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4. Radioactive Decay radioactive transmutation and decay are synonymous expressions 4 main series 4n 232 Thorium 4n + 2 238 Uranium-Radium 4n + 3 235 Actinium.

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Presentation on theme: "4. Radioactive Decay radioactive transmutation and decay are synonymous expressions 4 main series 4n 232 Thorium 4n + 2 238 Uranium-Radium 4n + 3 235 Actinium."— Presentation transcript:

1 4. Radioactive Decay radioactive transmutation and decay are synonymous expressions 4 main series 4n 232 Thorium 4n Uranium-Radium 4n Actinium 4n Neptunium 4.1 Decay Series





6 4.2a Law and Energy of Radioactive Decay radioactive decay law follows Poisson statistics behaves as where: N is the number of atoms of a certain radionuclide; -dN/dt is the disintegration rate; and is the disintegration constant in sec -1

7 4.2a Law and Energy of Radioactive Decay law of radioactive decay describes the kinetics of a reaction Where A is the mother radionuclide; B is the daughter nuclide; X is the emitted particle; and  E is the energy set free by the decay process (also known as Q-value)

8 4.2a Law and Energy of Radioactive Decay radioactive decay only possible when  E > 0 which can be calculated as however decay may only arise if nuclide A surmounts an energy barrier with a threshold E S or through quantum mechanical tunneling

9 4.2b Kinetics of Radioactivity Half-Life the time for any given radioisotope to decrease to 1/2 of its original quantity range from a few microseconds to billions of years

10 t 1/2 = 5 years 4.2b Kinetics of Radioactivity

11 each isotope has its own distinct half-life (t 1/2 ) and in almost all cases no operation, physical or chemical, can alter the transformation rate 1 st half-life  50% decay 2 nd half-life  75% decay 3 rd half-life  87.5% decay 4 th half-life  93.75% decay 5 th half-life  96.87% decay 6 th half-life  98.44% decay 7 th half-life  99.22% decay 4.2b Kinetics of Radioactivity


13 4.2c Probability of Disintegration number of nuclei dN in a time interval dt will be proportional to that time interval and to the number of nuclei N that are present; or at any time t there are N nuclei  dN = - Ndt where is the proportionality constant and the -ve sign is introduced because N decreases

14 at t = 0: N = N 0 therefore lnN 0 = C  the fraction of any radioisotope remaining after n half-lives is given by 4.2c Probability of Disintegration

15 where N o is the original quantity and N is the quantity after n half lives 4.2c Probability of Disintegration

16 if the time t is small compared with the half- life of the radionuclide ( t< { "@context": "", "@type": "ImageObject", "contentUrl": "", "name": "if the time t is small compared with the half- life of the radionuclide ( t<

17 Average Life of an Isotope it is equally important to know the average life of an isotope  4.2c Probability of Disintegration

18 Decay Constant Problems what is the constant 52 V which has a t 1/2 = 3.74 min.? 4.2c Probability of Disintegration

19 what is the constant for 51 Cr which has a t 1/2 = 27.7 days? what is the constant for 226 Ra which has t 1/2 = 1622 yrs 4.2c Probability of Disintegration

20 Decay Problem what % of a given amount of 226 Ra will decay during a period of 1000 years? 1/2 life of 226 Ra = 1622 yr 4.2c Probability of Disintegration

21 therefore the percentage transformed during the 1000 year period is: 100% % = 35.5% 4.2c Probability of Disintegration

22 4.2d Activity Curie (Ci), originally defined as the activity of 1 gm of Ra in which 3.7  atoms are transformed per sec in S.I. units activity is measured in Becquerel (Bq), where 1 Bq = 1 tps -> the quantity of radioactive material in which one atom is transformed per sec

23 activity of a radionuclide is given by its disintegration rate 4.2d Activity

24 equal weights of radioisotopes do not give equivalent amounts of radioactivity 238 U and its daughter 234 Th have about the same no. of atoms per gm. However their half- lives are greatly different 238 U = 4.5  10 9 yr; 234 Th = 24.1 days therefore, 234 Th is transforming 6.8  faster than 238 U 4.2d Activity

