from 1910

The differential equation governing the above processes can be written as:

$\frac{dQ_1}{dt} = S_1 - k_{Q_1} \cdot Q_1$

$\frac{dQ_2}{dt} = S_2 + k_{Q_1,Q_2} \cdot Q_1 - k_{Q_2}\cdot Q_2$

..

$\frac{dQ_i}{dt} = S_i + k_{Q_{i-1},Q_i} \cdot Q_{i-1} - k_{Q_i}\cdot Q_i$

where Qn is the number of atoms of species n present at time t, kn is the decay constant (total removal constant) for species n (k = ln2/τ = 0.69315/τ), 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. The solution to this system of equations is

Bateman solution with source terms

(1):

$Q_n(t)=\prod_{j=1}^{n-1} k_{j,j+1} \times \sum_{i=1}^n \sum_{j=i}^n \left( \frac {Q_i(0)e^{-k_jt}} {\prod_{\overset{p=i}{\underset{p \ne j}{}}}^n(k_p - k_j)} + \frac {S_i (1-e^{-k_jt})} {k_j \prod_{\overset{p=i}{\underset{p \ne j}{}}}^n(k_p - k_j)} \right)$

Bateman solution for 1 parent nuclide without source terms, Si = 0 and Qi(0) = 0 for i>1:

(2):

$Q_n(t)= Q_1(0) \prod_{j=1}^{n-1} k_{j,j+1} \times \sum_{j=1}^n \frac {e^{-k_jt}} {\prod_{\begin{array} {}_{p=1} \\ _{p \ne j} \end{array}}^n(k_p - k_j)}$

It is of interest to construct the first few terms i.e.

(3):

$Q_1(t) = Q_1(0) e^{-k_1t}$

$Q_2(t) = Q_1(0)k_{1,2} \left\{ \frac {e^{-k_1t}} {k_2-k_1} + \frac {e^{-k_2t}} {k_1-k_2}\right\}$

$Q_3(t) = Q_1(0)k_{1,2} k_{2,3} \left\{ \frac {e^{-k_1t}} {\left(k_2-k_1\right) \left(k_3-k_1\right)} + \frac {e^{-k_2t}} {\left(k_1-k_2\right) \left(k_3-k_2\right)} + \frac {e^{-k_3t}} {\left(k_1-k_3\right) \left(k_2-k_3\right)} \right\}$

$.......$

These relations allow one to update the numbers of atoms from time t=0 to time t. It is also of interest to calculate the numbers at various times in the range 0,t (for example for plotting purposes). This can be done by specifying the total time t over which the calculation is to be made, and the number of time-steps L to reach t. The time interval for each calculation is then Δt = t/L. For L = 1 (the default value used in the calculation), the numbers are evaluated at the time t. For L = 2, the numbers are evaluated at t/2, and t. For L = 3, the Qs are evaluated at t/3, 2t/3, t etc. From above, the relation to be used is then

(4)

$Q_n\left( l \Delta t \right)= \sum_{i=1}^{i=n} Q_i(0) \prod_{j=1}^{n-1} k_{j,j+1} \times \sum_{j=i}^n \frac {e^{-k_jl \Delta t}} {\prod_{\begin{array} {}_{p=i} \\ _{p \ne j} \end{array}}^n(k_p - k_j)} ~~~~\mbox{for }l=1,2,3,...~L$

### Convergent and Divergent Branches

The solution to the differential equations given in equation (4) is valid for the various species produced in series i.e. in a chain. If branching occurs, as indicated in the figure 1, the solution (4) must be applied to all possible chains. As an example, consider the radioactive decay of Ac225. The breakdown into linear chains is shown in table 1. Equation (4) 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.

Table 1. The three "linear chains" for the decay of Ac-225 showing the various paths by which the nuclide can decay.

Consider a simplified radioactive decay process involving only three nuclides, Q1, Q2, and Q3. The nuclide 1 decays into nuclide 2 which in turn decays to nuclide 3. Nuclide 1 is the parent of nuclide 2 (or nuclide 2 is the daughter of nuclide 1). From the relations given above, the number of atoms of nuclide 2 is given by

(5) $Q_2 = \frac {k_1} {k_2-k_1} \cdot Q_1(0) \cdot \left( e^{-k_1t}-e^{-k_2t}} \right) = \frac {k_1} {k_2-k_1} \cdot Q_1(0) \cdot e^{-k_1t} \left( 1-e^{- \left( k_2-k_1 \right)t}} \right)$

From equation (5) it can be seen that the time required to reach equilibrium depends on the half-life of both the parent and the daughter. Four cases can be distinguished:

τ1 >> τ2. The halflife of the parent is much longer than that of the daughter.

τ1 > τ2. The halflife of the parent is longer than that of the daughter.

