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textbook:nrctextbook:chapter6 [2025-02-24 12:59]
Merja Herzig 		textbook:nrctextbook:chapter6 [2025-05-07 12:01] (current)
Merja Herzig
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 Chapter 6 from //BASICS OF NUCLEAR PHYSICS AND OF RADIATION DETECTION AND MEASUREMENT – An open-access textbook for nuclear and radiochemistry students// by Jukka Lehto Chapter 6 from //BASICS OF NUCLEAR PHYSICS AND OF RADIATION DETECTION AND MEASUREMENT – An open-access textbook for nuclear and radiochemistry students// by Jukka Lehto
  
 +{{anchor:activity}}
 ===== 6.1. Decay law – activity - decay rate ===== ===== 6.1. Decay law – activity - decay rate =====
  
 ### ###
-The rate of radioactive decay is a characteristic feature for each radionuclide. Decay rate, also called activity, is the number of nuclear transformations (decays) (//dN//) at a defined time difference (unit time) (//dt//) and it is referred to as //A//  (activity{{anchor:activity}}):+The rate of radioactive decay is a characteristic feature for each [[textbook:nrctextbook:chapter4|radionuclide]]. Decay rate, also called activity, is the number of nuclear transformations (decays) (//dN//) at a defined time difference (unit time) (//dt//) and it is referred to as //A// (activity):
 ### ###
  
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 [VI.I] [VI.I]
 ;;# ;;#
 +{{anchor:decay_constant}}
 ### ###
-Radioactive decay is a stochastic phenomenon and we cannot know when a single nucleus will decay or what is the exact number of decays in unit time. If a fairly large number of radioactive nuclei are considered we can, however, know what fraction of nuclei will probably decay in unit time. This fraction, the probability of radioactive decay events, is characteristic for each radionuclide and it is called decay constant (//λ//). If, for example, the decay constant is 0.0001 s<sup>-1</sup> it means that among 100000 radioactive nuclei probably 10 nuclei will decay in one second and accordingly among 1000000 nuclei probably 100 nuclei. Thus radioactive decay rate is directly proportional to the number of radioactive nuclei.+Radioactive decay is a stochastic phenomenon and we cannot know when a single nucleus will decay or what is the exact number of decays in unit time. If a fairly large number of [[textbook:nrctextbook:chapter4|radioactive nuclei]] are considered we can, however, know what fraction of nuclei will probably decay in unit time. This fraction, the probability of radioactive decay events, is characteristic for each radionuclide and it is called decay constant (//λ//). If, for example, the decay constant is 0.0001 s<sup>-1</sup> it means that among 100000 radioactive nuclei probably 10 nuclei will decay in one second and accordingly among 1000000 nuclei probably 100 nuclei. Thus radioactive decay rate is directly proportional to the number of radioactive nuclei.
 ### ###
 +{{anchor:number_of_radioactive_nuclei}}
  
 $$A = \left| -\frac{dN}{dt} \right| = \lambda \cdot N$$;;# $$A = \left| -\frac{dN}{dt} \right| = \lambda \cdot N$$;;#
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 ### ###
-which equation answers the question what is the number of radioactive nuclei (//N//) at certain time point (//t//) when we know the initial number on nuclei //(N<sub>0</sub>)// at time point //t<sub>0</sub>//.  Thus, to calculate this only the value of the decay constant (//λ//) is to be known.+which equation answers the question what is the number of radioactive nuclei (//N//) at certain time point (//t//) when we know the initial number on nuclei //(N<sub>0</sub>)// at time point //t<sub>0</sub>//.  Thus, to calculate this only the value of the [[textbook:nrctextbook:chapter6#decay_constant|decay constant]] (//λ//) is to be known.
 ### ###
  
  
 ### ###
-The number of radioactive nuclei is not usually known and their number is also difficult to directly determine. Usually we are, however, more interested in development of activities with time. As seen from Equation VI.II the activity is directly proportional to the number of decaying nuclei, thus we can transform Equation VI.III to calculate activity (//A//) at a time point (//t//) just by replacing //N// with //A//:+The number of [[textbook:nrctextbook:chapter4|radioactive nuclei]] is not usually known and their number is also difficult to directly determine. Usually we are, however, more interested in development of [[textbook:nrctextbook:chapter6#activity|activities]] with time. As seen from Equation VI.II the [[textbook:nrctextbook:chapter6#activity|activity]] is directly proportional to the number of decaying nuclei, thus we can transform Equation VI.III to calculate activity (//A//) at a time point (//t//) just by replacing //N// with //A//:
 ### ###
  
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 ;;# ;;#
  
 +{{anchor:half_life}}
 ===== 6.2. Half-life ===== ===== 6.2. Half-life =====
  
 ### ###
-Decay constants are known for all radionuclides and they are tabulated in various textbooks and databases. They are, however, not used in calculations of activities but instead half-lives (//t½//) are used for this purpose. Half-life is defined as the time in which half of the initial radioactive nuclei have decayed. Since the activity is directly proportional to the number of radioactive nuclei this means that also activity decreases to half within the time of half-life. In the following, the relation between the decay constant and the half-life will be shown. In addition, an equation by which activities can be calculated at desired time points using half-lives will be derived.+[[textbook:nrctextbook:chapter6#decay_constant|Decay constants]] are known for all [[textbook:nrctextbook:chapter4|radionuclides]] and they are tabulated in various textbooks and databases. They are, however, not used in calculations of [[textbook:nrctextbook:chapter6#activity|activities]] but instead half-lives (//t½//) are used for this purpose. //Half-life is defined as the time in which half of the initial radioactive nuclei have decayed.// Since the activity is directly proportional to the number of radioactive nuclei this means that also activity decreases to half within the time of half-life. In the following, the relation between the decay constant and the half-life will be shown. In addition, an equation by which [[textbook:nrctextbook:chapter6#activity|activities]] can be calculated at desired time points using half-lives will be derived.
 ### ###
  
