64.2a Law and Energy of Radioactive Decay radioactive decay law follows Poisson statisticsbehaves aswhere:N is the number of atoms of a certain radionuclide;-dN/dt is the disintegration rate; and is the disintegration constant in sec-1
74.2a Law and Energy of Radioactive Decay law of radioactive decay describes the kineticsof a reactionWhereA is the mother radionuclide;B is the daughter nuclide;X is the emitted particle; andE is the energy set free by the decay process(also known as Q-value)
84.2a Law and Energy of Radioactive Decay radioactive decay only possible whenE > 0 which can be calculated ashowever decay may only arise if nuclide A surmounts an energy barrier with a threshold ES or through quantum mechanical tunneling
94.2b Kinetics of Radioactivity Half-Lifethe time for any given radioisotope to decrease to 1/2 of its original quantityrange from a few microseconds to billions of years
114.2b Kinetics of Radioactivity each isotope has its own distinct half-life (t1/2) and in almost all cases no operation, physical or chemical, can alter the transformation rate1st half-life 50% decay2nd half-life 75% decay3rd half-life 87.5% decay4th half-life % decay5th half-life % decay6th half-life % decay7th half-life % decay
134.2c 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 = - Ndtwhere is the proportionality constant and the -ve sign is introduced because N decreases
144.2c Probability of Disintegration at t = 0: N = N0 therefore lnN0 = C the fraction of any radioisotope remaining after n half-lives is given by
154.2c Probability of Disintegration where No is the original quantity and N is the quantity after n half lives
164.2c Probability of Disintegration if the time t is small compared with the half-life of the radionuclide ( t<<t1/2) then we can approximate
174.2c Probability of Disintegration Average Life of an Isotopeit is equally important to know the averagelife of an isotope
184.2c Probability of Disintegration Decay Constant Problemswhat is the constant 52V which has a t1/2 = 3.74 min.?
194.2c Probability of Disintegration what is the constant for 51Cr which has a t1/2 = 27.7 days?what is the constant for 226Ra which has t1/2 = 1622 yrs
204.2c Probability of Disintegration Decay Problemwhat % of a given amount of 226Ra will decay during a period of 1000 years?1/2 life of 226Ra = 1622 yr
214.2c Probability of Disintegration therefore the percentage transformed during the 1000 year period is:100% % = 35.5%
224.2d ActivityCurie (Ci), originally defined as the activity of 1 gm of Ra in which 3.7 1010 atoms are transformed per secin 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
23activity of a radionuclide is given by its disintegration rate 4.2d Activityactivity of a radionuclide is given by its disintegration rate
244.2d Activityequal weights of radioisotopes do not give equivalent amounts of radioactivity238U and its daughter 234Th have about the same no. of atoms per gm. However their half- lives are greatly different238U = 4.5 109 yr; Th = 24.1 daystherefore, 234Th is transforming 6.8 1010 faster than 238U
284.2d Activity since activity A is proportional to N, the number of atoms, we getA = A0e-tthe mass m of radioactive atoms can becalculated from their number N; activity A; Mmass of nuclide; and Nav Avogadro’s number( 6.02 X 1023)
294.2d ActivityProblem● how much time is required for 5 mg of 22Na (t1/2 = 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
304.2d Activity Specific Activity the relationship between mass of the material and activity or AS (SA) = no. of Bq's/unit mass or volume
314.2d ActivitySA can also be represented in combined mathematical known terms
324.2d ActivitySA may also be derived by using the fact that there are 3.7 1010 tps in 1 gm of 226Ra
334.2d Activity Problem calculate the specific activity of 14C (t1/2 = 5730 yrs)
344.2d Activity Problem potassium (atomic weight = 39.102 AMU) contains: 93.10 atom % 39K, having atomic mass AMUatom % 40K, which has a mass of 40.0 AMU and is radioactive with:t1/2 = 1.3 109 yr6.88 atom % 41K having a mass of AMU
354.2d Activityestimate the specific activity of naturally occurring potassiumspecific activity refers to the activity of 1 g material1 g of naturally occurring potassium contains: 1.18 10-4 g 40K plus non-radioactive isotopes
374.2d ActivityProblemprior to use of nuclear weapons, the SA of 14C in soluble ocean carbonates was found to be 16 dis/min ·g carbonamount of carbon in these carbonates has been estimated as 4.5 1016 kghow many MCi of 14C did the ocean carbonates contain?