25 60 Co 60 Ni , MeV , MeV , MeV 1 Bq with 3 emissions 4.2d Activity

26 1 Bq with 1.18 emissions 42 K 42 Ca , 2.04 MeV 18% , 1.53 MeV 4.2d Activity

27 1 kilobecquerel (kBq) = 10 3 Bq 1 megabecquerel (MBq) = 10 6 Bq 1 gigabecquerel (GBq) = 10 9 Bq 1 terabecquerel (TBq) = Bq 1 millicurie (mCi) = Ci 1 microcurie (μCi = Ci 1 nanocurie (nCi) = Ci 1 picocurie (pCi) = Ci 1 femtocurie (fCi) = Ci 1 Ci = 3.7  Bq 4.2d Activity

28 since activity A is proportional to N, the number of atoms, we get A = A 0 e - t the mass m of radioactive atoms can be calculated from their number N; activity A; M mass of nuclide; and N av Avogadro’s number ( 6.02 X ) 4.2d Activity

29 Problem ● how much time is required for 5 mg of 22 Na (t 1/2 = 2.60 y) to reduce to 1 mg? ● since the mass of a sample will be proportional to the no. of atoms in the sample get 4.2d Activity

30 Specific Activity the relationship between mass of the material and activity or  A S (SA) = no. of Bq's/unit mass or volume 4.2d Activity

31 SA can also be represented in combined mathematical known terms 4.2d Activity

32 SA may also be derived by using the fact that there are 3.7  tps in 1 gm of 226 Ra 4.2d Activity

33 Problem calculate the specific activity of 14 C (t 1/2 = 5730 yrs)

34 Problem potassium (atomic weight = AMU) contains:  atom % 39 K, having atomic mass AMU  atom % 40 K, which has a mass of 40.0 AMU and is radioactive with: t 1/2 = 1.3  10 9 yr  6.88 atom % 41 K having a mass of AMU 4.2d Activity

35 estimate the specific activity of naturally occurring potassium specific activity refers to the activity of 1 g material 1 g of naturally occurring potassium contains: 1.18  g 40 K plus non- radioactive isotopes 4.2d Activity


37 Problem prior to use of nuclear weapons, the SA of 14 C in soluble ocean carbonates was found to be 16 dis/min ·g carbon amount of carbon in these carbonates has been estimated as 4.5  kg how many MCi of 14 C did the ocean carbonates contain? 4.2d Activity

38 Problem a mixture of 239 Pu and 240 Pu has a specific activity of 6.0  10 9 dps the half-lives of the isotopes are 2.44  10 4 and 6.58  10 3 y, respectively calculate the isotopic composition 4.2d Activity

39 for 239 Pu for 240 Pu 4.2d Activity

40 number of seconds in a year is for 239 Pu: A = 2.27  10 9 /s  g for 240 Pu: A = 8.37  10 9 /s  g let the fraction of 239 Pu = x; then the fraction 240 Pu = 1 - x 4.2d Activity

41 (2.27  10 9 )x+(1 – x)(8.37  10 9 ) = 6.0  10 9 (8.37  10 9 ) – (6.10  10 9 ) x = 6.0   10 9 = (6.1  10 9 ) x x = 0.39 = 39% 239 Pu 4.2d Activity

42 Problem if 3  kg of radioactive 200 Au has an activity of 58.9 Ci, what is its half-life? no. of atoms in 3  kg of 200 Au is

43 decay constant is found from A = N finally 4.2d Activity

44 4.3 Radioactive Equilibria net production of nuclide 2 is given by decay rate of nuclide 1 less the decay rate of nuclide 2

45 4.3 Radioactive Equilibria solution of first order differential equation given that:

46 4.3 Radioactive Equilibria if nuclide 1 and 2 are separated at t = 0; then nuclide 2 is not produced and

47 4.3 Radioactive Equilibria after substitution for λ: the exponent term can be written to show the influence the ratio of

48 4.3 Radioactive Equilibria


50 4.4 Secular Equilibrium in secular equilibrium t 1/2 (1)>> t 1/2 (2) so reduces



53 β 64.1 h β 2.74 m β 28.8 y β 33 S 90 Kr 90 Rb 90 Sr 90 Y Zr 4.4 Secular Equilibrium after 10 half-lives