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 ### ###
-With the latter equation we can calculate activities using half-lives at any time points when we know the initial activity. When we want to calculate the initial activity at an earlier time point we use the inverse equation+With the latter equation we can calculate [[textbook:nrctextbook:chapter6#activity|activities]] using half-lives at any time points when we know the initial [[textbook:nrctextbook:chapter6#activity|activity]]. When we want to calculate the initial activity at an earlier time point, we use the inverse equation
 ### ###
  
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 ;;# ;;#
  
 +{{anchor:becquerel}} 
 +{{anchor:activity_unit}}
 ===== 6.3. Activity unit ===== ===== 6.3. Activity unit =====
  
 ### ###
-The official SI unit of activity is Becquerel (Bq) and it means one decay in SI unit time, i.e. one second:+The official SI unit of activity is //Becquerel// (Bq) and it means one decay in SI unit time, i.e. one second:
 ### ###
    
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 ;;# ;;#
  
 +{{anchor:curie_ci}}
 ### ###
-Earlier Curie (Ci) was used as the activity unit. One Curie is  3.7×10<sup>10</sup> decays in second and thus+Earlier //Curie// (Ci) was used as the activity unit. One Curie is  3.7×10<sup>10</sup> decays in second and thus
 ### ###
    
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 ### ###
  
 +{{anchor:dps}} 
 +{{anchor:dpm}}
 ### ###
-Sometimes activities are expressed as a dps unit, meaning disintegrations per second which are equal to activities presented as Bequerels. In some instances, for example in liquid scintillation counting, activity is also presented as dpm units (disintegrations per minute). One dpm is 1/60 dps or 16.7 mBq.+Sometimes [[textbook:nrctextbook:chapter6#activity|activities]] are expressed as a dps unit, meaning disintegrations per second which are equal to activities presented as [[textbook:nrctextbook:chapter6#becquerel|Becquerels]]. In some instances, for example in [[textbook:nrctextbook:chapter12|liquid scintillation counting]], activity is also presented as //dpm// units (disintegrations per minute). One dpm is 1/60 dps or 16.7 mBq.
 ### ###
  
 +{{anchor:specific_activity}} 
 +{{anchor:activity_concentration}}
 ===== 6.4. Specific activity - activity concentration ===== ===== 6.4. Specific activity - activity concentration =====
  
 ### ###
-Specific activity is often used as a synonym to activity concentration, but strictly speaking they have different meanings. Specific activity refers to concentration of a radionuclide with respect to the total amount of the same element as the radionuclide. Thus, specific activity is its concentration in a unit mass or mole of the same element, for example, 5 kBq of <sup>137</sup>Cs per 1 g of Cs or 0.038 kBq of <sup>137</sup>Cs per 1 mole of Cs.+Specific activity is often used as a synonym to activity concentration, but strictly speaking they have different meanings. Specific activity refers to concentration of a [[textbook:nrctextbook:chapter4|radionuclide]] with respect to the total amount of the same element as the radionuclide. Thus, specific activity is its concentration in a unit mass or mole of the same element, for example, 5 kBq of <sup>137</sup>Cs per 1 g of Cs or 0.038 kBq of <sup>137</sup>Cs per 1 mole of Cs.
 ### ###
  
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 ### ###
  
 +{{anchor:activity_vs_mass}} 
 +{{anchor:radioactive_decay_law}}
 ===== 6.5. The relation between the activity and the mass ===== ===== 6.5. The relation between the activity and the mass =====
  
-Conversion of activities to masses or vice versa is based on the radioactive decay law+Conversion of [[textbook:nrctextbook:chapter6#activity|activities]] to masses or vice versa is based on the //radioactive decay law//
  
 $$A = \lambda \times N$$ $$A = \lambda \times N$$
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 ### ###
-which shows the direct dependence of the activity on the number of decaying nuclei. Replacing 𝜆 by $\frac{\ln 2}{t_{1/2}}$ ln2/t<sub>1/2</sub> and N by $\left( \frac{m}{M} \right) \times N_A$ (m/M)×N<sub>A</sub> (where m is the mass in grams, M the molar mass of the element and N<sub>A</sub> the Avogadro number) yields+which shows the direct dependence of the activity on the number of decaying nuclei. Replacing 𝜆 by $\frac{\ln 2}{t_{1/2}}$ ln2/t<sub>1/2</sub> and N by $\left( \frac{m}{M} \right) \times N_A$ (m/M)×N<sub>A</sub> (where m is the mass in grams, M the molar mass of the element and N<sub>A</sub> [[textbook:nrctextbook:chapter3#avogadro_number|the Avogadro number]]) yields
 ### ###
  
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 which can be used to convert activities to masses or vice versa. which can be used to convert activities to masses or vice versa.
  
 +{{anchor:determination_of_half_lives}}
 ===== 6.6. Determination of half-lives ===== ===== 6.6. Determination of half-lives =====
  
-The determination of half-lives can be accomplished in two ways:+The determination of [[textbook:nrctextbook:chapter6#half_life|half-lives]] can be accomplished in two ways:
  
-  * For radionuclides decaying with such a fast rate that we can observe the decrease in a reasonable time the half-lives can be determined from their activities as a function of time as shown below in Fig. VI.1.+  * For [[textbook:nrctextbook:chapter4|radionuclides]] decaying with such a fast rate that we can observe the decrease in a reasonable time the [[textbook:nrctextbook:chapter6#half_life|half-lives]] can be determined from their [[textbook:nrctextbook:chapter6#activity|activities]] as a function of time as shown below in [[textbook:nrctextbook:chapter6#figure_61|Figure VI.1]]. 
  
