FLAP P9.2 Radioactive decay COPYRIGHT 1998 THE OPEN UNIVERSITY S570 V1.1

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Charleen Quinn
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1 Atoms of a given substance with differing atomic masses are said to be isotopes of that substance. The various isotopes of an element all contain the same number of protons but different numbers of neutrons.

2 Marie Curie ( ), who had two daughters, was responsible for coining the term daughter for the product of a radioactive decay.

3 The total energy of a particle of rest mass m m 0 and speed v is E = 0 c 2 1 (v 2 c 2 ). The rest energy of such a particle is E 0 = m 0 c 2 and the relativistic kinetic energy is E kin = E E 0 = m 0 c (v 2 c 2 ) Throughout this module all references to mass will be to rest mass, and such masses will generally be denoted by m rather than m 0.

4 The familiar quantity m 0 v 2 /2 provides an approximate value for E kin when v is small compared with the speed of light, c, but not in general.

5 The atomic nucleus and the nuclear binding energy graph is discussed more fully elsewhere in FLAP, as is an explanation of the stability line on Figure 1 in terms of the strong interaction between nucleons.

6 X-rays are also a type of ionizing radiation, but they are not a direct result of radioactivity. They differ from γ-rays only in having a lower frequency.

7 The charge on the proton e = C The atomic mass unit u = kg = MeV/c 2

8 The binding energy of a nucleus is the difference between the rest energy of the separate nuclear constituents (N neutrons and Z protons) and the rest energy of the nucleus. The binding energy of a nucleus multiplied by c 2 is known as the mass defect of the nucleus.

9 In this example we use K for kinetic energy, rather than E kin, in order to avoid having too many subscripts.

10 The Newtonian formula for kinetic energy is an adequate approximation in situations where speeds are much less than that of light (and hence the rest energy is much greater than the kinetic energy). This condition is satisfied by α-particles, since their rest energy is about 411GeV.

11 We have used a bold typeface p to represent momentum to remind you that it is a vector quantity with magnitude and direction.

12 According to Einstein s theory of relativity the momentum of a particle of mass m moving with speed v is of magnitude p = mv 1 (v 2 c 2 ) However, mv provides an approximate value for p in this case in the same way that 1 2 mv2 provided an approximate value for K.

13 The symbol is read as is approximately equal to.

14 The transfer of kinetic energy in an elastic collision is greatest if the colliding particles have comparable mass. Such a collision can be analysed in terms of energy conservation and linear momentum conservation. These ideas are discussed elsewhere in FLAP, see the Glossary for details.

15 11µm (micrometre) = m

16 The weak interaction is not the same as the weak (van der Waals) intermolecular force, which is electromagnetic in origin. It should also be noted that the term interaction is used to describe not only the forces that subatomic particles exert on each other but also the way they decay.

17 Electron antineutrino is often abbreviated to antineutrino. ν is the Greek letter nu.

18 There are two other pairs of neutrinos (particle and antiparticle) which complete the family of six. Just as neutrinos involved in β-decay are associated with the electron, each of the other pairs is associated with another particle (the muon and the tauon) which in many respects behaves like the electron, but which is unstable and much heavier than the electron.

19 The term half-life here is equal to the time for half a sample of free neutrons to decay into protons. The meaning of half-life will be discussed more fully in Section 3 of this module.

20 It has been verified experimentally that the proton, if it were unstable, would have a half-life of at least yr, but it has been postulated to be unstable by some theorists.

21 Spin is a property of particles which you may have met first in relation to electron spin. Many other particles have spin and this is associated with their magnetic interactions, but a general discussion of spin is beyond the scope of FLAP.

22 Remember, the rest masses of the neutrinos are negligible or zero.

23 501pm = m = 0.051nm.

24 Nuclei, like atoms, can exist in a variety of different discrete states that may be distinguished from one another by their characteristic energy. For any particular nucleus or atom the state of lowest energy is called the ground state. States of higher energy are called excited states.