54 Practical Applications determination of long-half-life of a mother nuclide by measuring the mass ratio of the daughter and mother nuclides providing the half-life of the daughter is known calculation of mass ratios of radionuclides calculation of the mass of a mother nuclide from the measured activity of a daughter nuclide or the reverse 4.4 Secular Equilibrium

55 Problem how many grams of 90 Y are in secular equilibrium with 1 mg of 90 Sr thus, the amount of 90 Y having the same activity of 1 mg of 90 Sr 4.4 Secular Equilibrium

56 1 mg of sample is 4.4 Secular Equilibrium specific activity of 90 Y is

57 therefore mass of 90 Y is 4.4 Secular Equilibrium

58 4.5 Transient Equilibrium in transient equilibrium the half-life of the mother is longer than the daughter t 1/2 (1)> t 1/2 (2) 


60 in secular equilibrium the mother and daughter have the same activities in transient equilibrium the the daughter activity is always higher 4.5 Transient Equilibrium

61 Practical Applications the same applications as in secular equilibrium except the following equation is used 4.5 Transient Equilibrium

62 4.6 Half-Life of Mother Nuclide Shorter than Half-Life of Daughter t 1/2 (1)< t 1/2 (2) no radioactive equilibrium attained fission product 141 Ce has a half-life of 13.9 minutes and its daughter product 146 Pr has a half-life of 24.4 mi


64 4.7 Similar Half-Lives and Attainment of Maximum Activity of Daughter Nuclide an important aspect in radiochemistry and health physics is the knowledge when daughter and granddaughters’ products reach their maximum activity by differentiating with respect to time and setting it equal to zero we get

65 4.7 Similar Half-Lives and Attainment of Maximum Activity of Daughter Nuclide

66 in the following decay sequence when will the maximum activity of 135 Xe occur? in 11.1 hours

67 4.8 Branching Decay branching decay is often seen in odd-odd nuclei or in decay series for example, 40 K decays into 40 Ca by  - emission with a probability of 89.5% and into 40 Ar by electron capture with a probability of 10.7%

68 4.8 Branching Decay

69 two probabilities of decay are independent and thus the decay rate is given as 4.8 Branching Decay

70 integration of the equation yields rates of production of nuclides B and C 4.8 Branching Decay

71 decay rates of nuclides B and C 4.8 Branching Decay

72 net rate of production of nuclide B using 4.8 Branching Decay

73 integrating and setting N B =0 at t = 0 the same holds for nuclide C 4.8 Branching Decay

74 in secular equilibrium we get  b +  C <<  b _  but only one half-life 4.8 Branching Decay

75 in secular equilibrium we get placing these terms into 4.8 Branching Decay

76 if the daughter nuclides are long-lived or stable (as 40 K)  4.8 Branching Decay

77 if the time t is small compared with the half- life of the mother nuclide A (t<< t 1/2 (A)) we get 4.8 Branching Decay

78 4.9 Successive Transformations nuclide 1  nuclide 2  nuclide 3  nuclide 4  nuclide n

79 Solution of the series of differential equations with n= 1, 2, 3, 4, …n yields 4.9 Successive Transformations with the coefficients given as:

80 4.9 Successive Transformations

81 for example if n = 3 we get 4.9 Successive Transformations

82 if nuclide 3 is stable then 4.9 Successive Transformations

83 if t 1/2 of the mother nuclide is much longer than the successive ones 1 << 2, 3, n we get 4.9 Successive Transformations and

84 under these conditions we get Successive Transformations and


86 16. Dating by Nuclear Methods General Aspects Cosmogenic Radionuclides Terrestrial Mother/Daughter Nuclide Pairs Natural Decay Series Ratios of Stable Isotopes Radioactive Disequilibria Fission Tracks

87 16.1 General Aspects the laws of radioactive decay are the basis of chronology by nuclear methods two kinds of dating by nuclear methods can be distinguished: 1) Measuring radioactive decay of cosmogenic radionuclides, such as 3 H or 14 C 2) Measuring the daughter nuclides formed by decay of primordial mother nuclides (e.g. K/Ar, Rb/Sr, U/Pb ….)

