Radiation in Environment

Transcription

1 Riku Ernesti Järvinen Radiation in Environment FYSZ46 Advanced Laboratory work Abstract Radiation from two environmental samples, a water sample from a drilled well and a fish (a powan), was measured with a germanium detector and further analyzed with gamma spectroscopy. The gamma peaks obtained from the measurement of the water sample showed that the sample contained radon, 222 Rn. The result for the average 1 specific activity of the sample was A sc avg = (71 ± 3) Bq/l and for the initial 2 activity A sc = (87±4) Bq/l. The results were somewhat smaller than the actual specific activities since part of the radon gas within the sample was lost just before the measurement was started. For the powan, a small gamma peak from the decay of 137 Cs was identified and the result for the specific activity of the sample was A spec = (2,±,1) Bq/kg. 1 Supposes that the activity was constant during the measurement time. 2 Activity in the beginning of the measurement.

2 Contents 1 Overview of Radioactive decays Types of Radioactive Decays Internal Conversion Phenomenology of Natural Decay Chains Activity, Specific Activity and Units of Activity Mathematics of Radioactive Decay The Universal Decay Law Branching Decays Internal Conversion Coefficients A Long-lived Parent Nucleus Consecutive decays and SDT Approximation* Gamma Spectroscopy and the Experiment Interactions Between Radiation and Matter The Experiment The Gamma Spectrum: Practical Analysis Error Analysis Negligible Sources of Error Non-negligible Sources of Error Results Energy calibration of the MCA Efficiency Calibration of the Detector Energy Resolution Calculation Powan Measurement Background Radon Measurement The Water Sample Parametric Analysis of the 295 kev Gamma Peak Equilibrium Value Analysis Conclusions 6 7 Appendix Table of Measurements Background Approximation Gamma Spectra Decay Schemes i

3 Introduction Radioactive decay is a spontaneous, random process in which an unstable (parent) nucleus transforms into another (daughter) nucleus in the absence of any external effects. Radioactivity can be either natural or artificial (man-made). There are two main reasons for natural radioactivity. First, the unstable nuclei that have amazingly long half-lives (t 1/2 > 1 8 years) and which were present when the Earth was formed have been decaying for a long time and have produced decay chains. There are three of such chains present. Secondly, there is a continuous production of unstable nuclei by cosmic rays entering the Earth s atmosphere. Natural radioactivity was discovered by Henri Becquerel in 1896 when he put two photographic plates (unexposed to light) next to uranium sulfide crystals. He noticed that the plates changed their colour to black [BRS]. In 193 Becquerel shared the Nobel Prize in Physics with Pierre and Marie Curie for the work on the natural radioactivity of the element radium. Artificial radioactivity means that radioactive nuclei are produced using some experimental methods. Generally it is necessary to take a stable nucleus and add or substract nucleons or transmute protons to neutrons or vice versa. Common methods for producing radioactive nuclei are called the chargedparticle- and neutron-activation [BRS]. Artificial radioactivity was first discovered by Irene Curie 3 and Pierre Joliot in 1934 and in 1935 they received the Nobel Prize in Chemistry for their achievement [KK88]. In this work, the main focus was on two different sources of radioactivity. The first one was a powan bought from a local supermarket. Because of the nuclear catastrophe of Chernobyl in 1986, some animals living in Scandinavia contain considerable amounts of 137 Cs. However, the origin of this fish was unknown. The element 137 Cs beta decays to the excited state of energy 661,66 kev of 137 Ba which further de-excites to the ground state. The second point of interest was to acquire a water sample from a local drilled well in Myllyjärvi, Jyväskylä and measure the 222 Rn radon activity of the sample. The isotope 222 Rn of radon is found in many drilled wells in Finland. Some wells contain so much radon that it may be dangerous to drink the water. Radon itself is a part of the natural decay chain of 238 U 92. By analyzing the gamma transitions of the daughters of radon, the activity can be determined. The motivation for doing experiments related to sources of radioactivity in nature is that with the development of technology the public health issues have 3 Daughter of Pierre and Marie Curie. ii

4 gained more support. It is essential to make sure that people are not exposed to radiation that may be dangerous. In this work we give a brief theoretical introduction to radioactivity (chapters 1 and 2) and gamma spectroscopy (chapter 3). The experimental setup is also described in chapter 3. In chapter 4, the principles of our error analysis are discussed. Chapter 5 is the core of this report since it presents all the calculations. Chapter 6 summarizes the main results. The appendix (chapter 7) contains the table of measurements, some relevant graphs and gamma spectra and pictures of decay schemes. iii

5 1 Overview of Radioactive decays In this section, different types of radioactive decays, internal conversion and phenomenology of the natural decay chains are discussed. 1.1 Types of Radioactive Decays In general, the term radioactive decay refers to five types of processes: alpha decay, beta decay, gamma decay, spontaneous fission and nucleon emission. The last two were not of interest in this work and are discussed elsewhere 4. In alpha and beta decay, the nucleus emits an alpha or beta particle, respectively, as it tries to achieve a more stable state [KK88]. In gamma decay, an excited state of a nucleus de-excites towards the ground state and energy is released in the form of radiation. The chart of the stable nuclides is shown in figure 1. Figure 1: The chart of stable nuclides [BRS]. Beta decay is common for almost all mass numbers and alpha decay for large mass numbers. Alpha decay The alpha decay process is described by the equation 4 See e.g. [KK88] and [BRS]. A ZX N A 4 Z 2 X N 2 +α, (1) 1

6 where X is the parent nucleus and X is the daughter nucleus. The alpha particle is a helium nucleus ( 4 He 2 ). The helium nucleus has a very large binding energy, approximately 7,1 MeV [STUK]. Normally it would require an energy input to the system to remove a nucleon from the atom. In the case of alpha decay, the binding energy of the helium nucleus can be used to release the nucleons. In practice, however, the alpha particle must have enough energy to overcome the Coulomb barrier and the binding energy alone is not sufficient. Therefore, the alpha emission is ultimately based on the quantum mechanical tunneling phenomenon [STUK]. Alpha particles emitted by an atom all have the same energy or they are divided into groups having the same energy (i.e. the energy spectrum 5 is discrete, see figure 2). The energy of alpha particles produced in decays is usually a few MeV. Figure 2: An alpha decay energy spectrum of 251 Fm [KK88]. Beta decay In beta decay, the nucleus spontaneously transforms into another nucleus in a way that changes the number of protons but leaves the mass number unchanged. There are three different beta decay processes. (1) β decay is described by the equation A ZX N A Z+1X N 1 +β +ν e, (2) 5 The spectrum analysis is described in detail in section