 ### ###
-When representing graphically the equation $A = A_0 \cdot 2^{-\frac{t}{t_{1/2}}}$ we get an exponential curve (Figure VI.1,left side). Taking logarithms from both sides yields the equation $\ln A = -\frac{\ln 2}{t_{1/2}} \cdot t + \ln A_0$, the graphical representation of which is line with a slope of $-\frac{\ln 2}{t_{1/2}}$ and the y-axis intersection is $\ln A_0$, i.e. activity at time t<sub>0</sub> (Figure VI.1, right side). The half-life is obtained by fitting a line to the logarithms of observed activity values and calculating the half-life from the slope. If, for example, in Figure VI.1 time were in years, the half-life of the nuclide would result by solving the equation $-0.693 = -\frac{\ln 2}{t_{1/2}}$ into 1 year.+When representing graphically the equation $A = A_0 \cdot 2^{-\frac{t}{t_{1/2}}}$ we get an exponential curve ([[textbook:nrctextbook:chapter6#figure_61|Figure VI.1]],left side). Taking logarithms from both sides yields the equation $\ln A = -\frac{\ln 2}{t_{1/2}} \cdot t + \ln A_0$, the graphical representation of which is line with a slope of $-\frac{\ln 2}{t_{1/2}}$ and the y-axis intersection is $\ln A_0$, i.e. activity at time t<sub>0</sub> ([[textbook:nrctextbook:chapter6#figure_61|Figure VI.1]], right side). The [[textbook:nrctextbook:chapter6#half_life|half-life]] is obtained by fitting a line to the logarithms of observed [[textbook:nrctextbook:chapter6#activity|activity]] values and calculating the [[textbook:nrctextbook:chapter6#half_life|half-life]] from the slope. If, for example, in [[textbook:nrctextbook:chapter6#figure_61|Figure VI.1]] time were in years, the [[textbook:nrctextbook:chapter6#half_life|half-life]] of the nuclide would result by solving the equation $-0.693 = -\frac{\ln 2}{t_{1/2}}$ into 1 year.
 ### ###
 +{{anchor:figure_61}}
 {{:textbook:nrctextbook:determination_of_activity_fig_6_1.png |}} {{:textbook:nrctextbook:determination_of_activity_fig_6_1.png |}}
 Figure VI.1. Activity (A) as a function of time (t). Left side presents activity in linear scale and right side in logarithmic scale. Figure VI.1. Activity (A) as a function of time (t). Left side presents activity in linear scale and right side in logarithmic scale.
  
 ### ###
-The method described above in Fig. VI.1 can also be used to determine half-lives of two coexisting radionuclides supposing that they differ enough from each other. Figure VI.2 shows the total activity curve of two radionuclides as a function of time both in a linear and a logarithmic activity scale. In the first phase, when there are still both radionuclides present, the logarithmic curve shape resembles an exponential one. As the shorter-lived radionuclide has decayed the curve turns into a line. This line represents the decay of the longer-lived radionuclide and its half-life can be  +The method described above in [[textbook:nrctextbook:chapter6#figure_61|Figure VI.1]] can also be used to determine [[textbook:nrctextbook:chapter6#half_life|half-lives]]  of two coexisting [[textbook:nrctextbook:chapter4|radionuclides]] supposing that they differ enough from each other. [[textbook:nrctextbook:chapter6#figure_62|Figure VI.2]] shows the total [[textbook:nrctextbook:chapter6#activity|activity]] curve of two [[textbook:nrctextbook:chapter4|radionuclides]] as a function of time both in a linear and a logarithmic [[textbook:nrctextbook:chapter6#activity|activity]] scale. In the first phase, when there are still both radionuclides present, the logarithmic curve shape resembles an exponential one. As the shorter-lived radionuclide has decayed the curve turns into a line. This line represents the decay of the longer-lived radionuclide and its half-life can be calculated from the slope of this line. To calculate the half-life of the shorter-lived radionuclide the line is extrapolated to time point zero and the extrapolated activity values of the line are subtracted from total activity curve. This yields another line representing the decay of the shorter-lived radionuclide for which the half-life is calculated from its slope.
-calculated from the slope of this line. To calculate the half-life of the shorter-lived radionuclide the line is extrapolated to time point zero and the extrapolated activity values of the line are subtracted from total activity curve. This yields another line representing the decay of the shorter-lived radionuclide for which the half-life is calculated from its slope.+
 ### ###
  
  
 +{{anchor:figure_62}}
 {{:textbook:nrctextbook:individual_and_total_activities_of_two_coexisting_radionuclides_as_a_function_of_time_fig_6_2.png |}} {{:textbook:nrctextbook:individual_and_total_activities_of_two_coexisting_radionuclides_as_a_function_of_time_fig_6_2.png |}}
  
-Figure VI.2. Individual and total activities of two coexisting radionuclides as a function of time. Top: activity in linear scale. Bottom: activity in logarithmic scale. Diamond (blue): Radionuclide with a half-life of 3 hours. Square (red): radionuclide with a half-life of 12 hours. Triangle (green): total activity.+Figure VI.2. Individual and total activities of two coexisting radionuclides as a function of time. Top: activity in linear scale. Bottom: activity in logarithmic scale. Diamond (blue): Radionuclide with a [[textbook:nrctextbook:chapter6#half_life|half-life]] of 3 hours. Square (red): radionuclide with a half-life of 12 hours. Triangle (green): total activity.
  