25 A photon can also be called a quantum (plural quanta) of electromagnetic radiation. Both terms are used widely in FLAP.

26 1 1 ev = J

27 Exponential functions are covered in the maths strand of FLAP. The origin of this kind of exponential behaviour is discussed in the block on differential equations.

28 It is worth pointing out that the ratio of the absorption coefficient for 21MeV γ-rays in lead and in water is approximately equal to the ratio of the densities of the two materials, as implied above.

29 Although the first step in the decay of U involves only α-emission, the subsequent steps involve β - and γ-emission and hence uranium cannot be used as a pure α-source.

30 The curie was named after Marie Curie1 another early investigator of radioactivity, and discoverer of radium.

31 Fluctuations in random events, such as radioactive decay or random errors in an experiment are described elsewhere in FLAP.

32 If there is a mixture of nuclides, the activity will be the sum of the contributions from the various nuclides.

33 Note that the decay constant λ has no connection with the wavelength λ of electromagnetic radiation.

34 For further discussion of units and dimensions, refer to the Glossary.

35 The calculus notation is introduced in the maths strand of FLAP.

36 The symbols t 1/2 and T 1/2 are also commonly used for half-life.

37 Differential equations, exponential decay and exponential functions are discussed more fully in the maths strand of FLAP. See the Glossary for details.

38 11µg (microgram) = g = kg

39 11u = kg

40 41.51MBq = 1.11mCi

41 This is only an approximation, because N 0 is only approximate.

42 Cosmic rays are mainly very high energy protons originating from the Sun or from outside our solar system. The particles enter the Earth s upper atmosphere and, because of their extreme energies, can induce nuclear processes1 1such as are needed in the production of 14 6 C. The full topic lies beyond the scope of FLAP.

43 Note that we are using the same symbol, R, for activity and for specific activity. The meaning should be clear from the context.

44 Because of the difference in energies, the atomic electron will be ejected from the atom which, as a result, will become a positive ion.3

45 It does so because originally there are more neutrons than protons in the heavy nuclei that are potential α-decay candidates. The loss of two protons is proportionally a larger change than the loss of two neutrons and therefore emitting an α-particle increases the ratio of neutron number to proton number and thus increases the stability.3

46 If there were just two particles as products of the decay then the argument would parallel that for α-decay. Since the masses of the β -particle and the daughter nucleus are fixed, the available energy (the Q-value) would split in a definite proportion between the two and only one definite kinetic energy for the β -particle would be possible. If there is an additional third particle involved then the energy can be split between the three in an infinite number of ways, limited only by the total energy available1 1and all kinetic energies for the β -particle, up to some maximum (total energy available), become possible.

47 The maximum β -particle kinetic energy corresponds to those decays in which the β -particle takes almost all the available energy with the third particle having a negligible amount1 1but the converse can also happen, with the β -particle having little or no kinetic energy. All values of β -particle kinetic energy between these two extremes are possible.3

48 Changes in the neutron/proton ratio within the nucleus result in changes in the nuclear binding energy. In β + -decay the net positive charge on the nucleus decreases and so the electrostatic repulsion is reduced and this tends to increase the binding energy. So there can still be a net gain in nuclear binding in β + -decay despite the required increase in the rest mass of the nuclear constituents as a proton becomes a neutron.3

49 The activity R(t) is the number of decays per second and the number of nuclei N(t) is dimensionless, so λ will have the unit s -1. 3

50 R(t) is directly proportional to N(t), so the graph would have the same shape (though the scales on the vertical axes would differ by a factor of λ).3

51 N(t) decreases to one-half its initial value. If R(t) halves in a given time interval, then Equation 11 R(t) = λn(t) (Eqn 11) shows that N(t) will also halve in this same time interval.3

52 Since 57301years is the half-life of 6 14 C, you would expect the average specific activity to be 7.51Bq1g 1.3

Sources of Radiation

Radioactivity Sources of Radiation Natural Sources Cosmic Radiation The Earth is constantly bombarded by radiation from outside our solar system. interacts in the atmosphere to create secondary radiation