7 where X is the parent nucleus, X the daughter nucleus and ν e the electron antineutrino. The β particle is an electron that is emitted from the nucleus. (2) β + decay has the following form: A ZX N A Z 1X N+1 +β + +ν e. (3) This process has a threshold energy of 1,22 MeV. The β + particle is the antiparticle of the electron, positron. (3) Electron capture process is given by A ZX N +e A Z 1X N+1 +ν e. (4) In this process, the nucleus absorbs an orbiting electron. When another electron from an outer shell takes the place of the captured electron, characteristic X-rays 6 are emitted. The process is an alternative for β + decay. Beta decay is regarded as a process of creating an electron (positron) from the available decay energy at the instant of the decay [KK88]. It differs from alpha decay in sense that the alpha particle is already present at the time of the decay. In contrast to alpha energy spectra, the spectrum of beta decay is always continuous. In figure 3, the kinetic energy of the electron may have any value between zero and the maximum energy referred to as the Q value. This follows from the fact that energy released in beta decay can be distributed in whatever way between the electron and the neutrino: beta decay is essentially a three-body process. Gamma decay Gamma decay means that an excited state of a nucleus decays to a lower excited state or to the ground state of the same nucleus. A photon of energy approximately equal 7 to the difference in energy between the initial nuclear state and the final state is emitted. It can be shown 8 that the energy of the emitted gamma ray photon is E γ = Mc 2 [ 1± 1+ 2 E Mc 2 ], (5) where M is the mass of the nucleus and E = E i E f is the energy difference between the initial and final states. Since E Mc 2, the square root can be 6 Section The nucleus has a small recoil energy which can usually be neglected. 8 See [KK88, ]. 3

8 Figure 3: A typical energy spectrum of beta decay [KK88]. On the vertical scale is the number of electrons produced in the decay that correspond to different energies on the horizontal scale. expanded and taking the first three terms in the series gives E γ E ( E)2 2Mc 2. The second term is usually of the order of1 5 in comparison with the first term; in most cases, this is much less than the experimental accuracy. Therefore, the second term can be neglected and E γ E. The energy range of gamma ray photons is from,1 MeV to 1 MeV. Since E = hf = hc/λ, where h is Planck s constant and λ is the wavelength of the photon, we get λ fm. In comparison with the wavelength of visible light, which is about nm, these wavelengths are times smaller. A typical gamma energy spectrum looks like an alpha spectrum but the observed energies are smaller. The details of gamma spectra are discussed in section 3.3. Gamma radiation may proceed as electric or magnetic multipole transitions [KK88]. These are labeled as E 1,E 2,... and M 1,M 2,..., respectively. Experimental observations have verified the selection rules which determine whether the radiation has electric or magnetic character. These rules restrict the values of angular momentum, nuclear isospin 9 and parity. (I) I i I f L I i +I f. (II) If the initial and final nuclear states have the same parity, the transitions in descending probability order are as follows: M 1,E 2,M 3,E 4,... If the initial and the final nuclear states have different parity, the transitions in descending probability order are as follows: E 1,M 2,E 3,M 4,... 9 Discussed in some detail in [KK88, 7 71]. 4

9 In the first rule, I i and I f are the initial and final isospin states and L is the angular momentum of the nucleus. The lower bound for angular momentum determines which multipole transition is the most probable to occur. The first rule is always used in conjunction with the second. As an example, let us look at the gamma decay of an excited state of 137 Ba to the ground state 1. From figure 4, the excitation energy is E i = 661,66 kev. Furthermore, I i = 11/2, π i = ( ) and I f = 3/2,π f = (+) where π describes parity. Therefore, 4 L 7 and the transitions proceed as M 4,E 5,M 6,... with M 4 being the most probable. Figure 4: The excited state of 137 Ba [FS86]. The state is populated from the β decay of 137 Cs. 1.2 Internal Conversion Gamma decay competes with a process called internal conversion in which the nucleus de-excites by transferring its energy directly to an atomic electron. Quantum mechanically this may occur when the wavefunction of the electron (usually from the K shell) overlaps with the nucleus wavefunction. The essential point is that there is no force carrier (a photon) but the energy is directly transferred from the nucleus to the electron, which leaves the nucleus and then appears in the lab as a free electron. The hole the electron leaves behind is filled with an electron from an outer shell and characteristic X-rays are emitted. When the activities of the measured samples are calculated, the contribution of internal conversion must be known. The effect of conversion can be estimated by calculating internal conversion coefficients, a process briefly discussed in section This process is also examined in section 5.4 where the activity of the fish sample is calculated. 5

10 1.3 Phenomenology of Natural Decay Chains Most of the radioactive nuclei were produced in supernovae that generate a variety of neutron-rich nuclei [BRS]. However, this happened some million years before our solar system condensed from an interstellar cloud. The condensation took place about 4,5 1 9 years ago, so only elements with a half-life 11 greater than 1 8 years are still found on Earth in significant numbers. These nuclei have started series of consecutive decays called the natural decay chains. There are three of such chains: (1) The uranium series is given as 238 U Th Pa U Th Ra Rn Po Pb Bi Po Pb Bi Po Pb 82. This series is of interest in this work and is discussed in more detail shortly. (2) The arctinium series is 235 U Th Pa Ac Th Ra Rn Po Pb Bi Tl Pb 82. (3) The thorium series is 232 Th Th Ra Ac Th Ra Rn Po Pb Bi Po 84 or 28 Tl Pb 82. All these chains proceed via alpha and beta decays. Since the activity of 222 Rn 86 in our sample was measured, let s take a closer look at chain 1. It is described in table 1 with the corresponding half-lives and decay types. From table 1 we acknowledge that the half-life of radon is much longer than any of those of its decay products. This helps us in the calculations of section 5. In our analysis, we calculate the activity of the sample containing radon by investigating the gamma transitions from the excited states of 214 Bi 83 and 214 Po 84 From figure 5 we see that the most important transitions are the ones with energies of approximately 295, 352 and 69 kev. 1.4 Activity, Specific Activity and Units of Activity The activity of a radioactive sample is the number of radioactive disintegrations per unit time for the sample as a whole. Specific activity is defined as the 11 The time that it takes for half of the nuclei in a sample to decay, see section