-  * For long-lived radionuclides for which the activity decreases so slowly that we cannot observe its decrease the half-life is determined by measuring both the activity and the mass of the radionuclide and calculating half-life from the equation $t_{1/2} = \frac{\ln 2 \times m \times N_A}{A \times M}$.+  * For [[textbook:nrctextbook:chapter4#long_lived_radionuclides|long-lived radionuclides]] for which the [[textbook:nrctextbook:chapter6#activity|activity]] decreases so slowly that we cannot observe its decrease the [[textbook:nrctextbook:chapter6#half_life|half-life]] is determined by measuring both the [[textbook:nrctextbook:chapter6#activity|activity]] and the mass of the radionuclide and calculating half-life from the equation $t_{1/2} = \frac{\ln 2 \times m \times N_A}{A \times M}$.
  
 ### ###
-For example, when the half-life is 106 years the activity decreases by 0.00007% in one year. Small differences in activity like this cannot be measured. To determine the half-life of long-lived radionuclides we need to measure the mass (m) of the radionuclide and count rate (R) obtained from the activity measurement. In addition, we also need to accurately know the counting efficiency (E) of the measurement system. If, for example, we have 1.27 mg of <sup>232</sup>Th and the count rate obtained from its measurement is 2.65 cps and counting efficiency of the measurement system +For example, when the [[textbook:nrctextbook:chapter6#half_life|half-life]] is 106 years the [[textbook:nrctextbook:chapter6#activity|activity]] decreases by 0.00007% in one year. Small differences in activity like this cannot be measured. To determine the half-life of [[textbook:nrctextbook:chapter4#long_lived_radionuclides|long-lived radionuclides]] we need to measure the mass (m) of the radionuclide and count rate (R) obtained from the activity measurement. In addition, we also need to accurately know the counting efficiency (E) of the measurement system. If, for example, we have 1.27 mg of <sup>232</sup>Th and the count rate obtained from its measurement is 2.65 cps and counting efficiency of the measurement system 
 is 51.5% (0.515) the activity of the sample is $A = \frac{R}{E} = \frac{2.65 \, \text{s}^{-1}}{0.515} = 5.15 \, \text{Bq}$. The number of thorium atoms in the sample is $1.27 \cdot 10^{-3} \, \text{g} \times \frac{6.023 \cdot 10^{23} \, \text{atoms/mole}}{232.0 \, \text{g/mole}} = 3.295 \cdot 10^{18}$. is 51.5% (0.515) the activity of the sample is $A = \frac{R}{E} = \frac{2.65 \, \text{s}^{-1}}{0.515} = 5.15 \, \text{Bq}$. The number of thorium atoms in the sample is $1.27 \cdot 10^{-3} \, \text{g} \times \frac{6.023 \cdot 10^{23} \, \text{atoms/mole}}{232.0 \, \text{g/mole}} = 3.295 \cdot 10^{18}$.
 Now the half-life can be calculated from the equation $t_{1/2} = \left( \frac{\ln 2}{A} \right) \times N = \frac{0.693 \times 3.295 \cdot 10^{18}}{5.15 \, \text{s}^{-1}} = 4.44 \cdot 10^{17} \, \text{s} = 1.41 \cdot 10^{10} \, \text{a}$. This method can also in principle be used to measure half-lives of shorter lived radionuclides but in their case the accurate measurement of the mass may either completely exclude the use of this method or at least results in inaccurate value. Now the half-life can be calculated from the equation $t_{1/2} = \left( \frac{\ln 2}{A} \right) \times N = \frac{0.693 \times 3.295 \cdot 10^{18}}{5.15 \, \text{s}^{-1}} = 4.44 \cdot 10^{17} \, \text{s} = 1.41 \cdot 10^{10} \, \text{a}$. This method can also in principle be used to measure half-lives of shorter lived radionuclides but in their case the accurate measurement of the mass may either completely exclude the use of this method or at least results in inaccurate value.
 ### ###
  
 +{{anchor:activity_equilibria}}
 ===== 6.7. Activity equilibria of consecutive decays ===== ===== 6.7. Activity equilibria of consecutive decays =====
  
 ### ###
-In the following we discuss two consecutive decay processes and their activity equilibria. Equilibrium means that the activities of the parent and daughter nuclides are the same. Consecutive decays and their equilibria are especially important in natural decay chains of uranium and thorium and in beta decays chains following fissions. In these chains there are typically more than two radionuclides present at the same time. Their equilibrium calculations are rather complicated and  +In the following we discuss two consecutive decay processes and their activity equilibria. //Equilibrium means that the activities of the parent and daughter nuclides are the same//. Consecutive decays and their equilibria are especially important in [[textbook:nrctextbook:chapter4#decay_chains|natural decay chains]] of uranium and thorium and in [[textbook:nrctextbook:chapter6#beta_decay_chains|beta decay chains]] following fissions. In these chains there are typically more than two [[textbook:nrctextbook:chapter4|radionuclides]] present at the same time. Their equilibrium calculations are rather complicated and require computer programs. An example of such calculation is given in [[textbook:nrctextbook:chapter6#figure_67|Figure VI.7]] (at the end of this chapter). Here we, however, focus on equilibrium between two [[textbook:nrctextbook:chapter4|radionuclides]], the //parent nuclide////Italic Text// and the //daughter nuclide//. When considering two consecutive decays the number of the parent nuclei (N<sub>1</sub>) depends only on its characteristic decay rate, i.e. [[textbook:nrctextbook:chapter6#decay_constant|decay constant]] (λ<sub>1</sub>). The number of the daughter nuclei (N<sub>2</sub>) 
-require computer programs. An example of such calculation is given at the end of the chapter. Here we, however, focus on equilibrium between two radionuclides, the parent nuclide and the daughter nuclide. When considering two consecutive decays the number of the parent nuclei (N<sub>1</sub>) depends only on its characteristic decay rate, i.e. decay constant (λ<sub>1</sub>). The number of the daughter nuclei (N<sub>2</sub>) +
 in turn is dependent both on its own decay rate (λ<sub>2</sub>) and on the parent's decay rate (λ<sub>1</sub>). The former determines the decay (decrease) of the daughter nuclides while the latter determines the ingrowth from the parent (increase). Thus the number of daughter nuclei is in turn is dependent both on its own decay rate (λ<sub>2</sub>) and on the parent's decay rate (λ<sub>1</sub>). The former determines the decay (decrease) of the daughter nuclides while the latter determines the ingrowth from the parent (increase). Thus the number of daughter nuclei is
 ### ###
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 ;;# ;;#
  