88 16.1 General Aspects Rutherford was first to see the potential of determining the age of uranium minerals from the amount of helium formed by radioactive decay this potential was realized soon after the elucidation of the natural decay series of uranium and thorium Ernest Rutherford Nobel Prize in Chemistry 1908

89 16.1 General Aspects time scale of applicability for naturally occurring radionuclides depends on the half-life (t 1/2 ) age to be determined and t 1/2 should be on roughly the same order: 0.1* t 1/2 < age < 10* t 1/2

90 16.1 General Aspects dating on the basis of radioactive equilibrium is possible after about 10 half- lives of the longest-lived daughter nuclides the longest lived nuclides are: (4n+2) → 234 U (t 1/2 = 2.44 x 10 5 years) (4n) → 228 Ra (t 1/2 = 5.75 years) (4n+3) → 231 Pa (t 1/2 = 3.28 a 10 4 years)

91 16.1 General Aspects stable decay products, such as 4 He, 206 Pb, 207 Pb, 208 Pb, 40 Ar, and 87 Sr, increase continuously with time. if one stable atom is formed per radioactive decay of the mother nuclide, the number of stable radiogenic atoms is: (1.1)

92 16.1 General Aspects N 1 0 is the number of atoms of the mother nuclide at t=0. for dating, N 2 and N 1 have to be determined (16.1)

93 16.1 General Aspects if several stable atoms are formed per radioactive decay of the mother nuclide, as in the case of 4 He formed by radioactive decay of 238 U, Th, 235 U and their daughter nuclides, the number of stable radiogenic atoms is: (16.2) where n is the number of 4He atoms produced in the decay series.

94 16.1 General Aspects the following methods of dating by nuclear methods can be distinguished 1. measurement of cosmogenic radionuclides 2. measurement of terrestrial mother/daughter nuclide pairs 3. measurement of members of the natural decay series

95 16.1 General Aspects 4. measurement of isotope ratios of stable radiogenic isotopes 5. measurement of radioactive disequilibria 6. measurement of fission tracks

96 16.1 General Aspects there are some problems with the methods outlined here, and these will be discussed separately in detail one major problem with most methods is whether the system is open or closed. If it is open, then the nuclides of interest could be lost or enter the system during the time period of interest

97 16.2 Cosmogenic Radionuclides cosmogenic radionuclides are produced by the interaction of cosmic rays with the components of the atmosphere, mainly in the stratosphere. if the intensity of cosmic rays (protons and neutrons) can be assumed to be constant, then the production rate of the radionuclides is constant.

98 16.2 Cosmogenic Radionuclides

99 as these radionuclides take part in various natural cycles on the surface of the earth, they are incorporated in various organic and inorganic products, such as plants, sediments and glacial ice if no exchange takes place, the activity of the radiounculides is a measure of the age.

100 16.2 Cosmogenic Radionuclides tritium (T) atoms formed in the stratosphere are transformed into HTO and enter the water cycle as well as the various water reservoirs, such as surface waters, groundwaters and polar ice large quantities of T have been released into the atmosphere due to nuclear weapons testing, causing an increase in the T:H ratio by about 1000 times T dating is thus restricted appreciably for all but glacier and polar ice samples, in which the influence of nuclear explosions is negligible

101 16.2 Cosmogenic Radionuclides Libby proved the formation of 14 C by the interaction of cosmic rays with the nitrogen in the atmosphere in C atoms are quickly oxidized in the atmosphere to CO 2, which is incorporated by the process of assimilation into plants and via the food chain into animals and humans

102 16.2 Cosmogenic Radionuclides death of living things signifies the end of 14 C uptake. 14 C activity decreases with the half-life, provided no exchange of carbon atoms with the environment takes place. half-life of 14 C is very favorable for dating of archaeological samples in the range of about ,000 years.

103 16.2 Cosmogenic Radionuclides 14 C dating basic assumptions C: 12 C ratio in living things is identical with that in the atmosphere C: 12 C ratio has been constant in the atmosphere during the period of time considered.

