4. Radioactive Decay 4.1 Decay Series synonymous expressions

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

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

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)

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 ES or through quantum mechanical tunneling

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

t1/2 = 5 years

each isotope has its own distinct half-life (t1/2) and in almost all cases no operation, physical or chemical, can alter the transformation rate 1st half-life  50% decay 2nd half-life  75% decay 3rd half-life  87.5% decay 4th half-life  % decay 5th half-life  % decay 6th half-life  % decay 7th half-life  % decay

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

at t = 0: N = N0 therefore lnN0 = C  the fraction of any radioisotope remaining after n half-lives is given by

where No is the original quantity and N is the quantity after n half lives

if the time t is small compared with the half-life of the radionuclide ( t<<t1/2) then we can approximate

Average Life of an Isotope it is equally important to know the average life of an isotope 

Decay Constant Problems what is the  constant 52V which has a t1/2 = 3.74 min.?

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

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

therefore the percentage transformed during the 1000 year period is: 100% % = 35.5%

4.2d Activity Curie (Ci), originally defined as the activity of 1 gm of Ra in which 3.7  1010 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

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

4.2d Activity equal weights of radioisotopes do not give equivalent amounts of radioactivity 238U and its daughter 234Th have about the same no. of atoms per gm. However their half- lives are greatly different 238U = 4.5  109 yr; Th = 24.1 days therefore, 234Th is transforming 6.8  1010 faster than 238U

4.2d Activity 60Co , 0.314 MeV , 1.1173 MeV , 1.332 MeV 60Ni
1 Bq with 3 emissions

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

4.2d Activity 1 Ci = 3.7  1010 Bq 1 megabecquerel (MBq) = 106 Bq
1 kilobecquerel (kBq) = 103 Bq 1 megabecquerel (MBq) = 106 Bq 1 gigabecquerel (GBq) = 109 Bq 1 terabecquerel (TBq) = 1012 Bq 1 millicurie (mCi) = 10-3 Ci 1 microcurie (μCi = 10-6 Ci 1 nanocurie (nCi) = 10-9 Ci 1 picocurie (pCi) = Ci 1 femtocurie (fCi) = Ci

4.2d Activity since activity A is proportional to N, the
number of atoms, we get A = A0e-t the mass m of radioactive atoms can be calculated from their number N; activity A; M mass of nuclide; and Nav Avogadro’s number ( 6.02 X 1023)

4.2d Activity Problem ● 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

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

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

4.2d Activity SA may also be derived by using the fact that there are 3.7  1010 tps in 1 gm of 226Ra

4.2d Activity Problem calculate the specific activity of 14C
(t1/2 = 5730 yrs)

4.2d Activity Problem potassium (atomic weight = 39.102 AMU) contains:
93.10 atom % 39K, having atomic mass AMU atom % 40K, which has a mass of 40.0 AMU and is radioactive with: t1/2 = 1.3  109 yr 6.88 atom % 41K having a mass of AMU

4.2d Activity 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  10-4 g 40K plus non-radioactive isotopes

4.2d Activity

4.2d Activity Problem prior to use of nuclear weapons, the SA of 14C 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  1016 kg how many MCi of 14C did the ocean carbonates contain?

4.2d Activity Problem a mixture of 239Pu and 240Pu has a specific activity of 6.0  109 dps the half-lives of the isotopes are 2.44  104 and 6.58  103y, respectively calculate the isotopic composition

4.2d Activity for 239Pu for 240Pu

4.2d Activity number of seconds in a year is
for 239Pu: A = 2.27 109/s  g for 240Pu: A = 8.37 109/s  g let the fraction of 239Pu = x; then the fraction 240Pu = 1 - x

4.2d Activity (2.27  109)x+(1 – x)(8.37  109) = 6.0  109 (8.37  109) – (6.10 109) x = 6.0  109 2.37  109 = (6.1  109) x x = 0.39 = 39% 239Pu

4.2d Activity Problem no. of atoms in 3  10-9 kg of 200Au is
if 3  10-9 kg of radioactive 200Au has an activity of 58.9 Ci, what is its half-life? no. of atoms in 3  10-9 kg of 200Au is

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

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

given that: solution of first order differential equation

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

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

4.4 Secular Equilibrium in secular equilibrium t1/2 (1)>> t1/2 (2) so reduces

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

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

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

4.4 Secular Equilibrium therefore mass of 90Y is

4.5 Transient Equilibrium
in transient equilibrium the half-life of the mother is longer than the daughter t1/2 (1)> t1/2 (2) 

in secular equilibrium the mother and daughter have the same activities in transient equilibrium the the daughter activity is always higher