 +{{anchor:equation_614}}
 The solution of this equation with respect to N<sub>2</sub> is   The solution of this equation with respect to N<sub>2</sub> is  
  
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 ### ###
-where //N<sub>1</sub><sup>0</sup>// and //N<sub>2</sub><sup>0</sup>// are the numbers of parent and daughter, respectively, at time point zero (t=0). The first term in the equation presents the number of daughter nuclei due to ingrowth and the decay of ingrown nuclei while the second term represents decay of those daughter nuclei that were present at time point zero. Figure VI.3 gives a graphical presentation of the Eq. VI.XIV for a case where the half-life of the daughter is clearly shorter than that of the parent nuclide, i.e. its shows the ingrowth of the daughter nuclide activity as a function of the number half-lives of the daughter nuclide. Activity is here presented as the percentage of the maximum activity obtainable.+where //N<sub>1</sub><sup>0</sup>// and //N<sub>2</sub><sup>0</sup>// are the numbers of parent and daughter, respectively, at time point zero (t=0). The first term in the equation presents the number of daughter nuclei due to ingrowth and the decay of ingrown nuclei while the second term represents decay of those daughter nuclei that were present at time point zero. [[textbook:nrctextbook:chapter6#figure_63|Figure VI.3]] gives a graphical presentation of the [[textbook:nrctextbook:chapter6#equation_614|Eq. VI.XIV]] for a case where the [[textbook:nrctextbook:chapter6#half_life|half-life]] of the daughter is clearly shorter than that of the parent nuclide, i.e. its shows the ingrowth of the daughter nuclide [[textbook:nrctextbook:chapter6#activity|activity]] as a function of the number half-lives of the daughter nuclide. Activity is here presented as the percentage of the maximum activity obtainable.
 ### ###
- + {{anchor:figure_63}}
  
  
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-   * secular equilibrium, in which the half-life of the parent nuclide is very long and the half-life of the daughter nuclide is considerably shorter than that of the parent   +   * [[textbook:nrctextbook:chapter6#secular_equilibrium|Secular equilibrium]], in which the [[textbook:nrctextbook:chapter6#half_life|half-life]] of the parent nuclide is very long and the half-life of the daughter nuclide is considerably shorter than that of the parent   
-  * transient equilibrium, in which the half-life of the parent is so short that we observe decrease in its activity in a reasonable time and the half-life of the daughter nuclide is shorter than that of the parent   +  * [[textbook:nrctextbook:chapter6#trancient_equilibrium|Trancient equilibrium]], in which the half-life of the parent is so short that we observe decrease in its activity in a reasonable time and the half-life of the daughter nuclide is shorter than that of the parent   
-  * no-equilibrium, in which the half-life of the parent nuclide is shorter than that of the daughter  +  * [[textbook:nrctextbook:chapter6#non_equilibrium|Non-equilibrium]], in which the half-life of the parent nuclide is shorter than that of the daughter 
  
 +{{anchor:secular_equilibrium}}
 ==== 6.7.1. Secular equilibrium ==== ==== 6.7.1. Secular equilibrium ====
  
 ### ###
-An example of a secular equilibrium is case where the parent nuclide is the fission product <sup>137</sup>Cs which decays by beta decay process to 137mBa which in turn decays by internal transition process to stable <sup>137</sup>Ba. The half-life of <sup>137</sup>Cs is 30 years while the half-life of <sup>137</sup>Ba is only 2.6 minutes. Figure VI.3 shows the development of activities in a case when <sup>137m</sup>Ba has been chemically separated from its parent <sup>137</sup>Cs with BaSO<sub>4</sub> precipitation and both <sup>137m</sup>Ba-bearing precipitate and remaining solution containing only <sup>137</sup>Cs are measured for their <sup>137</sup>Cs and 137mBa activities immediately after chemical separation and measurements are repeated as a function of time. <sup>137m</sup>Ba in precipitate +An example of a secular equilibrium is case where the parent nuclide is the [[textbook:nrctextbook:chapter5#fission|fission]] product <sup>137</sup>Cs which decays by [[textbook:nrctextbook:chapter5#beta_decay|beta decay]] process to <sup>137m</sup>Ba which in turn decays by [[textbook:nrctextbook:chapter5#internal_transition|internal transition]] process to stable <sup>137</sup>Ba. The [[textbook:nrctextbook:chapter6#half_life|half-life]] of <sup>137</sup>Cs is 30 years while the half-life of <sup>137</sup>Ba is only 2.6 minutes. Figure VI.3 shows the development of [[textbook:nrctextbook:chapter6#activity|activities]] in a case when <sup>137m</sup>Ba has been chemically separated from its parent <sup>137</sup>Cs with BaSO<sub>4</sub> precipitation and both <sup>137m</sup>Ba-bearing precipitate and remaining solution containing only <sup>137</sup>Cs are measured for their <sup>137</sup>Cs and <sup>137m</sup>Ba activities immediately after chemical separation and measurements are repeated as a function of time. <sup>137m</sup>Ba in precipitate 
 decays following its half-life on 2.6 minutes (red). The activity of <sup>137</sup>Cs in the solution phase (blue) remains practically constant since observation time (30 min) is extremely short compared to the half-life of <sup>137</sup>Cs (30 years). <sup>137m</sup>Ba in the solution (green) starts immediately after chemical separation to grow in and attains equilibrium with <sup>137</sup>Cs in about ten half-lives of the daughter, i.e. half an hour. Since the activities of <sup>137m</sup>Ba and <sup>137</sup>Cs are the same the total activity (curve 4) is twice the parent nuclide. decays following its half-life on 2.6 minutes (red). The activity of <sup>137</sup>Cs in the solution phase (blue) remains practically constant since observation time (30 min) is extremely short compared to the half-life of <sup>137</sup>Cs (30 years). <sup>137m</sup>Ba in the solution (green) starts immediately after chemical separation to grow in and attains equilibrium with <sup>137</sup>Cs in about ten half-lives of the daughter, i.e. half an hour. Since the activities of <sup>137m</sup>Ba and <sup>137</sup>Cs are the same the total activity (curve 4) is twice the parent nuclide.
 ### ###
      