11 Table 1: The decay chain of 238 U 92. Nucleus Half-life Decay type 238 U 92 4, a α 234 Th 9 24,1 d β 234 Pa 91 1,17 m β 234 U ,5 1 3 a α 23 Th 9 75, a α 226 Ra a α 222 Rn 86 3,8235 d α 218 Po 84 3,1 m α 214 Pb 82 26,8 m β 214 Bi 83 19,9 m β 214 Po ,3 µs α 21 Pb 82 22,3 a β 21 Bi 83 5,13 d β 21 Po ,376 d α 26 Pb 82 > 1 2 a Stable number of disintegrations per unit time per unit weight. The SI unit for the activity of a sample is Becquerel (Bq), which is equal to one disintegration per unit time (1 dps). Another unit that is sometimes used (due to historical conventions) is Curie (Ci), which is defined as [IAEA] 1 Ci = 3,7 1 1 dps. 7

12 Figure 5: The decay scheme of 222 Rn [CoN98]. 8

13 2 Mathematics of Radioactive Decay The formulae derived in this section apply to all types of radioactive decay. 2.1 The Universal Decay Law The law of radioactive decay is based on the fact that the decay is a purely statistical process [IAEA]. This means that it cannot be predicted when an individual nucleus decays; it only has some decay probability per unit time, σ, also known as the decay constant. If there are N radioactive nuclei present at time t = and no new nuclei are introduced into the sample, the number dn decaying in time dt is equal to N times the decay probability σ: Nσ = dn dt. (6) Solving this differential equation is trivial: the solution is N(t) = N e σt, (7) where N = N(t = ) is the number of parent nuclei in the beginning of the measurement. If we set N(t) =,5 N for some t, we obtain the half-life t 1/2, which is the time it takes for half of the nuclei to decay: t 1/2 = ln(2) σ. (8) For gamma transitions, the half-life is usually of the order of 1 9 s or less. However, there are also states that have significantly longer half-lives, even minutes or hours. These states are called isomers. In this work, no isomers were encountered. The activity of the sample is given as A(t) = dn dt = σn = σn e σt. (9) If we want to calculate the activity of the sample when we have measured the number of decayed nuclei to be N in time t, we write N = A(t) dt. (1) t If the activity does not change much in the measured time (i.e. if t t 1/2 ), equation (1) reduces to A = N t. (11) This simple formula gives a rough estimate of something like the average activity in time t. 9

14 2.2 Branching Decays There are many radioactive nuclei that show two or more competing decay modes. The relative probabilities of each of these modes are called the branching ratios. We assign a partial decay probability σ i for each possible mode i and have σ total = i Similarly for half-lives we obtain σ i, or 1 = 1 σ i. (12) σ total i t 1/2 = i t (i)1/2, (13) where t (i)1/2 = ln(2)/σ i. It is important to clarify that the partial decay probabilities never appear in any exponential factor, since we cannot just turn off the other decay mode while observing the other [KK88]. When a nucleus alpha or beta decays to the excited states of another nucleus, the excited states are populated according to different probabilities. These probabilities cannot be treated in the same way as the branching ratios because a higher excited state may gamma decay to a lower excited state and thus increase the probability of the lower state. To calculate all these probabilities each time would be extremely tedious. Therefore, the absolute and relative gamma intensities are defined. The absolute intensity or abundance of an excited state tells how many gammas are emmitted from the specific state for a hundred decays (beta or alpha) of the parent nucleus that populate the excited states of the daughter nucleus. Absolute intensity is always smaller than 1 %. Relative intensity, on the other hand, can be larger than 1 % since it measures the intensity of an excited state in comparison to another excited state for which we know the absolute intensity. Relative intensities are sometimes used because they can be determined with greater accuracy than the absolute intensities [ToI78]. In this work, we use the absolute intensities given in [FS86]. 2.3 Internal Conversion Coefficients In section 1.2 we stated that we must know the effect of internal conversion. In general, the total decay probability σ has two components, one arising from gamma emission and the other from internal conversion: σ = σ γ +σ int. (14) 1

15 The internal conversion coefficient is defined as u = σ int σ γ σ = σ γ (1+u). (15) One can also define partial coefficients representing individual atomic shells: σ = σ γ (1+u K +u L +u M +...). (16) The explicit calculation of the coefficients is cumbersome 12 and beyond the scope of this work. 2.4 A Long-lived Parent Nucleus As we can see from table 1 in section 1.3, the half-life of radon is many times longer than those of its decay products. Mathematically, we have λ 1 λ i, i > 1. (17) This is the case of secular equilibrium where, after some time, the rate of change in the number of daughter nuclei (that are active) becomes close to zero: dn i /dt. This means that the daughter nuclei are produced at the same rate they decay. The differential equation governing the decay chain thus becomes d dt N i(t) = = σ i N i (t)+σ i 1 N i 1 (t) N i (t) = σ i σ i 1 N i 1 (t). Calculating down the decay chain we observe that [BRS] N 1 (t) = N 1 e λ1t N i (t) = N 1 (t) λ 1 λ i, (18) (19) where the i th nucleus is anyone but the last one in the decay chain of 238 U 92. From equations (9) and (19) it follows that the activity of each daughter is equal to the activity of the parent, A i (t) = A 1 (t). In conjunction with equation (11), this result gives a relatively easy way to calculate 222 Rn activities (sections 5.5 and 5.6). 2.5 Consecutive decays and SDT Approximation* In this section, the formulae for decay chains of two, three and four are presented. Some approximations are made, as will be discussed in more detail shortly. An approximation method called the short decay time approximation is introduced. 12 See articles [HLS69] and [BSA91]. 11