384.2d ActivityProblema mixture of 239Pu and 240Pu has a specific activity of 6.0 109 dpsthe half-lives of the isotopes are 2.44 104 and 6.58 103y, respectivelycalculate the isotopic composition
534.4 Secular Equilibrium after 10 half-lives β β β β Kr Rb Sr Y Zr 33 S 90Kr90Rb90Sr90Y90Zr33 S2.74 m28.8 y64.1 h
544.4 Secular Equilibrium 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 knowncalculation of mass ratios of radionuclidescalculation of the mass of a mother nuclide from the measured activity of a daughter nuclide or the reverse
554.4 Secular Equilibrium Problem how many grams of 90Y are in secular equilibrium with 1 mg of 90Srthus, the amount of 90Y having the same activity of 1 mg of 90Sr
564.4 Secular Equilibrium1 mg of sample isspecific activity of 90Y is
604.5 Transient Equilibrium in secular equilibrium the mother and daughter have the same activitiesin transient equilibrium the the daughter activity is always higher
614.5 Transient Equilibrium Practical Applicationsthe same applications as in secular equilibrium except the following equation is used
624.6 Half-Life of Mother Nuclide Shorter than Half-Life of Daughter t1/2 (1)< t1/2 (2)no radioactive equilibrium attainedfission product 141Ce has a half-life of 13.9 minutes and its daughter product 146Pr has a half-life of 24.4 mi
644.7 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 activityby differentiating with respect to time and setting it equal to zero we get
654.7 Similar Half-Lives and Attainment of Maximum Activity of Daughter Nuclide
664.7 Similar Half-Lives and Attainment of Maximum Activity of Daughter Nuclide in the following decay sequence when will the maximum activity of 135Xe occur?in 11.1 hours
674.8 Branching Decaybranching decay is often seen in odd-odd nuclei or in decay seriesfor example, 40K decays into 40Ca by -emission with a probability of 89.5% and into 40Ar by electron capture with a probability of 10.7%
8616. Dating by Nuclear Methods General AspectsCosmogenic RadionuclidesTerrestrial Mother/Daughter Nuclide PairsNatural Decay SeriesRatios of Stable IsotopesRadioactive DisequilibriaFission Tracks
8716.1 General Aspectsthe laws of radioactive decay are the basis of chronology by nuclear methodstwo kinds of dating by nuclear methods can be distinguished:1) Measuring radioactive decay of cosmogenic radionuclides, such as 3H or 14C2) Measuring the daughter nuclides formed by decay of primordial mother nuclides (e.g. K/Ar, Rb/Sr, U/Pb ….)
8816.1 General AspectsRutherford was first to see the potential of determining the age of uranium minerals from the amount of helium formed by radioactive decaythis potential was realized soon after the elucidation of the natural decay series of uranium and thoriumErnest RutherfordNobel Prize in Chemistry 1908
8916.1 General Aspectstime scale of applicability for naturally occurring radionuclides depends on the half-life (t1/2)age to be determined and t1/2 should be on roughly the same order:0.1* t1/2 < age < 10* t1/2
9016.1 General Aspectsdating on the basis of radioactive equilibrium is possible after about 10 half-lives of the longest-lived daughter nuclidesthe longest lived nuclides are:(4n+2) → 234U (t1/2 = 2.44 x 105 years)(4n) → 228Ra (t1/2 = 5.75 years)(4n+3) → 231Pa (t1/2 = 3.28 a 104 years)
9116.1 General Aspectsstable decay products, such as 4He, 206Pb, 207Pb, 208Pb, 40Ar, and 87Sr, 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)
9216.1 General Aspects(16.1)N10 is the number of atoms of the mother nuclide at t=0.for dating, N2 and N1 have to be determined
9316.1 General Aspectsif several stable atoms are formed per radioactive decay of the mother nuclide, as in the case of 4He formed by radioactive decay of 238U, Th, 235U 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.