τ1 < τ2. The halflife of the parent is shorter than that of the daughter.

τ1 ≈ τ2. The halflifes of the parent and daughter are similar.

These will be discussed in more detail in the following sections.

#### (τ1>>τ2): Secular Equilibrium:

In secular equilibrium, the half-life of the parent is much longer than that of the daughter i.e. τ1 >>τ2 (k1 <<k2). In this case equation (5) reduces to

(6) Q2 = (k1/k2)Q1(0)(1-e-k2t)

For times t >>τ1, radioactive equilibrium is established and the following relation holds

Secular Equilibrium: Q2/Q1 = k1/k2 = τ21, and A1 =A2

where A is the activity defined by k×Q. Hence in radioactive equilibrium the ratio of the numbers, and the masses are constant whereas the activities are equal.

#### (τ1>τ2) : Transient Equilibrium

In transient equilibrium the half-life of the daughter is of the same order but smaller than that of the parent i.e.τ1 > τ2 (k1 < k2). The general equation for the daughter is from equation (5)

(7) Q2 = (τ2/(τ21))Q1(0)(e-k1t - e-k2t)

As an example, consider the decay of Ba140 as shown in fig 2. For times t << τ1 (12.75d), the first exponential term is very close to 1 and A2 increases according to (1-e1-kt) (rising part of activity of La-140 in fig. 2). For times t >> τ2 (1.68d), the second exponential becomes smaller than the first one with A2 decreasing according to e-kt (see fig. 2). For this decreasing part of the curve, one obtains

Transient Equilibrium: Q2 = (τ2/(τ21))Q1

where the relation Q1 = Q1(0) e-k·t has been used. This is the condition for transient equilibrium.

Half-lives of Ba140 and daughters.

Figure 2. Transient equilbrium in the Ba140 decay chain.

#### (τ1 < τ2): The Half-life of the Parent is Shorter than that of the Daughter

When the parent has a shorter half-life that that of the daughter, the daughter activity grows to some maximum and then decays with its own characteristic half-life. An example is shown in fig. 3 for Po-218.

Figure 3. No equilbrium: decay of Po-218.

#### (τ1 ≈ τ2): The Half-lives of the Parent and Daughter are Similar

As the half-lives of the parent and daughter become more equal, the attainment of equilibrium becomes more delayed. An example of this is shown in fig. 4 for the decay of I-135.

Figure 4. Similar half-lives: decay of I-135.

The time required for the maximum daughter activity is tmax = (k2-k1)-1×ln(k2/k1). As can be seen from fig. 4, this occurs around 11 hours.

## Using the Decay Engine Module

To make a decay calculation within NUCLEONICA, the user must first select the Decay Engine from the Application Centre on the main page. Thereafter, the nuclide of interest can be selected using the drop down menus. Alternatively, the nuclide can be selected from the nuclide chart, and then with the right mouse button, the Decay Engine can be selected.

#### Input User Interface

The resulting Decay Engine input interface is shown in the figure for Po-210. The user can either accept the default input values shown or enter input into the boxes. In the main Decay Engine tab, a number of default input parameters are shown. These are described in more detail in the following sections.

Decay calculation window input interface

##### Time units + Decay time

A default time shown with units corresponds to 10 half-lives of the selected nuclide.

##### Starting quantities / Final quantities

The default activity is 1 MBq, and by default the activity will be plotted in the graph (see Type of graph). The red question marks in the Final quantity box indicate this is the quantity to be calculated. Also shown in red is the number of linear chains - also a result of the calculation. The number of such chains depends on the Accurcy factor (described later).

##### Calculation details
###### Number of timesteps

This input box is used to set the number of intermediate times at which the decay is evaluated between t=0 and t ( t is the value in the "Decay Time" box). The default number of timesteps is 40 such that a graph can be drawn. The maximum number of timesteps is 40.

###### Accuracy factor

This input box is used to set the accuracy of the calculation. As explained in section Convergent and Divergent Branches, a parent nuclide may decay to different daughter nuclides depending on the branching ratios. The importance of each of these pathways in the decay process can be evaluated by the product of the branching ratios. If there is only one decay pathway, i.e. no branching, the product of all branching ratios is 1. If there is a pathway with a small branching ratio, then this will result in a branching ratio product determined by this low branching ratio. The input box Accuracy Factor sets the minimum product of branching ratios which are accounted for in the calculation. To evaluate all possible paths, the Accuracy Factor should be set to zero. This is discussed in more detail in section Details. The number of pathways or "Chains" accounted for in the calculation is given in the Number of linear chains box.

###### Distance (cm)

The Distance (cm) is required for the evaluation of the gamma dose rate. It is also required for plotting the gamma dose rates in the graph. The default value is 100 cm.

###### Number of linear chains

This is not an input variable but depends on the Accuracy factor. On pressing the Start button to make a calculation, this box shows the number of pathways or chains taken into account in the calculation.