16 A Chain of Two Decays Sometimes the daughter nucleus of a radioactive nucleus is also active. In this case, the formula for the amount of daughter nuclei at time t is a bit complex since new daughter nuclei are produced as a function of time. As starters, we have a pair of equations: N 1 (t) = N 1 e σ1t dn 2 dt = σ 2 N 2 (t)+σ 1 N 1 (t). (2) In equation (2), labels 1 and 2 refer to the parent and daughter nucleus, respectively and N 1 is the number of parent nuclei at time t =. Multiplying the second equation by e σ2t gives e σ2t d dt N 2(t) = σ 2 e σ2t N 2 (t)+σ 1 e (σ2 σ1)t N 1. (21) Now we identify the time derivative of the term e σ2t N 2 (t) and have Integrating both sides yields d ( e σ 2t N 2 (t) ) = σ 1 e (σ2 σ1)t N 1. (22) dt t d ( e σ 2t N 2 (t) ) t dt = σ 1 e (σ2 σ1)t N 1 dt dt e σ2t N 2 (t ) N 2 () = σ ( ) 1N 1 e (σ2 σ1)t 1 σ 2 σ 1 N 2 (t) = σ 1N 1 σ 2 σ 1 ( e σ 1t e σ2t) +N 2 e σ2t. In the last line we changed the variable from t to t. (23) SDT Approximation To determine the activity of the sample containing radon, we may use the formula derived in section 2.4. We now present an alternative way to evaluate the activity of the sample to which we later refer to as the short decay time (STD) approximation. In the following derivations of this chapter we assume that the excited states decay instantly once they are populated. Since no isomers were encountered in this work, the half-lives of the excited states are more than ten orders of magnitude shorter than the half-lives of the examined alpha and beta decays. The main philosophical problem with this analysis is that it hinders the statistical nature of gamma decay process. The results obtained this way are somewhat larger than the actual values. This is not a big problem, though, 12

17 because we can interpret the results as upper limits for the activity. In the case of a drilled well that contains radon, it is important to obtain a result that does not underestimate the activity. The main reasons for this analysis are the following: With equation (11) we only get a number that describes the activity. With SDT approximation analysis we also obtain a function A(t) of the activity. Equation (11) supposes that t t 1/2. To a good approximation, this is true for the fish sample but not for the water sample. The activity calculated with (11) gives a result that is somewhat smaller than the actual value. In short, we use chains of three and four decays in the following way: 222 Rn 86 alpha decays to 218 Po 84 which further alpha decays to 214 Pb 82 with a branching ratio of, Pb 82 beta decays to the excited states 214 Bi 83. Two of these excited states, namely the ones with energies E 1 = 295,21 kev and E 2 = 351,923 kev, have abundances large enough so that they are observed. 214 Bi 83 beta decays to the excited states of 214 Po 84 with a branching ratio of,9998. The state with E 3 = 69,312 kev has a significant abundance and is thus recognized. A Chain of Three Decays Starting with the decay chain of three, we have the equations N 2 (t) = σ 1N 1 ( e σ 1t e σ2t) +N 2 e σ2t σ 2 σ 1 dn 3 = σ 3 N 3 (t)+σ 2 N 2 (t), dt (24) where the labels 2 and 3 refer to the 218 Po 84 and 214 Pb 82 nuclei, respectively. Multiplying the second equation by e σ3t and rearranging the terms gives The integration of the LHS yields t d ( e σ 3t N 3 (t) ) = σ 2 e σ3t N 2 (t). (25) dt d ( e σ 3t N 3 (t) ) dt = e σ3t N 3 (t ) N 3. (26) dt 13

18 The integration of the RHS is a bit more tedious: t σ 2 t e σ3t N 2 (t) dt = σ 2 e σ3t σ 1N 1 ( e σ 1t e σ2t) +N 2 e σ2t dt σ 2 σ 1 t = σ 2σ 1 N 1 e (σ3 σ1)t e (σ3 σ2)t dt+ σ 2 σ 1 t +σ 2 N 2 e (σ3 σ2)t dt = σ t 2σ 1 N 1 ( e (σ 3 σ 1)t σ 2 σ 1 e(σ3 σ2)t σ 3 σ 1 σ 3 σ 2 t ( e (σ 3 σ 2)t) +σ 2 N 2 σ 3 σ 2 = σ 2σ 1 N [ 1 e (σ3 σ1)t 1 σ 2 σ 1 σ 3 σ 1 [ e (σ 3 σ 2)t 1 ] +σ 2 N 2. σ 3 σ 2 ) + e(σ3 σ2)t 1 ] + σ 3 σ 2 (27) Now we have e σ3t N 3 (t) = RHS+N 3 (28) Using equation (27) and changing the dummy variable t back to t we get N 3 (t) = σ 2σ 1 N [ 1 e σ1t e σ3t e σ2t e σ3t ] + σ 2 σ 1 σ 3 σ 1 σ 3 σ 2 [ e σ 2t e σ3t ] +σ 2 N 2 +N 3 e σ3t. σ 3 σ 2 (29) The activity is given by the equation (9). Now A 3 (t) = σ 3 N 3 (t) and (with equation (1)) we have t N = σ 3 N 3 (t) dt σ 3 σ 2 σ 1 N 1 = (σ 2 σ 1 )(σ 3 σ 1 ) σ 3 σ 2 σ 1 N 1 (σ 2 σ 1 )(σ 3 σ 2 ) + σ 3σ 2 N 2 σ 3 σ 2 t = σ 3σ 2 σ 1 N 1 σ 2 σ 1 t t e σ1t e σ3t dt e σ2t e σ3t dt+ e σ2t e σ3t dt+σ 3 N 3 t [ 1 ( 1 e σ1t σ 3 σ 1 σ 1 1 e σ2t )] + σ 2 1 ( 1 e σ3t σ 3 σ 2 σ 3 + σ [ 3σ 2 N 2 1 e σ 2t 1 e σ3t σ 3 σ 2 σ 2 σ 3 e σ3t dt 1 e σ3t ) σ 3 ] +N 3 (1 e σ3t ). (3) 14