104 16.2 Cosmogenic Radionuclides 3. Periodic variation of the 14 C : 12 C ratio (~9 x 10 3 y at an amplitude of ~±5%) is correlated with the variation of the magnetic field of the earth causing changes in the intensity and composition of the cosmic radiation and consequently in the production rate of 14 C

105 16.2 Cosmogenic Radionuclides humans have caused drastic changes in the 14 C: 12 C ratio since the beginning of the industrial age. fossil Fuel combustion has diluted the 14 CO 2 by releasing 14 C-free CO 2 nuclear explosions liberated neutrons in the upper atmosphere that sharply increased 14 C production these changes should not influence dating of samples more than 100 years old.

106 16.2 Cosmogenic Radionuclides ratio of carbon isotopes 14 C: 13 C: 12 C in samples of recent origin is about 1:0.9 X :0.8 x ratio cannot be measured by classical mass spectrometry because ions of the same mass are found at practically the same position.

107 16.2 Cosmogenic Radionuclides accelerator mass spectrometry (AMS) has been successful at identifying some nuclides. 26 Al, 32 Si, 36 Cl, 41 Ca, and 129 I have all been identified typically dating by these nuclides is not favored for several reasons such as: low concentrations low production rates technical challenges associated with detection

108 16.2 Cosmogenic Radionuclides Radiocarbon dating, the use of long-lived radioisotopes in climate research, and new developments in accelerator mass spectrometry are the main research activities of the laboratory. Ion beams are also applied to materials analysis and modification.

109 16.3 Terrestrial Nuclide Pairs where, N 2 is the total number of atoms of the stable nuclide (2), N 2 0 is the number of atoms of this nuclide present at t=0, and N 1 (e λt -1) is the number of radiogenic atoms formed by decay of the mother nuclide (16.3) dating by this method requires evaluation of the following equation:

110 16.3 Terrestrial Nuclide Pairs

111 there are two methods for sample analysis: Independent determination of N 2 and N 1 Simultaneous determination of N 2 and N 1 by mass spectrometry properties of mother and daughter must be similar for simultaneous determination both methods require additional determination of N 2 0, but it can be neglected in some special cases.

112 16.3 Terrestrial Nuclide Pairs in the 40 K/ 40 Ar method, the mass spectrometry is complicated because of the necessary 40 Ar isotope dilution the time required for this process may introduce additional 40 Ar from atmosphere, and lead to a false dating

113 16.3 Terrestrial Nuclide Pairs simultaneous determination of N 2 and N 1 is performed by measuring the ratios with a stable non-radiogenic nuclide as reference nuclide (N r ) by using the following equation: (16.4)

114 16.3 Terrestrial Nuclide Pairs rearranging this equations leads to the following equation for the age of the sample where t 1/2 is the half-life of the radioactive mother nuclide (16.5)

115 16.3 Terrestrial Nuclide Pairs simultaneous determination of mother and daughter nuclide by MS is applied in the 87 Rb/ 87 Sr and 147 Sm/ 134 Nd methods. These methods have had applications in geochronology in the dating of minerals, magmatic rocks, and sedimentary rocks of various origins applications of the 176 Lu/ 176 Hf and 187 Re/ 187 Os methods have no advantages over the two previous methods major drawbacks are low concentrations of Lu (<1mg/kg) and Re (~1ng/kg) found in the minerals

116 16.4 Natural Decay Series

117 taking into account the long-lived radionuclides, radioactive equilibrium is established after about 10 6 y in the case of the uranium and actinium series and after about 10 y in the case of the thorium series variations in the ratio 207 Pb: 206 Pb indicate geological processes since 204 Pb is not radiogenic, it is commonly used as a reference nuclide

118 16.4 Natural Decay Series three kinds of systems can be distinguished: 1.losing parts of the members of the decay chains or the radiogenic Pb by diffusion or recrystallization processes (i.e. open systems)

119 16.4 Natural Decay Series applications of this technique are summarized in the following table

120 16.4 Natural Decay Series 2. the loss of members of decay chains can be neglected and in which the concentration of the mother nuclide can be taken as a measure of age (equation (16.4) applies)

121 16.4 Natural Decay Series applicable forms of equation (16.4) for case number 2. (16.6) (16.7) (16.8)

