Quantum Physical Phenomena in Life (and Medical) Sciences

Quantum Physical Phenomena in Life (and Medical) Sciences Nuclear reactions and radiations Péter Maróti Professor of Biophysics, University of Szeged, Hungary Suggested texts: S. Damjanovich, J. Fidy and J. Szőlősi: Medical Biophysics, Semmelweis, Budapest 2006 P. Maróti, L. Berkes and F. Tölgyesi: Biophysics Problems, A Textbook with Answers, Akadémiai Kiadó, Bp. 1998.

The first experimental evidence for the existence of the nucleus of the atom: the Rutherford’s experiment N: the number of atoms/m3 that scatter the alpha particles d: the thickness of very thin gold layer Z1: the number of elementary electric charges of the particle to be scattered Z2: the number of elementary electric charges of the scattering centers Ekin: kinetic energy of the incident particles The scattering experiment prooves that the majority of the mass of the atom is concentrated into an extremely small volume which has not only very high mass density but positive electric charges, as well. This entity is called nucleus of the atom. Back scattering: rare but observable phenomenon

Comparison of orders of magnitudes of the linear sizes of electron, nucleus and atom The radius of the electron is: relectron≈ 1.3·10-15 m = 1.3 fm. The radius of the nucleus can be determined from the Rutherford’s scattering experiment: where M is the atomic mass number. For example, the atomic mass number of the most common isotop of the uran is M = 238 and thus rnucleus ≈ 8·10-15 m = 8 fm. The radius of the whole atom (nucleus + electron shells) is in the order of magnitude of 0.1 nm (= 1 Å = 1·10-10 m): ratom ≈ 10-10 m = 105 fm

Cartoon-representation of the linear sizes constituents of the nucleus: nukleons (neutrons and protons) quarks atom nucleus

Of the electron, proton, and neutron, only the electron is today considered as „fundamental particle”. The proton and neutron, like other hadrons, are composed of various combinations of QUARKS. LEPTONS: e-, μ-, τ-, νe, νμ, ντ QUARKS: u, d, c, s, t, b Their ANTIPARTICLES Fundamental particles Constituents of the matter Mediators of the interactions

Basic interactions of the nature Weak interaction. The β¯ -decay of the radioactivity is the best known appearence of the weak interaction. It is orders of magnitude weaker than the Strong interaction that is effective among the nucleons (protons and neutrons) of the nucleus. This interaction holds the nucleus together. Electromagnetic interaction, for examplethe electrostatic (Coulomb) interaction among charged parciles. The electromagnetic and weak interactions are called electroweak interaction. Gravitational interaction, acts among masses of the substances and is always attractive (and never repulsing).

Nuclear force that keeps the nucleus together • The nuclear force isan attractive force, that acts among protons and neutrons. It is stronger than the repulsive Coulombic force between two protons; at a distance of r > 0.7 fm between two protons, the nuclear force is 100 times greater than the Coulombic interaction. • The nuclear force is independent on the electric charge of the nucleon; it is the same for all nucleon pairs (proton-proton, neutron-neutron or proton-neutron). • The range of the nuclear force is short. The nuclear force is negligible above a certain separation (about r > 2 fm) of the nucleons. Therefore, the attractive nuclear force does not spread out to the whole nucleus, it acts practically for nucleon pairs (neighbors) only. • If the distance is too short (r < 0.7 fm), the nuclear force becomes repulsive, i.e. it keeps the nucleons at definite distances. This is why the density of the nucleus does not depend on the number of the nucleons. Exchange interaction: The strong coupling (interaction) among nucleons in the nucleus is transmitted by pions (π+-mesons). These are exchange particles and hold the nucleus together. The nucleons (protons and neutrons) consist of even more elementary particles, the quarks. The nucleons are hold together by exchange particles (quarks and glüons). interaction