19 Now we solve for N 1 in equation (3) and change the variable t back to t: [ σ N N 3σ 2 1 e σ 2 t 2 σ 3 σ 2 σ 2 1 e σ 3t σ 3 ] N 3 (1 e σ3t ) N 1 = [ ( ) ( ) ] (31) σ 3σ 2σ 1 1 e σ 1 t σ 2 σ 1 σ 1 e σ 3 t 1 e 1(σ 3 σ 1) σ 3(σ 3 σ 1) σ 3 t σ 1 e σ 2 t 3(σ 3 σ 2) σ 2(σ 3 σ 2) The denominator is constant and has no error 13. Thus we define ξ σ [ 3σ 2 σ 1 N ( 1 1 e σ 1t σ 2 σ 1 σ 1 (σ 3 σ 1 ) 1 e σ3t ) σ 3 (σ 3 σ 1 ) ( 1 e σ 3t σ 3 (σ 3 σ 2 ) 1 e σ2t ) ] (32) σ 2 (σ 3 σ 2 ) and have N 1 = [ σ N N 3σ 2 1 e σ 2 t 2 σ 3 σ 2 σ 2 1 e σ 3t σ 3 ] N 3 (1 e σ3t ) The radon activity of the sample is now given by ξ (33) A 1 (t) = σ 1 N 1 e σ1t. (34) In equation (31) we identify the parameters N 2 and N 3. These can be adjusted to see how the activity of the sample changes 14. In practical calculations of this work, however, we use the approximation N 2 = = N 3. (35) This is not strictly valid in our case, as will be discussed later. We now arrive at the formula A Chain of Four Decays A 1 (t) = Nσ 1e σ1t. (36) ξ The following derivation supposes that there are no daughter nuclei present in the beginning of the measurement 15 We start with the equations N 3 (t) = σ [ 2σ 1 N 1 e σ 1t e σ3t e σ2t e σ3t ] σ 2 σ 1 σ 3 σ 1 σ 3 σ 2 dn 4 (t) = σ 4 N 4 (t)+σ 3 N 3 (t) dt (37) Using the same mathematical tricks as in the previous derivations, we arrive at d ( e σ 4t N 4 (t) ) = σ 3 N 3 (t)e σ4t (38) dt 13 The error in the measurement time is negligible. 14 The details are presented in section Had we not done this approximation, the result would have been extremely tedious. 15

20 Next we integrate the LHS: t d ( e σ 4t N 4 (t) ) dt = e σ4t N 4 (t) N 4. (39) dt The integration of the RHS need a bit of work: t σ 3 N 3 e σ4t dt = σ [ 3σ 2 σ 1 N t 1 σ 2 σ 1 e (σ4 σ1)t e (σ4 σ3)t σ 3 σ 1 e(σ4 σ2)t e (σ4 σ3)t ] dt σ 3 σ 2 t σ 3 σ 2 σ 1 N 1 = e (σ4 σ1)t (σ 2 σ 1 )(σ 3 σ 1 ) e(σ4 σ3)t σ 4 σ 1 σ 4 σ 3 t σ 3 σ 2 σ 1 N 1 e (σ4 σ2)t (σ 2 σ 1 )(σ 3 σ 2 ) e(σ4 σ3)t σ 4 σ 2 σ 4 σ 3 [ σ 3 σ 2 σ 1 N 1 e (σ 4 σ 1)t 1 = (σ 2 σ 1 )(σ 3 σ 1 ) σ 4 σ 1 ] 1 σ e(σ4 σ3)t 3 σ 2 σ 1 N 1 σ 4 σ 3 (σ 2 σ 1 )(σ 3 σ 2 ) [ e (σ 4 σ 2)t 1 e(σ4 σ3)t ] 1 σ 4 σ 2 σ 4 σ 3 (4) Combining the results from the integrations of LHS and RHS, changing the variable t to t and using the approximation N 4 =, we get [ σ 3 σ 2 σ 1 N 1 e σ 1t e σ4t N 4 (t) = (σ 2 σ 1 )(σ 3 σ 1 ) σ 4 σ 1 [ σ 3 σ 2 σ 1 N 1 e σ 2t e σ4t e σ3t e σ4t (σ 2 σ 1 )(σ 3 σ 2 ) σ 4 σ 2 σ 4 σ 3 e σ3t e σ4t σ 4 σ 3 ] ]. (41) 16