##### Start / Reset

On pressing the Start button, the input data is posted to the web-server where the decay calculations are made. Thereafter, the output data is returned to the user computer.

The Reset button can be used to clear the input data used in the calculation.

##### Type of graph

In the "Type of graph" list box, the user can choose which quantity should be shown i.e. numbers, masses, activities, and gamma emission rate.

#### Results

If the default values shown in the input interface are accepted, pressing the Start button will lead to the results shown in the figure below.

Results of the Decay calculation

The results show the final activity and the number of linear decay chains in red. Had the accuracy factor been set to 0 (all chains calculated), the number of linear chains would be 23. The details in the output grid show the parent and all the daughter of Po-218 in the second column. In the following columns, the halflives, number of atoms, masses, activities, gamma dose rates at 1 m, and the number of disintegrations of the parent and all daughters are given. The total values in each case are also given. The choice of which properties to show in the Results grid can be set by the user in the Options tab (discussed later in this section).

##### Time units + Decay time

It is assumed that at time t = 0, only a single radioactive parent nuclide is present. Such a situation can arise for example after chemical separation of a parent nuclide from its daughter(s). The number entered in the input box is the time at which the decay is evaluated with reference to t=0. The default time is ten half-lives of the parent nuclide. After this time the parent nuclide has almost entirely decayed to its daughter(s).

##### Starting quantities / Final quantities

(Mass-Activity Calculator)

The source strength is set by specifying first the unit i.e. becquerel, curie, grams, or the number of atoms, and then the quantity. The quantity should be entered in scientific notation in the form 1, 10, 1e2 1e-6 etc. The default value 1 MBq. The source strength box functions also as an activity calculator which allows the conversion of one unit into another. As an example, for the nuclide Co-60 select, the default activity is 1 MBq. On changing the entry in the unit list box, one obtains 2.7e-5 Ci, 2.4e-8 g, 2.4e14 atoms.

The Mass-Activity-Number Calculator

The conversion from mass to number of atoms N, and vice versa etc., is described in the Basic quantities and relations

At the bottom of the Results table, the user has the possibility to download the table results to an Excel file or to a comma separated value (CSV) file. In this latter case (CSV) the user needs additionally to select the character separator.

Choosing the Excel option, the resulting Excel file is shown.

##### Rescale Feature

The Rescale tool is a very simple yet powerful feature in the Decay Engine. Very often the one knows the activity (e.g. total activity) one would like to have after a certain time. The Rescale tool allows the input activity in the calculation to be rescaled such that this value is obtained after the selected decay time. Without this tool, a number of iterations would be necessary to obtain the desired result.

To use the tool, copy any value from the table or the graph into the 'Rescale from' box. This value can than be rescaled to any other value (given in the 'Rescale to' box) by starting the calculation again.

Rescale feature in the Decay Engine

Example: The radionuclide F-18 is used in positron emission tomography (PET) in hospitals. The radionuclide is created in an accelerator which is located approximately 100 km from the hospital where it will be used. The journey time is approximately 2 hours for the transport. How much F-18 needs to be produced such that upon arrival at the hospital the total activity is 300 MBq?

With the decay engine, select F-18. The default activity is 1 MBq. Based on the transport time, the decay time is set to 2 h. The resulting activity after two hours is 469 kBq. So starting with 1 MBq, the activity remaining after two hours is 469 kBq. We now wish to rescale these results. Instead of 469 kBq we would like to have 300 MBq after the two hour period.

Rescale feature in the Decay Engine

Through the use of the rescale tool this is straightforward. We rescale from 469 kBq to 300 MBq. On pressing the Start button, all results are rescaled such that after 2 hours there are 300 MBq F-18 remaining. It can then be see that the Starting quantity should be 640 kBq.

A related example of the use of the rescale tool is given in the nuclide mixtures module.

##### Graph

A full description of the details and settings of the graph can be found in the webGraph wiki page.

Graph of the decay of Po-210

A graph of the results is shown below the results table. By default, the number of timesteps calculated in 40. The maximum number of timesteps is 40. Below the graph, the Show Graph Settings allow the user to select the type of graph to be plotted plus a variety of additonal settings. Both logarithmic and linear plots are available. In addition the region of interest can be specified for both x and y axes.

Various options for plotting graphs of the radioactive decay.

A full description of the details and settings of the graph can be found in the webGraph wiki page.

Print: From the Show Graph Settings, the graph can be printed directly to a printer.

Save Configuration The graph configuration (data plus graph settings) can be saved in xml format. This is described in more detail in the webGraph section.

#### Options

As mentioned in the previous section, the choice of which properties to show in the Results grid can also be set by the user from the Options tab shown below.

