# Presentation: 223369

## 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 **
**	****4n + 1 ****	****237****Neptunium**** **

**4.1  Decay Series**

## Slide 2

## Slide 3

## Slide 4

## Slide 5

## 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**

## 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)**

## 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**

## 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 **

## 4.2b Kinetics of Radioactivity

**t****1/2**** = 5 years**

**4.2b 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 rate

		1st  half-life     50% decay
		2nd  half-life    75% decay		
		3rd  half-life    87.5% decay
		4th  half-life    93.75% decay
		5th  half-life    96.87% decay
		6th  half-life    98.44% decay
		7th  half-life    99.22% decay 

**	****	****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**

## Slide 12

## 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**

**4.2c Probability of Disintegration**

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

**4.2c Probability of Disintegration**

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

**4.2c Probability of Disintegration**

## 4.2c Probability of Disintegration 

**it is equally important to know the average     **
**    ****life of an isotope ******

**4.2c Probability of Disintegration**

## Decay Constant Problems

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

** ****what is the ******** constant ****52****V which has a t****1/2**** = 3.74 min.?**

**4.2c Probability of Disintegration**

## what is the  constant for 51Cr which has a t1/2 = 27.7 days?

**what is the ******** constant for ****226****Ra which has t****1/2**** = 1622 yrs**

**4.2c Probability of Disintegration**

## Decay Problem

what % of a given amount of 226Ra will decay during a period of 1000 years?       

                  1/2 life of 226Ra = 1622 yr

**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**

## therefore the percentage transformed during the 1000 year period is:  

                     100% - 64.5% = 35.5%

**                     ****100% - 64.5% = 35.5%**

**4.2c Probability of Disintegration**

## 4.2d Activity

**Curie (Ci), originally defined as the activity of 1 gm of Ra in which 3.7 ******** 10****10**** 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**

## 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;     234Th = 24.1 days

 therefore, 234Th is transforming 6.8  1010 faster than 238U

** ****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 ******** 10****10**** faster than ****238****U**

**4.2d Activity**

## 4.2d Activity

**1 Bq with 3 emissions**

**4.2d Activity**

## 4.2d Activity

**4.2d Activity**

##     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)		        = 10-12 Ci
	1 femtocurie (fCi) 		        = 10-15 Ci

**	****1 megabecquerel (MBq)  = 10****6****  Bq**
**	****1 gigabecquerel (GBq)    = 10****9****  Bq**
**	****1 terabecquerel (TBq)      = 10****12**** Bq**
**	****1 millicurie (mCi) ****	****	****        = 10****-3**** Ci**
**	****1 microcurie (μCi ****	****	****        = 10****-6**** Ci **
**	****1 nanocurie (nCi)****	****	****        = 10****-9**** Ci**
**	****1 picocurie (pCi)****	****	****        = 10****-12**** Ci**
**	****1 femtocurie (fCi) ****	****	****        = 10****-15**** Ci**

**1 Ci = 3.7 ******** 10****10**** Bq**

**4.2d Activity**

**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 10****23****)**

**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

**● ****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**

**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**

## 4.2d Activity

**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**

## 4.2d Activity

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

## 4.2d Activity

**potassium (atomic weight = 39.102 AMU) contains:**
**93.10 atom % ****39****K, having atomic mass 38.96371 AMU**
**0.0118 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 40.96184 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

** ****specific activity refers to the activity of 1 g material**
** ****1 g of naturally occurring potassium contains: 1.18 ******** 10****-4**** g ****40****K 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?

**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 ******** 10****16**** kg**
**how many MCi of ****14****C 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

**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**

##  for 239Pu

** ****for ****240****Pu**

**4.2d Activity**

## 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**

##  (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

** ****(8.37 ******** 10****9****) – (6.10 ********10****9****) x = 6.0 ******** 10****9**
** ****2.37 ******** 10****9**** = (6.1 ******** 10****9****) x**
**  ****x = 0.39 = 39% ****239****Pu**