21 The number of decayed nuclei is calculated as in the case of three consecutive decays: N = t A 4 (t) dt σ 4 σ 3 σ 2 σ 1 N 1 = (σ 2 σ 1 )(σ 3 σ 1 )(σ 4 σ 1 ) σ 4 σ 3 σ 2 σ 1 N 1 (σ 2 σ 1 )(σ 3 σ 1 )(σ 4 σ 3 ) σ 4 σ 3 σ 2 σ 1 N 1 (σ 2 σ 1 )(σ 3 σ 2 )(σ 4 σ 2 ) t t t t e σ1t e σ4t dt e σ3t e σ4t dt e σ2t e σ4t dt σ 4 σ 3 σ 2 σ 1 N 1 e σ3t e σ4t dt (σ 2 σ 1 )(σ 3 σ 2 )(σ 4 σ 3 ) = σ { [ 4σ 3 σ 2 σ 1 N e σ 1t (σ 2 σ 1 ) (σ 3 σ 1 ) σ 1 (σ 4 σ 1 ) 1 e σ4t σ 4 (σ 4 σ 1 ) ( 1 e σ 3t σ 3 (σ 4 σ 3 ) 1 e σ4t )] [ 1 1 e σ 2t σ 4 (σ 4 σ 3 ) σ 3 σ 2 σ 2 (σ 4 σ 2 ) ( 1 e σ4t 1 e σ 3t σ 4 (σ 4 σ 2 ) σ 3 (σ 4 σ 3 ) 1 e σ4t )]} σ 4 (σ 4 σ 3 ) (42) Now, with the change of the dummy variable t into t, the initial number of 222 Rn nuclei is given by { { [ σ4 σ 3 σ 2 σ e σ 1t N 1 = N (σ 2 σ 1 ) (σ 3 σ 1 ) σ 1 (σ 4 σ 1 ) 1 e σ4t σ 4 (σ 4 σ 1 ) ( 1 e σ 3t )] [ σ 3 (σ 4 σ 3 ) 1 e σ4t 1 1 e σ 2t σ 4 (σ 4 σ 3 ) σ 3 σ 2 σ 2 (σ 4 σ 2 ) (43) ( 1 e σ4t 1 e σ 3t )]}} 1 σ 4 (σ 4 σ 2 ) σ 3 (σ 4 σ 3 ) 1 e σ4t σ 4 (σ 4 σ 3 ) Again we see that the denominator is constant for each measurement time and we define { { σ4 σ 3 σ 2 σ 1 η (σ 2 σ 1 ) ( 1 e σ 3t σ 3 (σ 4 σ 3 ) 1 e σ4t 1 e σ4t σ 4 (σ 4 σ 2 ) [ 1 1 e σ 1t (σ 3 σ 1 ) σ 1 (σ 4 σ 1 ) 1 e σ4t σ 4 (σ 4 σ 1 ) )] [ 1 1 e σ 2t σ 4 (σ 4 σ 3 ) σ 3 σ 2 σ 2 (σ 4 σ 2 ) ( 1 e σ 3t )]}} σ 3 (σ 4 σ 3 ) 1 e σ4t σ 4 (σ 4 σ 3 ) (44) The activity is given by A 1 (t) = Nσ 1e σ1t. (45) η 17

22 3 Gamma Spectroscopy and the Experiment Each radioactive nuclei has a characteristic distribution of gamma radiation [STUK]. The purpose of gamma spectroscopy is to identify different nuclei according to the measured energy spectra. Observing of gammas is relatively easy since they have negligible absorption and scattering in air (contrary to alpha and beta particles) and their energies can be measured with very high accuracy [KK88]. 3.1 Interactions Between Radiation and Matter Measurement of a gamma spectrum is based on three types of interactions between radiation and matter: photoelectric effect, Compton scattering and pair production. The photon cross-sections for the three processes in carbon and lead are shown in figure 6. Figure 6: The cross-sections for photon interactions in C and Pb [BRS]. 18

23 Rayleigh scattering shown in figure 6 means the elastic scatterings of photons from atoms. This interaction does not appear in our gamma spectra because the detector used in the experiment only measures the current created by the electrons that are kicked out of their atomic orbits by the interactions of photons 16. Photoelectric effect is the most important interaction for low-energy (1-15 kev) photons. When a photon strikes the surface of the material (usually metal), an electron near the surface may absorb enough energy to overcome the attraction of the positive ions in the material and escape into the surrounding space [YF4]. The threshold energy for this process is the ionization energy of the electron shell. In Compton scattering, a photon scatters from a nearly free atomic electron [KK88]. The scattered photon has a longer wavelength (and a smaller frequency) than the initial one. The energy of the scattered photon depends on the scattering angle between the electron and the photon. The threshold energy for the process is the ionization energy of the electron shell. In pair production, a photon transforms into an electron-positron pair. Thus, the process has a threshold energy of 122 kev and does not occur at low energies. In our experiment, the gamma energies analysed are all below the threshold for pair production. However, it is possible to identify escape gammas that are related to pair production. Possible interaction processes in the detector are shown in figure 7. When a positron annihilates with an electron, two gammas are produced. In processes 4 and 5 we see the escape gammas. Depending on whether both, one or neither of them is later photo-absorbed in the detector, pair production contributes to the full-energy gamma or single-escape or double-escape gamma of 511 kev or 122 kev below the full-energy gamma. 3.2 The Experiment The principles for detecting radiation are quite similar in all nuclear radiation experiments. Radiation enters a detector, interacts with the atoms of the detector material and releases a large number of low-energy electrons from their atomic orbits. These electrons are collected and formed into a voltage or current pulse. What is actually registered by the detector are the pulses, not the radiation that caused them. 16 See section

24 Figure 7: Interaction processes in the detector [KK88]. On the top of the figure in the middle a photon (not a proton) scatters from an atomic electron and leaves the detector crystal. Process 2 shows photoelectric absorptions. In processes 4 and 5 we identify the annihilation of a photon into an electron-positron pair and the escaping gamma photons. The measurements we performed are listed below. They are described accurately later. (1) Energy calibration of the MCA (section 5.1). (2) Measurement of the activity of a known radiation sample (section 5.2). (3) Measurement of the 137 Cs activity of the fish sample (section 5.4). (4) Measurement of background 222 Rn activity (section 5.5). (5) Measurement of the 222 Rn activity of the water sample (section 5.6). Germanium Semiconductor Detector The principles of a semiconductor detector are discussed next. Germanium is a valence 4 element and it forms solid crystals in which the atoms form four covalent bonds with the neighboring atoms. Since germanium 2

25 is a semiconductor material, thermally excited electrons from the valence band may move to the conduction band and leave behind holes which migrate through the crystal [CK96]. To control this current flow, other elements called dopants with valence 3 or 5 are added to the germanium crystal. Valence 5 atoms form four covalent bonds with neighboring atoms while the fifth electron is able to move through the lattice relatively easily. The moving electrons form a set of discrete donor states right below the conduction band. This type of semiconductor is called the n-type semiconductor because of the excess of negative charge [KK88]. If valence 3 atoms are added to the germanium crystal, they form three covalent bonds and leave a hole that can migrate through the crystal. The holes form a set of discrete acceptor states just above the valence band. Because the charge carriers are holes with no negative charge, this semiconductor type is called the p-type semiconductor. When the n- and p-type materials are brought together, a junction diode is formed. The electrons from the n-type material can diffuse across the junction into the p-type material and combine with the holes. The electrons leave behind ionized fixed donor sites. Similarly, the holes that diffuse from the p-side to the n-side leave behind fixed acceptor sites. An electric field from the n-side to the p-side is formed to stop any further migration (see figure 8) [KK88]. Figure 8: Junction diode and the depletion region [KK88]. The additional potential difference ev ext is caused by the applied reverse bias voltage. This potential difference makes the depletion region larger and thus increases the active area of the detector. In the vicinity of the junction there are only few electrons left. This region is called the depletion region. In the experiment, a relatively large reverse bias 21