9416.1 General Aspectsthe following methods of dating by nuclear methods can be distinguishedmeasurement of cosmogenic radionuclidesmeasurement of terrestrial mother/daughter nuclide pairs3. measurement of members of the natural decay series
9516.1 General Aspects4. measurement of isotope ratios of stable radiogenic isotopes5. measurement of radioactive disequilibria6. measurement of fission tracks
9616.1 General Aspectsthere are some problems with the methods outlined here, and these will be discussed separately in detailone 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
9716.2 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.
9916.2 Cosmogenic Radionuclides 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 iceif no exchange takes place, the activity of the radiounculides is a measure of the age.
10016.2 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 icelarge 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 timesT dating is thus restricted appreciably for all but glacier and polar ice samples, in which the influence of nuclear explosions is negligible
10116.2 Cosmogenic Radionuclides Libby proved the formation of 14C by the interaction of cosmic rays with the nitrogen in the atmosphere in 194714C atoms are quickly oxidized in the atmosphere to CO2, which is incorporated by the process of assimilation into plants and via the food chain into animals and humans
10216.2 Cosmogenic Radionuclides death of living things signifies the end of 14C uptake.14C activity decreases with the half-life, provided no exchange of carbon atoms with the environment takes place.half-life of 14C is very favorable for dating of archaeological samples in the range of about ,000 years.
10316.2 Cosmogenic Radionuclides 14C dating basic assumptions14C: 12C ratio in living things is identical with that in the atmosphere2. 14C: 12C ratio has been constant in the atmosphere during the period of time considered.
10416.2 Cosmogenic Radionuclides 3. Periodic variation of the 14C : 12C ratio (~9 x 10 3y 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 14C
10516.2 Cosmogenic Radionuclides humans have caused drastic changes in the 14C: 12C ratio since the beginning of the industrial age.fossil Fuel combustion has diluted the 14CO2 by releasing 14C-free CO2nuclear explosions liberated neutrons in the upper atmosphere that sharply increased 14C productionthese changes should not influence dating of samples more than 100 years old.
10616.2 Cosmogenic Radionuclides ratio of carbon isotopes 14C: 13C: 12C in samples of recent origin is about1:0.9 X 1010:0.8 x 1012.ratio cannot be measured by classical mass spectrometry because ions of the same mass are found at practically the same position.
10716.2 Cosmogenic Radionuclides accelerator mass spectrometry (AMS) has been successful at identifying some nuclides.26Al, 32Si, 36Cl, 41Ca, and 129I have all been identifiedtypically dating by these nuclides is not favored for several reasons such as:low concentrationslow production ratestechnical challenges associated with detection
10816.2 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.
10916.3 Terrestrial Nuclide Pairs dating by this method requires evaluationof the following equation:(16.3)where, N2 is the total number of atoms of the stable nuclide (2), N20 is the number of atoms of this nuclide present at t=0, and N1 (eλt-1) is the number of radiogenic atoms formed by decay of the mother nuclide
11116.3 Terrestrial Nuclide Pairs there are two methods for sample analysis:Independent determination of N2 and N1Simultaneous determination of N2 and N1 by mass spectrometryproperties of mother and daughter must be similar for simultaneous determinationboth methods require additional determination of N20 , but it can be neglected in some special cases.