122 16.4 Natural Decay Series 3. the loss of members of decay chains can be neglected, but in which the concentration of the mother nuclide cannot be taken as a measure of the age

123 16.4 Natural Decay Series a practical application of equations (16.6) through (16.8) is the calculation of the age of the solar system mass spectrometry analysis of meteorites gives isotope ratios of the Pb isotopes 206 Pb: 204 Pb=9.4 and 207 Pb: 204 Pb=10.3 assuming these values are the initial isotope ratios at the time of formation of the solar system, the age is found by application of equation (16.5): (16.9)

124 16.4 Natural Decay Series dating with 210 Pb is of interest for the dating of glacier and polar ice, and climatology. the source of 210 Pb is 222 Rn emitted into the air some 222 Rn is emitted from volcanos. annual amounts of 210 Pb brought down with precipitations is relatively constant the easiest method of detection of 210 Po is by α spectrometry (detection limit ~ Bq) after attainment of radioactive equilibrium and chemical separation

125 16.4 Natural Decay Series early stages of dating by nuclear methods were by measurement of 4 He formed by α decay in the natural decay series it was difficult to ensure the prerequisites of dating by U/ 4 He method, because neither 4 He nor α -emitting members of the decay series can be lost or produced by any other means beside alpha decay of U

126 16.5 Ratios of Stable Isotopes there are four stable isotopes of lead: 204 Pb, 206 Pb, 207 Pb, and 208 Pb. primordial Pb is what was formed in the course of the genesis of the elements. Radiogenic Pb is the additional amounts formed by decay of 235 U, 238 U, and/or 232 Th.

127 16.5 Ratios of Stable Isotopes mineral dating is possible by taking 204 Pb as a reference nuclide, and comparing the ratios of each other stable nuclide to it by mass spectrometry if the contents of U or Th are known and losses can be neglected, eqs. (16.6, 16.7, and 16.8) can be applied.

128 16.5 Ratios of Stable Isotopes measurement of the Pb/Pb ratio offers the possibility of dating without knowledge of the contents of U and Th. basis for the Pb/Pb method is given by equations (16.6), (16.7), and (16.8) knowledge of the ratio 235 U: 238 U as a function of time fact that the ratio Th:U is practically constant for minerals of the same genesis.

129 16.5 Ratios of Stable Isotopes the 39 Ar/ 40 Ar method is a variant of the 40 K/ 40 Ar method. neutron activation analysis is applied to determine the amount of K present in the sample sample and a standard of known age are irradiated under the same conditions for about 1 day

130 16.5 Ratios of Stable Isotopes Ar is produced and measured by mass spectrometry age of the sample is calculated by the relation (16.10)

131 16.6 Radioactive Disequilibria useful for providing information about separation processes in minerals and ores, and sediments in oceans or lakes by measuring the decay of the separated daughter nuclide or the growth of the daughter in the phase containing the mother, the time of separation can be determined

132 16.6 Radioactive Disequilibria prerequisite is that the mother and daughter nuclide exhibit different chemical behavior under the given conditions may be caused by different solubility of mother and daughter nuclide, by different probabilities of escape or by different leaching rates due to recoil effects examples are U/Th and U/Pa

133 16.6 Radioactive Disequilibria Example with 234 U/ 230 Th UO 2 2+ ions are found in natural waters, in the form of [Uo 2 (CO 3 ) 3 ] 4- ions Th ions are completely hydrolyzed and easily sorbed on particulates, and thus settle in sediments corals and other inhabitants form skeletons by uptake of elements dissolved in the sea

134 16.6 Radioactive Disequilibria applications geochemistry for dating of crystallization processes by measuring the ratio 238 U: 230Th excess of 230 Th or 231 Pa found in marine sediments allows dating of these sediments and determination of the sedimentation rate archaeology, the 234 U: 230 Th method is applied for dating of carbonates used by humans or for dating of bones or teeth.

135 16.7 Fission Tracks in this way, 234 Th (daughter of 238 U) and the long-lived 230 Th are separated and if the skeletons can be considered to be closed systems, the ingrowth of 230 Th is a measure of the age.