26 voltage of 2 kv in the same direction as the initial electric field was used to increase the electric field and to make the depletion region larger. When radiation enters the depletion region and creates electron-hole pairs, the total number of electrons flowing in one direction in the detector form an electronic charge pulse. The amplitude of the pulse is proportional to the energy of the radiation. The detector we used was a p-type HIgh Purity Germanium (HPGe)) detector [ORTEC]. The poptop detector capsule was attached to a cryostat to cool it down with liquid nitrogen (77 K temperature). A photograph of the detector capsule is shown in figure 9. Figure 9: The germanium detector capsule. Carrying out the Experiment The detailed descriptions of the measured samples are given in section 5 when each experiment is considered independently. The equipment used in the experiment is listed in section 7.1. A schematic view of the equipment is shown in figure 1. Figure 1: A schematic view of the experimental setup [KK88]. The sample and the detector were placed in a hollow circular low-background lead shield with an inner radius of 18 cm and an outer radius of 28 cm (figures 22

27 11 and 12). The inner surface of the cylinder was covered with layers of tin (Sn) and copper (Cu) to decrease the amount of characteristic X-rays of lead entering the detector. Figure 11: The experimental setup. The low-background lead shielding was placed on the top of the steel rack. The cryostat shown in the lower right corner was placed under the steel rack with the poptop detector capsule attached. The cryostat was cooled with liquid nitrogen. Figure 12: A schematic view of the lead chamber. The Sn and Cu linings were used to decrease the amount of characteristic X-rays of lead from entering the detector. The sample was placed on top of the detector cylinder. In the energy calibration measurement, the samples were put on a cardboard box about 1 cm high 23

28 to decrease the amount of random summation 17. In all other measurements, the samples were put inside a,58 l polypropylene Marinelli beaker and measured in the corresponding geometry. Marinelli geometry was used to maximize the counting efficiency 18. When performing a measurement in Marinelli geometry, the sample has to be homogenous. The distribution of the radioactive material must be isotropic. We assumed the samples to be homogenous but there is no real way of knowing whether this was really the case. A beaker similar to the one used in the measurements is shown in figure 13. Figure 13: A Marinelli beaker [DRCT]. The electronic signal from the detector was driven into the preamplifier, which converted the charge pulse into a voltage pulse and drived it into the amplifier. The amplifier increased the strength of the signal by a linear factor. After the amplifier the signal was directed to a scaler which was used to control the shaping time of the signal. By increasing the shaping time, the energy resolution of the detector can be improved. However, if the shaping time is increased too much, the probability for coincidence summation 19 becomes too large and makes it difficult to determine the amount of actual processes in the detector. From the scaler the signal continued its way to a computer-based multichannel analyzer (MCA) which was integrated into a data acquisition card. In the MCA, the input pulses were divided into different digital values (channels) according to their energies; the pulses with approximately the same energy were given the same digital number. In general, the accuracy is determined by the amount of numbers available: in our analysis, the MCA had 496 channels cor- 17 Discussed in more detail later. 18 It can be shown that, in Marinelli geometry, the counting efficiency is the highest possible. 19 A phenomenon where two gammas are registered at the same instant. We will discuss this later. 24

29 responding to different energy values. Graphically, the MCA gives the number of pulses on the vertical axis and the channel numbers on the horizontal axis. The graphical representation was seen on computer screen and it was used to analyze the gamma transitions in nuclei. Next we discuss how a gamma spectrum is analyzed in practice. 3.3 The Gamma Spectrum: Practical Analysis There are over 65 known gamma transitions in the energy range of 3 ev 27 kev [STUK]. In general, a gamma spectrum of a nucleus may be very complex: a computer is needed for the analysis in most cases. In this research, however, we simply compared the background measurement and the measurement of the sample in question. The apparent gamma transitions were identified and the identities of the radioactive nuclei were confirmed. A simplified gamma spectrum is shown in figure 14. On the horizontal axis are the channels that correspond to different photon energies: the correspondence is determined by energy calibration (section 5.1). On the vertical axis is the photon intensity, i.e. the number of photon counts (charge pulses) the detector has registered in the measurement time. It is to be noted that this intensity is NOT the same quantity as the absolute gamma intensity discussed in section 2.2. Figure 14: A simplified gamma spectrum [KK88]. Gammaspectroscopic analysis uses photon peaks caused by full absorption. 25

30 This happens when all Compton scattered electrons and the photoelectron from photoelectric effect are absorbed into the depletion region. If full absorption does not occur (i.e. the photon leaves the crystal), the photon that caused the interaction is not registered. Photons that interact via photoelectric effect are found in photopeaks that correspond to discrete energies. This is because the photon gives all of its energy to an electron in the crystal. The peaks are usually spread over a couple of channels like a normal distribution due to the statistical variation of charges produced in the detector. The area of the the peak is proportional to the number of photons registered in the peak. In this experiment we approximated the area as the sum of the counts registered in the channels where the photopeak appears. It is essential that the areas of the peaks are determined in the same way in both the energy calibration and the actual experiment [STUK]. This gives consistent results. Photons that scatter from electrons can have different energies depending on the scattering angle. Therefore, a Compton continuum appears, as seen in figure 14. The Compton continuum is not flat since the scattering probability depends on the scattering angle 2. Compton scattering is not a very good interaction mode from the perspective of gamma spectroscopy because the scattered photon may leave the crystal. In figure 14 we can identify the escape gammas mentioned in section 3.1. In practice, the form of the observed spectra rarely coincide with the simplified spectrum of figure 14. The spectrum we measured for the background radiation is shown in figure 15. There are a few processes that take place in nuclei which (alongside with the detector efficiency) give gamma spectra their characteristic form. Next we discuss bremsstrahlung radiation, X-rays and Auger electrons. Bremsstrahlung Radiation Monoenergetic electrons that move in a medium larger than the size of the electron radiate a fraction of their total energy [KJ95]. The average amount of radiation from an electron depends on the kinetic energy of the electron and the atomic number Z. We are talking about the average radiation which means that an individual electron has some probability to radiate part of its kinetic energy as bremsstrahlung radiation. The probability for the production of low-energy quanta is much higher than for high-energy quanta. 2 The probability for Compton scattering is given by the Klein-Nishina formula, see [KK88, 2]. 26