-  
- 
 ### ###
-Secular does not mean eternal. Looking at a very long-term all secular equilibria are transient. How long-term we need to look depends on the half-life of the parent. For example, if we looked the example describe above for a hundred years period the equilibrium would appear as transient equilibrium. For <sup>230</sup>Th (t<sub>1/2</sub> = 75000 y), for example, the transient equilibrium period with <sup>226</sup>Ra would be hundreds of thousands of years.+Secular does not mean eternal. Looking at a very long-term all secular equilibria are [[textbook:nrctextbook:chapter6#transient_equilibrium|transient]]. How long-term we need to look depends on the half-life of the parent. For example, if we looked the example describe above for a hundred years period the equilibrium would appear as [[textbook:nrctextbook:chapter6#transient_equilibrium|transient equilibrium]]. For <sup>230</sup>Th (t<sub>1/2</sub> = 75000 y), for example, the transient equilibrium period with <sup>226</sup>Ra would be hundreds of thousands of years.
 ### ###
    
 {{:textbook:nrctextbook:development_of_secular_equilibrium_fig_6_4.png |}} {{:textbook:nrctextbook:development_of_secular_equilibrium_fig_6_4.png |}}
-Figure VI.4. Development of a secular radioactive equilibrium in which the half-life of the parent nuclide very long and the half-life on the daughter nuclide (<sup>137m</sup>Ba, t<sub>½</sub> = 2.6 min) is considerably shorter than that of the parent (<sup>137</sup>Cs, t<sub>½</sub> = 30 a). Left: activity on linear scale. Right: activity on logarithmic scale. Blue: <sup>137</sup>Cs. Red: <sup>137m</sup>Ba if separated from <sup>137</sup>Cs. Green: ingrowth of <sup>137m</sup>Ba with <sup>137</sup>Cs after separation of <sup>137m</sup>Ba from <sup>137</sup>Cs.   +Figure VI.4. Development of a secular radioactive equilibrium in which the half-life of the parent nuclide very long and the half-life on the daughter nuclide (<sup>137m</sup>Ba, t<sub>½</sub> = 2.6 min) is considerably shorter than that of the parent (<sup>137</sup>Cs, t<sub>½</sub> = 30 a). Left: activity on linear scale. Right: activity on logarithmic scale. Blue: <sup>137</sup>Cs. Red: <sup>137m</sup>Ba if separated from <sup>137</sup>Cs. Green: ingrowth of <sup>137m</sup>Ba with <sup>137</sup>Cs after separation of <sup>137m</sup>Ba from <sup>137</sup>Cs.  
 +  
 +{{anchor:transient_equilibrium}}
 ==== 6.7.2. Transient equilibrium ==== ==== 6.7.2. Transient equilibrium ====
 +{{anchor:beta_decay_chain}}
 ### ###
-An example of transient equilibrium is a beta decay chain where <sup>140</sup>Ba decays to <sup>140</sup>La and the latter to stable <sup>140</sup>Ce.+An example of transient equilibrium is a [[textbook:nrctextbook:chapter5#beta_decay|beta decay]] chain where <sup>140</sup>Ba decays to <sup>140</sup>La and the latter to stable <sup>140</sup>Ce.
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 [VI.XV] [VI.XV]
 ;;# ;;#
 +{{anchor:figure_65}}
 {{:textbook:nrctextbook:development_of_transient_equilibrium_fig_6_5.png |}} {{:textbook:nrctextbook:development_of_transient_equilibrium_fig_6_5.png |}}
 Figure VI.5. Development of a transient radioactive equilibrium in which the half-life of the parent (<sup>140</sup>La, t<sub>½</sub> = 40.2 d) is so short that we observe decrease in its activity in a reasonable time and the half-life on the daughter nuclide (<sup>140</sup>Ba, t<sub>½</sub> = 12.8 d) is shorter than that of the parent. Left: activity on linear scale. Right: activity on logarithmic scale. Blue: <sup>140</sup>La. Red:  <sup>140</sup>Ba.   Figure VI.5. Development of a transient radioactive equilibrium in which the half-life of the parent (<sup>140</sup>La, t<sub>½</sub> = 40.2 d) is so short that we observe decrease in its activity in a reasonable time and the half-life on the daughter nuclide (<sup>140</sup>Ba, t<sub>½</sub> = 12.8 d) is shorter than that of the parent. Left: activity on linear scale. Right: activity on logarithmic scale. Blue: <sup>140</sup>La. Red:  <sup>140</sup>Ba.  
  