136 16.7 Fission Tracks fission tracks are observed in solids due to spontaneous or neutron-induced fission of heavy nuclei and can be made visible under an optical microscope. 238 U is the only spontaneous fission isotope that gives dense enough tracks for dating.

137 16.7 Fission Tracks the method is the same as that used with track detectors such as photographic emulsion and autoradiography, dielectric track detectors, cloud chambers, bubble chambers, and spark chambers XO particle passing through a bubble chamber

138 16.7 Fission Tracks track density (number of fission tracks/cm 2) in a mineral is a function of U concentration & the age of the mineral for the purpose of dating, a sufficient number of tracks must be counted, so the concentration of U or the age should be relatively high 238 U spontaneous fission track density is first measured, and then the sample is irradiated so that the neutron-induced fission of 235 U is obtained

139 16.7 Fission Tracks the age t of the mineral is calculated by the following formula: (16.11)

140 16.7 Fission Tracks where λ (238) is the decay constant of 238 U, x is the isotope ratio 235 U: 238 U, D(sf) and D(n,f) are the fission track densities due to spontaneous fission of 238 U and due to neutron-induced fission of 235 U, respectively, σ(n,f) is the cross section of fission of 235 U by thermal neutrons, and t i is the irradiation time.

141 16.7 Fission Tracks for homogeneous distribution of U in the sample, the values of D(sf) and D(n,f) can be determined in different aliquots of the sample for heterogeneous distribution of U, D(sf) and the sum D(sf) + D(n,f) must be counted in the same sample fission tracks are also influenced by recrystallization processes in solids, and is therefore useful in determining the temperature/pressure that the mineral was exposed to over time

142 Consider the decay series A  B  C  D, where the half-lives of A, B, and C are 3.45 h, 10.0 min, and 2.56 h, respectively. We first prepare some pure radionuclide A and exactly 2.75 h after this preparation we measure the activity of daughter C. What would the activity of daughter C be (in Bq) after the 2.75 h decay of pure A, if we started with 7.35 x 10 7 Bq’s of pure A (nuclidic mass of A = )? Problem



145 A sample of g of a pure radionuclide with a mass number of 244 was observed to have an absolute activity of 4.45 microcuries ( µ Ci). Calculate the half-life of this radionuclide and with the aid of a chart of the nuclides tentatively identify this radionuclide. Problem


147 Calculate the activity (in mCi) of a medical 60 Co source containing 1.00 mg of the isotope. Problem

148 Calculate the activity, in dps and Ci, expected for a 1.00 mg 252 Cf source that is 10.0 years old. The half-life of 252 Cf is 2.64 y. Problem

149 (a) Calculate the mass, in grams, of the 241 Am present in the smoke detector which has 1 µCi

150 (b) How long will it take to reduce the activity of 241 Am from 1.0 to 0.50 µCi? (b) From 1.0 µCi to 0.5 µCi is a reduction of one-half in the activity, so 1 half-life is required. For 241 Am this is y.

151 137 Cs decays via β - emission to 137m Ba. An experiment is begun with 5.00 x 10 6 Bq of pure 137 Cs. Calculate the activity due to 137m Ba after a decay period of 50 min. Problem


153 A sample contains a mixture of 239 Pu and 240 Pu in unknown proportions. The activity of the mixed sample was found to be 4.35 x 10 7 dpm for a sample of mg of Pu. Calculate the weight % of each Pu isotope present. Problem



156 In the decay chain A  B  C stable the half-lives of A and B are and 43.9 min, respectively. If we start with pure A, how long a decay period would be required for the activity of B to become equal to the activity of A? Problem


158 218 Po decays with a half-life of 3.10 min to 214 Pb, which in turn decays with a half-life of 26.8 min to 214 Bi. Assuming we have a source of pure 218 Po at the start of our experiment, what decay time will be required for the activity to 214 Pb to reach its maximum value? Problem


160 A piece of wood from the ruins of an ancient dwelling was found to have a 14C activity of 13 disintegrations per minute of carbon content. The 14 C activity of living wood is 16 disintegrations per minute per gram. How long ago did the tree die from which the wood sample came? Problem


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