File:DE Options1.png
Options window showing the modes of operation, calculation details and quantities to be shown in the Results grid.

##### Mode of operation

There are two modes of operation of the Decay Engine:

1. Time mode: this has already been described above. For an initial activity of a given nuclide or mixture, the final activity is calculated after a decay time.

2. Date mode: if an activity is known at some time (date), then the activity can be calculated at any other time (date). The Date mode uses universal time (see link for further details).

Input window for Date mode of the Decay Engine

In this date mode, the user can enter a date (e.g. when the sample was taken). The calculation result then shows the activity at any later time (date). This is useful for the calculations of standards.

See the blog article on Date calculator.

Example: Consider a standard of Pb-210 from 1.9.1987. What is the activity of this standard today. In the input one enters the activity and date of certificate. The results give the activity today and the precise time elapsed.

Activity today of a Pb-210 standard from 1.9.1987

Example: A radwaste samples was measured on a date in January and contained some medium half-life nuclides. What is the precise time elapsed and the activity today?

##### Select quantities to be shown in the decay Result grid

(under development)

#### Linear Decay Chains

The linear decay chains tab is shown in the Results window. In this tab all the individual decay chains can be seen together with the numbers obtained by using equation (2). The importance of a decay chain is determined by the product of the branching ratios. If the branching ratios were all 1, there would only be one chain and the Accuracy Factor = 1.0. If a particular branching ratio in a decay chain is small e.g. 1E-8, and all other branching ratios were 1, then the product would be 1E-8. This implies that this particular decay chain is relatively less important. One must be careful in neglecting such chains, however, since the concentration or mass of the relevant nuclide may be small but may have a very high activity and for example gamma emission rate. An example is the decay of U232.

If only the main chain were considered (i.e. the chain with the highest product of branching ratios) one would miss the daughter Tl208 which is a powerful gamma emitter.

Ideally, one would like to evaluate all the decay chains for a particular parent nuclide. The problem is, however, that some heavy nuclides have a large number of decay chains - U-238 has for example 52 separate decay chains! To apply equation (2) to all these chains would impose a considerable burden on the web server. Fortunately, in most cases of interest, this is not necessary. It is sufficient to restrict the number of chains to only a few. The parameter used to restrict the number of chains to be evaluated is the product of the branching ratios "Accuracy Factor" for the particular chain. The minimum value can be set in input window. The default value is 1E-2. When the Start button is pressed, the branching ratios of all decay chains are evaluated and compared to this value. If the product of the branching ratios is less than the Accuracy Factor, the chain is rejected as being unimportant. If greater, the chain is stored for further evaluations. The number of chains stored - Number of linear chains -, i.e. with the product of the branching ratios ≥ Accuracy factor, is given at the Calculation details tablein the input window.

If the Accuracy factor is set to 0, all chains are evaluated but as stated above, this may take some time.

## References

General :

R. Dreher, Modified Bateman solution for identical eigenvalues, Annals of Nuclear Energy, 53 (2013) 427–438. Article

H. Bateman, Proc. Camb. Phil. Soc. 16 (1910) 423.

K. Skrable et al., Health Physics, 27 (1974) 155-157.

K. H. Lieser, Nuclear and Radiochemistry: Fundamentals and Applications, VCH /Wiley, 1997.

J. Magill and P. Peerani, (Non-) Proliferation Aspects of Accelerator Driven Systems, J. Phys. IV France 9 (1999) 167-181.

paper

How "hot" is Am-242m:

J. Galy, J. Magill, H. van Dam, J, Valko A Neutron Booster for Spallation Sources- Application to Accelerator Driven Systems and Isotope Production, Nucl. Instr. Meth. Res. 2002.

paper

J. Magill, R. Schenkel, Non-Proliferation Issues for Generation IV Power Systems: Advanced Waste Management in Global Warming and Energy Policy, Eds B.N. Kursunoglu, S.L. Mintz, A. Perlmutter, Kluwer Academic/Plenum Publishers, 2001.

"Age" Determination of Plutonium Particles:

J. Magill, R. Schenkel, L. Koch, Verification of Nuclear Materials and Sites, Detection of Clandestine Activities, and Assessment of the Proliferation Resistance of New Reactors and Fuel Cycle Concepts, Fachsitzung zum Thema "Abrüstung und Verification" Dresden 22-23 März, 2000.

M.Wallenius, K. Mayer, "Age" Determination of Plutonium Material in Nuclear Forensics by Thermal Ionisation Spectrometry, Fresenius Journal of Analytical Chemistry 366 (2000) 234-238.

Alpha-Immunotherapy:

C. Apostlidus, R. Carlos-Marquez, W. Janssens, R. Molinet, A. Quadi, Cancer Treatment using Bi-213 and Ac-225 in Radioimmunotherapy, Nuclear News, Dec. 2001, 29-33.