Practical Applications the same applications as in secular equilibrium except the following equation is used

4.6 Half-Life of Mother Nuclide Shorter than Half-Life of Daughter
t1/2 (1)< t1/2 (2) no radioactive equilibrium attained fission product 141Ce has a half-life of 13.9 minutes and its daughter product 146Pr has a half-life of 24.4 mi

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

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

4.8 Branching Decay branching decay is often seen in odd-odd nuclei or in decay series for 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%

4.8 Branching Decay

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

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

4.8 Branching Decay decay rates of nuclides B and C

4.8 Branching Decay net rate of production of nuclide B using

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

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

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

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

4.8 Branching Decay if the time t is small compared with the half- life of the mother nuclide A (t<< t1/2(A)) we get

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

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

for example if n = 3 we get

if nuclide 3 is stable then

if t1/2 of the mother nuclide is much longer than the successive ones 1 << 2,  3, n we get and

under these conditions we get 0 and

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

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 3H or 14C 2) Measuring the daughter nuclides formed by decay of primordial mother nuclides (e.g. K/Ar, Rb/Sr, U/Pb ….)

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

16.1 General Aspects time 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

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) → 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)

16.1 General Aspects stable 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)

16.1 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

16.1 General Aspects if 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.

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

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

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

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.

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.

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

Libby proved the formation of 14C by the interaction of cosmic rays with the nitrogen in the atmosphere in 1947 14C 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

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.

14C dating basic assumptions 14C: 12C ratio in living things is identical with that in the atmosphere 2. 14C: 12C ratio has been constant in the atmosphere during the period of time considered.

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

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 CO2 nuclear explosions liberated neutrons in the upper atmosphere that sharply increased 14C production these changes should not influence dating of samples more than 100 years old.

ratio of carbon isotopes 14C: 13C: 12C in samples of recent origin is about 1:0.9 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.

accelerator mass spectrometry (AMS) has been successful at identifying some nuclides. 26Al, 32Si, 36Cl, 41Ca, and 129I 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

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.

16.3 Terrestrial Nuclide Pairs
dating by this method requires evaluation of 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

there are two methods for sample analysis: Independent determination of N2 and N1 Simultaneous determination of N2 and N1 by mass spectrometry properties of mother and daughter must be similar for simultaneous determination both methods require additional determination of N20 , but it can be neglected in some special cases.

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

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)

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

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 origins applications of the 176Lu/176Hf and 187Re/187Os 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

16.4 Natural Decay Series

16.4 Natural Decay Series taking 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 series variations in the ratio 207Pb:206Pb indicate geological processes since 204Pb is not radiogenic, it is commonly used as a reference nuclide

16.4 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)

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

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)

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

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

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 206Pb:204Pb=9.4 and 207Pb:204Pb=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)

16.4 Natural Decay Series dating with 210Pb is of interest for the dating of glacier and polar ice, and climatology. the source of 210Pb is 222Rn emitted into the air some 222Rn is emitted from volcanos. annual amounts of 210Pb brought down with precipitations is relatively constant the easiest method of detection of 210Po is by α spectrometry (detection limit ~ 10-4 Bq) after attainment of radioactive equilibrium and chemical separation

16.4 Natural Decay Series early stages of dating by nuclear methods were by measurement of 4He formed by α decay in the natural decay series it 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

16.5 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.

mineral dating is possible by taking 204Pb 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.

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 235U:238U as a function of time fact that the ratio Th:U is practically constant for minerals of the same genesis.

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 sample sample and a standard of known age are irradiated under the same conditions for about 1 day

Ar is produced and measured by mass spectrometry age of the sample is calculated by the relation (16.10)

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

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

Example with 234U/230 Th UO22+ ions are found in natural waters, in the form of [Uo2(CO3)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

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

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

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. 238U is the only spontaneous fission isotope that gives dense enough tracks for dating.

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

16.7 Fission Tracks track density (number of fission tracks/cm2) 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 238U spontaneous fission track density is first measured, and then the sample is irradiated so that the neutron-induced fission of 235U is obtained

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

16.7 Fission Tracks where 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.

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

Problem 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 107 Bq’s of pure A (nuclidic mass of A = )?

Problem 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 (µCi). Calculate the half-life of this radionuclide and with the aid of a chart of the nuclides tentatively identify this radionuclide.

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

Problem Calculate 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.

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

(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.

Problem 137Cs 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.

Problem A 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.

Problem In 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?

Problem 218Po 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?

Problem 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 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?

4. Radioactive Decay 4.1 Decay Series synonymous expressions

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