 ### ###
-The transient equilibrium is otherwise identical with the secular equilibrium except that the parent nuclide decays with such a short rate that we observe decrease in activity in a reasonable time. After attaining the equilibrium in about ten half-lives of the daughter, about two weeks in case of Fig.VI.5, both parent and the daughter decay at the rate of the parent nuclide. Also, after attaining the equilibrium the total activity is twice the activity of the parent nuclide.+The transient equilibrium is otherwise identical with the [[textbook:nrctextbook:chapter6#secular_equilibrium|secular equilibrium]] except that the parent nuclide decays with such a short rate that we observe decrease in [[textbook:nrctextbook:chapter6#activity|activity]] in a reasonable time. After attaining the equilibrium in about ten half-lives of the daughter, about two weeks in case of [[textbook:nrctextbook:chapter6#figure_65|Figure VI.5]], both parent and the daughter decay at the rate of the parent nuclide. Also, after attaining the equilibrium the total activity is twice the activity of the parent nuclide.
 ### ###
  
- +{{anchor:non_equilibrium}} 
 ==== 6.7.3. No-equilibrium ==== ==== 6.7.3. No-equilibrium ====
  
 ### ###
-An example of no-equilibrium case is the alpha decay pair <sup>218</sup>Po (t<sub>½</sub> = 3 min) → <sup>214</sup>Pb (t<sub>½</sub> = 26.8 min), where the half-life of the parent is shorter than that of the daughter. No equilibrium develops since the parent decays before the daughter.+An example of no-equilibrium case is the [[textbook:nrctextbook:chapter5#alpha|alpha decay]] pair <sup>218</sup>Po (t<sub>½</sub> = 3 min) → <sup>214</sup>Pb (t<sub>½</sub> = 26.8 min), where the [[textbook:nrctextbook:chapter6#half_life|half-life]] of the parent is shorter than that of the daughter. No equilibrium develops since the parent decays before the daughter.
 ### ###
    
 {{:textbook:nrctextbook:development_of_activities_no-equilibrium_fig_6_6.png |}} {{:textbook:nrctextbook:development_of_activities_no-equilibrium_fig_6_6.png |}}
 Figure VI.6. Development of activities in case of no radioactive equilibrium, in which the half-life of the parent nuclide (<sup>218</sup>Po, t<sub>½</sub> = 3 min) is shorter than that of the daughter (<sup>214</sup>Pb, t<sub>½</sub> = 26.8 min).  Figure VI.6. Development of activities in case of no radioactive equilibrium, in which the half-life of the parent nuclide (<sup>218</sup>Po, t<sub>½</sub> = 3 min) is shorter than that of the daughter (<sup>214</sup>Pb, t<sub>½</sub> = 26.8 min). 
-Left: activity on linear scale. Right: activity on logarithmic scale. Blue: <sup>218</sup>Po . Red: <sup>214</sup>Pb.  +Left: activity on linear scale. Right: activity on logarithmic scale. Blue: <sup>218</sup>Po . Red: <sup>214</sup>Pb. 
 +  
 +{{anchor:equilibria_in_natural_decay_chains}}
 ==== 6.7.4. Equilibria in natural decay chains ==== ==== 6.7.4. Equilibria in natural decay chains ====
 +{{anchor:long_lived_radionuclides}}
 ### ###
-In natural uranium and thorium decay chains there are individual pairs in which there would not be any equilibrium if they were present separately. An example of such pairs in the 238U decay chain is <sup>234</sup>Pa parent (t<sub>½</sub> = 6.7 h) and <sup>234</sup>U daughter (t<sub>½</sub> = 245000 y). They are, however, typically in  +In natural uranium and thorium [[textbook:nrctextbook:chapter4#decay_chains|decay chains]] there are individual pairs in which there would not be any equilibrium if they were present separately. An example of such pairs in the <sup>238</sup>U decay chain is <sup>234</sup>Pa parent (t<sub>½</sub> = 6.7 h) and <sup>234</sup>U daughter (t<sub>½</sub> = 245000 y). They are, however, typically in  
-equilibrium since the grandparent of <sup>234</sup>Pa, <sup>238</sup>U (t<sub>½</sub> = 4.4 × 10<sup>9</sup> y), feeds continually new <sup>234</sup>Pa and they are in equilibrium with each other. Since <sup>238</sup>U has the longest half-life in the whole chain and therefore the activities of all subsequent radionuclides in the chain have the same activity as <sup>238</sup>U supposing that the system has been closed millions of years. In the nature there are chemical processes, such as dissolution into groundwater, that remove some component of the chains which causes disequilibria in the chains.+equilibrium since the grandparent of <sup>234</sup>Pa, <sup>238</sup>U (t<sub>½</sub> = 4.4 × 10<sup>9</sup> y), feeds continually new <sup>234</sup>Pa and they are in equilibrium with each other. Since <sup>238</sup>U has the longest [[textbook:nrctextbook:chapter6#half_life|half-life]] in the whole chain and therefore the [[textbook:nrctextbook:chapter6#activity|activities]] of all subsequent [[textbook:nrctextbook:chapter4|radionuclides]] in the chain have the same activity as <sup>238</sup>U supposing that the system has been closed millions of years. In the nature there are chemical processes, such as dissolution into groundwater, that remove some component of the chains which causes disequilibria in the chains.
 ### ###
      