11216.3 Terrestrial Nuclide Pairs in the 40K/40Ar method, the mass spectrometry is complicated because of the necessary 40Ar isotope dilutionthe time required for this process may introduce additional 40Ar from atmosphere, and lead to a false dating
11316.3 Terrestrial Nuclide Pairs simultaneous determination of N2 and N1 is performed by measuring the ratios with a stable non-radiogenic nuclide as reference nuclide (Nr) by using the following equation:(16.4)
11416.3 Terrestrial Nuclide Pairs rearranging this equations leads to the following equation for the age of the sample(16.5)where t1/2 is the half-life of the radioactive mother nuclide
11516.3 Terrestrial Nuclide Pairs simultaneous determination of mother and daughter nuclide by MS is applied in the 87Rb/87Sr and 147Sm/134Nd methods. These methods have had applications in geochronology in the dating of minerals, magmatic rocks, and sedimentary rocks of various originsapplications of the 176Lu/176Hf and 187Re/187Os methods have no advantages over the two previous methodsmajor drawbacks are low concentrations of Lu (<1mg/kg) and Re (~1ng/kg) found in the minerals
11716.4 Natural Decay Seriestaking into account the long-lived radionuclides, radioactive equilibrium is established after about 106 y in the case of the uranium and actinium series and after about 10 y in the case of the thorium seriesvariations in the ratio 207Pb:206Pb indicate geological processessince 204Pb is not radiogenic, it is commonly used as a reference nuclide
11816.4 Natural Decay Series three kinds of systems can be distinguished: losing parts of the members of the decay chains or the radiogenic Pb by diffusion or recrystallization processes (i.e. open systems)
11916.4 Natural Decay Seriesapplications of this technique are summarized in the following table
12016.4 Natural Decay Series2. 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)
12116.4 Natural Decay Seriesapplicable forms of equation (16.4) for case number 2.(16.6)(16.7)(16.8)
12216.4 Natural Decay Series3. 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
12316.4 Natural Decay Seriesa practical application of equations (16.6) through (16.8) is the calculation of the age of the solar systemmass spectrometry analysis of meteorites gives isotope ratios of the Pb isotopes 206Pb:204Pb=9.4 and 207Pb:204Pb=10.3assuming 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)
12416.4 Natural Decay Seriesdating with 210Pb is of interest for the dating of glacier and polar ice, and climatology.the source of 210Pb is 222Rn emitted into the airsome 222Rn is emitted from volcanos.annual amounts of 210Pb brought down with precipitations is relatively constantthe easiest method of detection of 210Po is by α spectrometry (detection limit ~ 10-4 Bq) after attainment of radioactive equilibrium and chemical separation
12516.4 Natural Decay Seriesearly stages of dating by nuclear methods were by measurement of 4He formed by α decay in the natural decay seriesit was difficult to ensure the prerequisites of dating by U/ 4He method, because neither 4He nor α -emitting members of the decay series can be lost or produced by any other means beside alpha decay of U
12616.5 Ratios of Stable Isotopes there are four stable isotopes of lead: 204Pb, 206Pb, 207Pb, and 208Pb.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 235U, 238U, and/or 232Th.
12716.5 Ratios of Stable Isotopes mineral dating is possible by taking 204Pb as a reference nuclide, and comparing the ratios of each other stable nuclide to it by mass spectrometryif the contents of U or Th are known and losses can be neglected, eqs. (16.6, 16.7, and 16.8) can be applied.
12816.5 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 byequations (16.6), (16.7), and (16.8)knowledge of the ratio 235U:238U as a function of timefact that the ratio Th:U is practically constant for minerals of the same genesis.
12916.5 Ratios of Stable Isotopes the 39Ar/40Ar method is a variant of the 40K/40Ar method.neutron activation analysis is applied to determine the amount of K present in the samplesample and a standard of known age are irradiated under the same conditions for about 1 day
13016.5 Ratios of Stable Isotopes Ar is produced and measured by mass spectrometryage of the sample is calculated by the relation(16.10)
13116.6 Radioactive Disequilibria useful for providing information about separation processes in minerals and ores, and sediments in oceans or lakesby 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
13216.6 Radioactive Disequilibria prerequisite is that the mother and daughter nuclide exhibit different chemical behavior under the given conditionsmay be caused by different solubility of mother and daughter nuclide, by different probabilities of escape or by different leaching rates due to recoil effectsexamples are U/Th and U/Pa
13316.6 Radioactive Disequilibria Example with 234U/230 ThUO22+ ions are found in natural waters, in the form of [Uo2(CO3)3]4- ionsTh ions are completely hydrolyzed and easily sorbed on particulates, and thus settle in sedimentscorals and other inhabitants form skeletons by uptake of elements dissolved in the sea
13416.6 Radioactive Disequilibria applicationsgeochemistry for dating of crystallization processes by measuring the ratio 238U:230Thexcess of 230Th or 231Pa found in marine sediments allows dating of these sediments and determination of the sedimentation ratearchaeology, the 234U:230Th method is applied for dating of carbonates used by humans or for dating of bones or teeth.