31 Intensity (# of counts / meas. time) Counts Channel # Figure 15: Gamma spectrum of the background radiation. We notice the X-ray peak of lead in the low-energy end of the spectrum. The form of the spectrum is mainly caused by the exponentially decrasing detector efficiency and bremsstrahlung radiation. In many cases bremsstrahlung is a result from the electrons of a beta decay. The effect of bremsstrahlung on the gammaspectroscopic analysis is that it raises the intensity of the low end of the energy spectrum and thus makes lowenergy gammas practically invisible. In figure 16 we see a typical bremsstrahlung spectrum produced by a beta decay that resembles the form of the gamma spectra we measured. Figure 16: A typical beta decay energy spectrum and the corresponding bremsstrahlung spectrum [KJ95]. 27

32 X-rays and Auger electrons X-rays are produced when an inner electron shell (usually the K shell) vacancy is filled with an electron from a higher shell and the energy released in the process is carried away by an X-ray photon. The energy of the photon is equal to the difference in the binding energies of the shells minus the small recoil energy of the nucleus. Because of the nature of the process, X-rays are a characteristic property of an element and form a line spectrum. The energies of X-ray photons increase with mass number and for lead the energies are between 7 and 9 kev. In figure 15 we identify the most intense X-ray peak of lead. This peak appeared because of the lead shielding around the measured sample. In general, X-rays can cloud the low-end of the energy spectrum and make the low-energy gammas difficult to detect. In our case, however, the energies of the observed gammas were all well above the lead X-rays energies. In an Auger electron creation process, the original vacancy on an inner electron shell is filled with an electron from an outer shell and a third electron called the Auger electron is released from a still outer shell. The energy released is equal to the difference of the binding energies of the shells minus the recoil and it is carried away by the Auger electron which is released into the surrounding space. Summation If a radiation source sends two gammas at the same instant, it is possible that the detector crystal absorbs both. Subsequently there appears a charge in the crystal that is equal to the sum of the charges caused by the individual photons. There are two types of summation. (1) Random summation leads to a decrease of counts from the photopeak. The effect becomes negligible if the source is placed far enough from the detector; thus, random summation depends on the activity of the source. One reason for placing the samples used in the energy calibration on the top of a cardboard box was to decrease the count rate. Had we not done it, the detector would have lost many counts and the energy calibration would have failed. (2) Coincidence summation means that two or more gammas sent by the same nucleus are observed within the time resolution of the detector. The probability of coincidence summation depends on the distance between the source and the detector but not the activity. In accurate measurements the effect of coincidence summation can be determined by using a specific 28

33 experimental setup 21. In all our analysis we assumed that there is no summation. The Efficiency of the Detector The efficiency calibration of the detector is normally done to determine the ratio between the number photons that were created in the sample and the number of counts that were registered by the detector. The efficiency depends on the photon energy, geometry, detector type and the photon absorption in air between the sample and the detector [STUK]. There are various ways to describe the efficiency of which the following two are the most important. The latter of them was used in our analysis. (1) Total efficiency is the probability that a photon emitted by the source becomes registered [STUK]. Total efficiency is calculated from the equation P total (E) = N t N, (46) wheren t is the number of counts observed at photon energye andn is the total number of photons with energy E created in the measurement time. This equation is derived under the assumption that there is no summation. (2) Peak efficiency is the probability that a photon with some energy E strikes the detector crystal, is absorbed into the crystal and registered into the corresponding photopeak. Clearly the peak efficiency for a detector is always smaller than the total efficiency of the same detector. The peak efficiency is calculated as P peak (E) = N p N, (47) where N p is the number of observed counts in the photopeak of energy E. This equation also assumes no summation. Energy Resolution of the Detector Detector resolution is a crucial factor in gamma spectroscopy. The smaller the resolution, the better the accuracy for determining the gamma peaks. Figure 17 gives a qualitative picture of the energy resolutions of a germanium detector and a scintillator 22 detector. The resolution is determined as the full width at the half of the maximum (FWHM) of the observed gamma peak (see figure 17). The resolution depends 21 See [KK88, ] and [KJ95, ]. 22 For a discussion on scintillator detectors, see e.g. [STUK, ]. 29

34 Figure 17: A qualitative view of the energy resolution of a germanium and a scintillator detector [KK88]. The great resolution of the germanium detector comes, unfortunately, with financial cost. on the peak energy and the standard is to give the resolution with respect to the 1332 kev gamma ray of 6 Co. Electronic noise in the preamp and the amplifier increase the resolution. Can All Nuclei Be Identified with Gamma Spectroscopy? Despite its many good features, gamma spectroscopy cannot be used to identify all nuclei. Let s take as an example the decay scheme of 9 Sr. It is described in figure 18. As we can see, the 9 Sr nuclei first β decay to the ground state of 9 Y since there are no apparent gamma decays from the excited states of 9 Y. Then the 9 Y nuclei β decay to 9 Zr, which is stable. In the figure it is stated that some of the 9 Y nuclei decay to the excited state of 9 Zr which has an excitation energy of 2186 kev. However, the intensity of this gamma transition is less than 1 % [CoN98] and most likely it would not be possible to identify the peak from background. Thus, there are no eligible gamma transitions to be analyzed and gamma spectroscopy cannot be used to determine whether the sample contained any strontium. 3

35 Figure 18: The decay scheme of 9 Sr [CoN98]. 31