  
 ### ###
-In the geosphere in the natural decay chains beginning from <sup>238</sup>U, <sup>235</sup>U and <sup>232</sup>Th the activities of all members are the same in each series, identical with those of <sup>238</sup>U, <sup>235</sup>U and <sup>232</sup>Th, in systems which have been preserved without disturbances long enough. In such case the series is in equilibrium state. If some component of the series is removed, by dissolution for example, the equilibrium is disturbed and a disequilibrium state is created. If for example uranium is dissolved from a primary  +In the geosphere in the natural [[textbook:nrctextbook:chapter4#decay_chains|decay chains]] beginning from <sup>238</sup>U, <sup>235</sup>U and <sup>232</sup>Th the activities of all members are the same in each series, identical with those of <sup>238</sup>U, <sup>235</sup>U and <sup>232</sup>Th, in systems which have been preserved without disturbances long enough. In such case the series is in equilibrium state. If some component of the series is removed, by dissolution for example, the equilibrium is disturbed and a disequilibrium state is created. If for example uranium is dissolved from a primary uranium-bearing mineral by oxidation the remaining radionuclides in the series will be supported by its most [[textbook:nrctextbook:chapter4#long_lived_radionuclides|long-lived radionuclide]]  which is <sup>230</sup>Th in case of <sup>238</sup>U series. If the dissolved uranium will then be precipitated somewhere out of the system a new equilibrium will start to develop. The time required to attain the equilibrium is governed by the most long-lived daughter [[textbook:nrctextbook:chapter4|radionuclide]] in the series, <sup>230</sup>Th in case of <sup>238</sup>U series. The [[textbook:nrctextbook:chapter6#half_life|half-life]] of <sup>230</sup>Th is 75000 years and this time is required to attain 50% of the equilibrium, 150000 years for 75% equilibrium, 225000 years for 87.5% equilibrium and eight half-lives, 600000 years, for 99.6% equilibrium. The disequilibria can be utilized in dating geological events. If for example, the <sup>230</sup>Th/<sup>238</sup>U ratio is 0.5 in a uranium mineral we may calculate that this uranium mineral was precipitated 75000 years ago.  
-uranium-bearing mineral by oxidation the remaining radionuclides in the series will be supported by its most long-lived radionuclide which is <sup>230</sup>Th in case of <sup>238</sup>U series. If the dissolved uranium will then be precipitated somewhere out of the system a new equilibrium will start to develop. The time required to attain the equilibrium is governed by the most long-lived daughter radionuclide in the  +
-series, <sup>230</sup>Th in case of <sup>238</sup>U series. The half-life of <sup>230</sup>Th is 75000 years and this time is required to attain 50% of the equilibrium, 150000 years for 75% equilibrium, 225000 years for 87.5% equilibrium and eight half-lives, 600000 years, for 99.6% equilibrium. The disequilibria can be  +
-utilized in dating geological events. If for example, the <sup>230</sup>Th/<sup>238</sup>U ratio is 0.5 in a uranium mineral we may calculate that this uranium mineral was precipitated 75000 years ago.  +
  
 ### ###
  
 ### ###
-To calculate activities of all members in a series manually is a cumbersome task. Computer programs for this purpose have been fortunately developed. One of them is the Decservis-2 program developed at the Laboratory of Radiochemistry, University of Helsinki, Finland. An example of such calculation carried out by Decservis-2 is shown in Figure VI.6. Here, we assume separation of <sup>226</sup>Ra (1 Bq) from the system and development of equilibrium between<sup> 226</sup>Ra and its progeny in  +To calculate [[textbook:nrctextbook:chapter6#activity|activities]] of all members in a series manually is a cumbersome task. Computer programs for this purpose have been fortunately developed. One of them is the Decservis-2 program developed at the Laboratory of Radiochemistry, University of Helsinki, Finland. An example of such calculation carried out by Decservis-2 is shown in Figure VI.6. Here, we assume separation of <sup>226</sup>Ra (1 Bq) from the system and development of equilibrium between<sup> 226</sup>Ra and its progeny in 10000 years. We have to assume that the gaseous <sup>222</sup>Rn, the daughter of <sup>226</sup>Ra, is not escaped from the system. In the first phase, up to about a month, the equilibrium is attained with <sup>222</sup>Rn, <sup>218</sup>Po, <sup>214</sup>Pb, <sup>214</sup>Bi and <sup>214</sup>Po and the time required for equilibrium is governed by the most long-lived member of these, <sup>222</sup>Rn with a half-life of 3.8 days. In the second phase, up to about 200 years, the equilibrium is attained with <sup>210</sup>Pb, <sup>210</sup>Bi and <sup>210</sup>Po and the time required for equilibrium is governed by the most long-lived member of these, <sup>210</sup>Pb with a half-life of 22 years. The half-life of<sup> 226</sup>Ra is 1600 years and decrease in its activity and correspondingly activities of its progeny can be seen after about 1000 years. 
-10000 years. We have to assume that the gaseous <sup>222</sup>Rn, the daughter of <sup>226</sup>Ra, is not escaped from the system. In the first phase, up to about a month, the equilibrium is attained with <sup>222</sup>Rn, <sup>218</sup>Po,  +
-<sup>214</sup>Pb, <sup>214</sup>Bi and <sup>214</sup>Po and the time required for equilibrium is governed by the most long-lived member of these, <sup>222</sup>Rn with a half-life of 3.8 days. In the second phase, up to about 200 years, the equilibrium is attained with <sup>210</sup>Pb, <sup>210</sup>Bi and <sup>210</sup>Po and the time required for equilibrium is governed by the most long-lived member of these, <sup>210</sup>Pb with a half-life of 22 years. The half-life of<sup> 226</sup>Ra is 1600 years and decrease in its activity and correspondingly activities of its progeny can be seen after about 1000 years. +
  
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 +{{anchor:figure_67}}
 {{:textbook:nrctextbook:attainment_of_radioactive_equilibrium_of_226ra_progeny_fig_6_7.png |}} {{:textbook:nrctextbook:attainment_of_radioactive_equilibrium_of_226ra_progeny_fig_6_7.png |}}
  
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