13516.7 Fission Tracksin this way, 234Th (daughter of 238U) and the long-lived 230Th are separatedand if the skeletons can be considered to be closed systems, the ingrowth of 230Th is a measure of the age.
13616.7 Fission Tracksfission tracks are observed in solids due to spontaneous or neutron-induced fission of heavy nuclei and can be made visible under an optical microscope.238U is the only spontaneous fission isotope that gives dense enough tracks for dating.
13716.7 Fission Tracksthe 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 chambersXO particle passing through a bubble chamber
13816.7 Fission Trackstrack density (number of fission tracks/cm2) in a mineral is a function of U concentration & the age of the mineralfor the purpose of dating, a sufficient number of tracks must be counted, so the concentration of U or the age should be relatively high238U spontaneous fission track density is first measured, and then the sample is irradiated so that the neutron-induced fission of 235U is obtained
13916.7 Fission Tracksthe age t of the mineral is calculated by the following formula:(16.11)
14016.7 Fission Trackswhere λ (238) is the decay constant of 238U, x 10-3 is the isotope ratio 235U:238U,D(sf) and D(n,f) are the fission track densities due to spontaneous fission of 238U and due to neutron-induced fission of 235U, respectively,σ(n,f) is the cross section of fission of 235U by thermal neutrons, and ti is the irradiation time.
14116.7 Fission Tracksfor homogeneous distribution of U in the sample, the values of D(sf) and D(n,f) can be determined in different aliquots of the samplefor heterogeneous distribution of U, D(sf) and the sum D(sf) + D(n,f) must be counted in the same samplefission 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
142ProblemConsider the decay series A B C D, where the half-lives of A, B, and C are3.45 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 107 Bq’s of pure A (nuclidic mass of A = )?
145ProblemA sample of g of a pure radionuclide with a mass number of 244 was observed to have an absolute activity of 4.45 microcuries (µCi). Calculate the half-life of this radionuclide and with the aid of a chart of the nuclides tentatively identify this radionuclide.
147ProblemCalculate the activity (in mCi) of a medical 60Co source containing 1.00 mg of the isotope.
148ProblemCalculate the activity, in dps and Ci, expected for a 1.00 mg 252Cf source that is 10.0 years old. The half-life of 252Cf is 2.64 y.
149Problem(a) Calculate the mass, in grams, of the 241Am present in the smoke detector which has 1 µCi
150(b) How long will it take to reduce the activity of 241Am from 1 (b) How long will it take to reduce the activity of 241Am 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 241Am this is y.
151Problem137Cs decays via β- emission to 137mBa. An experiment is begun with 5.00 x 106 Bq of pure 137Cs. Calculate the activity due to 137mBa after a decay period of 50 min.
153ProblemA sample contains a mixture of 239Pu and 240Pu in unknown proportions. The activity of the mixed sample was found to be 4.35 x 107 dpm for a sample of mg of Pu. Calculate the weight % of each Pu isotope present.
156ProblemIn the decay chain A B Cstable 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?
158Problem218Po decays with a half-life of 3.10 min to 214Pb, which in turn decays with a half-life of 26.8 min to 214Bi. Assuming we have a source of pure 218Po at the start of our experiment, what decay time will be required for the activity to 214Pb to reach its maximum value?
160ProblemA 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 14C activity of living wood is 16 disintegrations per minute per gram. How long ago did the tree die from which the wood sample came?