**4.2d Activity**

## 4.2d Activity

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

##  decay constant is found from
				A = N

**	****	****	****	****A = ********N**

** ****finally**

**4.2d Activity**

## 4.3 Radioactive Equilibria

**net production of nuclide 2 is given by decay rate of nuclide 1 less the decay rate of nuclide 2**

## 4.3 Radioactive Equilibria

** ****solution of first order differential equation**

** ****given that:**

## 4.3 Radioactive Equilibria

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

## 4.3 Radioactive Equilibria

**after substitution for ****λ****:**

**the exponent term can be written to show the influence the ratio of**

## 4.3 Radioactive Equilibria

## Slide 49

## 4.4 Secular Equilibrium

-  **in secular equilibrium t****1/2 ****(1)>> t****1/2**** (2) so**

-  **reduces**

## Slide 51

## Slide 52

## 4.4 Secular Equilibrium

**   **

**β**

** **

** **

**64.1 h**

** **

**   **

**β**

** **

** **

**2.74 m**

** **

**   **

**β**

** **

** **

**28.8 y**

** **

**   **

**β**

** **

** **

**33 S**

** **

**90**

**Kr**

** **

**90**

**Rb**

** **

**90**

**Sr**

** **

**90**

**Y**

** **

**90**

**Zr**

** **

**4.4 Secular Equilibrium**

-  **after 10 half-lives**

## 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 

**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

**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**

## 1 mg of sample is

**4.4 Secular Equilibrium**

-  **specific activity of ****90****Y is**** **

##  therefore mass of 90Y is

**4.4 Secular Equilibrium**

## 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) ******

## Slide 59

##  in secular equilibrium the mother and daughter have the same activities

 in transient equilibrium the the daughter activity is always higher

** ****in transient equilibrium the the daughter activity is always higher**

**4.5 Transient Equilibrium**

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

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

**4.5 Transient Equilibrium**

## 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**

## Slide 63

## 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**** **

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

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

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

**in 11.1 hours**

## 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% **

## 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**

**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**

## 4.8 Branching Decay

**4.8 Branching Decay**

## 4.8 Branching Decay

**4.8 Branching Decay**

## 4.9 Successive Transformations

** ****nuclide 1 ******** nuclide 2 ********nuclide 3 ******** nuclide 4 ******** nuclide n**

## 4.9 Successive Transformations

**4.9 Successive Transformations**

** ****with the coefficients given as:**

## 4.9 Successive Transformations

## 4.9 Successive Transformations

**4.9 Successive Transformations**

## 4.9 Successive Transformations

**4.9 Successive Transformations**

## 4.9 Successive Transformations

**4.9 Successive Transformations**

** ****and**

## 4.9 Successive Transformations

**4.9 Successive Transformations**

** ****and            **

## Slide 85

## 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 ****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 ....)**

## 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 (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**

## 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:**
**	****	****(4*****n*****+2)  ****→****	****234****U (t****1/2**** = 2.44 x 10****5 ****years)**
**	****	****(4*****n*****)****	****   ****→****	****228****Ra (t****1/2**** = 5.75 years)**
**	****	****(4*****n*****+3)  ****→****	****231****Pa (t****1/2**** = 3.28 a 10****4**** years)**

## 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)

## 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)**

## 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.**

## 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.**

## 16.2 Cosmogenic Radionuclides

## 16.2 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 ice **
**if no exchange takes place, the activity of the radiounculides is a measure of the age.**

## 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**

## 16.2 Cosmogenic Radionuclides

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

## 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 250 - 40,000 years****.**

## 16.2 Cosmogenic Radionuclides

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

## 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 **

## 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.**

## 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 10****10****:0.8 x 10****12****.**
** ****ratio cannot be measured by classical mass spectrometry because ions of the same mass are found at practically the same position.**

## 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**

## 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. **

## 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****λ****t****-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:**

## 16.3 Terrestrial Nuclide Pairs

## 16.3 Terrestrial Nuclide Pairs

**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.**

## 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**

## 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)**

## 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)**

## 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**

## 16.4 Natural Decay Series

## 16.4 Natural Decay Series

**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**

## 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 ****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)**

## 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 ~ 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 ****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**

## 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.**

## 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.**

## 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.**

## 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 **

## 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)**

## 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**

## 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**

## 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 **

## 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.**

## 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.**

## 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.**

## 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/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**

## 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 ****238****U, 7.252 x 10****-3**** 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.**

## 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 

**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 = 158.9 )?**

**Problem **

## Slide 143

## Slide 144

## Problem

**Problem**

## Slide 146

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

**Problem**

## Problem

**Problem**

## Problem

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

## (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 ****241****Am this is 432.7 y.**

## Problem

**Problem**

## Slide 152

## Problem

**Problem**

## Slide 154

## Slide 155

## Problem

**Problem**

## Slide 157

## Problem

**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**

## Slide 159

## Problem

**Problem**

## Slide 161