Models of the Nucleus The liquid drop model. The nucleus is approximated by a drop of incompressible liquid with a surface tension and uniform density that is kept by short range forces together. After vonWeizsäcker,the binding energy of the drop of fluid EB consists of 5 different terms: Energy of condensation, which is liberated if the nucleons of the nucleus are all unified. As A is proportional to the volume of the nucleus, this term is sometimes called as volume energy. The surface energy is proportional to the surface of the nucleus. The nucleons at the surface are weaker bound than those in the bulk. The Coulomb-energy is a repulsive energy that comes from the electrosatic interactions of the positively charged protons in the nucleus. Energy of asymmetry. If the number of neutrons overweights that of the protons, than the binding energy becomes less than that of the symmetric nucleus. The symmetric nuclea (Z = A/2) are the most stable. Pair energy. The binding energy is large for the even nuclea (g,g) and small for odd (u,u) nuclea. (g,g): both Z and N are even (u,u): both Z and N are odd in the nucleus.

Models of the nucleus The liquid drop model (cont.) Pair energy:δ≈± aP·A-1/2 where the sign „+” refers to (g,g), and the sign „-” refers (u,u) nuclea; δ = 0 for both (u,g) and (g,u) nuclea. The proportionality factors of the terms aV, aS, aC, aA and aP for definite masses of nuclea can be obtained from experimental results. A frequently used set of parameters: aV = 15,56 MeV; aS = 17,23 MeV; aC = 0,72 MeV; aA = 23,29 MeV and aP = 12 MeV Shell model. The energy term system of nucleons in the nucleus is highly similar to the of electrons in the atoms. The electrons are arranged in shell (sub shell) system in the atom, the nucleons are arranged also according to similar hierarchy. The nucleons are moving independently from each other. Each nucleon is in the potential well of sphaerical symmetry established by the other A -1nucleons. The nucleons are set to different energy levels determined by the orbital momentum and the internal (self) momentum (spin). The momenta are coupled. The Pauli exclusion principle controls the number of nucleons on the energy levels. In ground state, the nucleons fill up the lowest energy nivous of the nucleus.

Models of the nucleus Shell model– reason for extremely high stability of some nuclea. If the nucleon shells are closed, the nucleus is very stable. Indeed, there are nuclea in the nature whose numbers of protons and neutrons are unique and their stability is much higher than that of their neighbors. These are the nuclei of magic numbers whose proton (Z) or neutron (N) numbers are out of the following series: 2, 8, 20, 28, 50, 82 and N = 126. The nuclei of double magic numbers are those nuclei whose both neutron number and proton number are magic numbers. For example: The nuclei of double magic numbers show extreme stability. The shell model describes well - the (ground and excited states of) nuclei of light elements and - the ground state of nuclei of all elements.

The nuclear binding energy is defined as the amount of energy that is required to disassemble a nucleus completely into its constituent protons and neutrons. Mass-energy equivalence (Einstein): Here c is the speed of light in vacuum, m is the mass deficit, which is used to liberate energy E (and vice versa). Energy can be ordered to every mass and oppositely: mass can be ordered to every energy. For example: the m = 9,11·10-31 kg mass of electron corresponds to E = 0.511 MeV energy. The mass of nucleus of A = Z + N mass number is always somewhat smaller than the sum of the masses of the N neutrons and Z protons. The mass deficit corresponds to the binding energy of the nucleus. So much energy is required to break the nucleus to individual nucleons, i.e. to remove them beyond the range of nuclear forces (> 2 fm).

Specific nuclear energy (binding energy per number of nucleons): total binding energy (Ebinding) devided by the atomic number (A). Nuclear energy The function B/A vs. Ahas a (flat) maximum around A ~ 60. The average value of B/A is between 7.5 and 8.5 MeV with exception of the lightest elements. Fission Specific binding energy, Ebinding / A Nuclear energy can be obtained by 1) fusion of the nuclei of light elements and 2) fission of the nuclei of the heaviest elements. Fusion

Specific nuclear energy (energy/# of nucleons) 56Fe 4He 8 238U Specific binding energy Especific binding energy (MeV) 4 Δm = 1 mass deficit (1.67·10-24 g) corresponds to Ebinding = 931.5 MeV binding energy. Z: number of protons N: number of neutrons M: mass of the nucleus Δm: mass deficit c: speed of light 3T The mass deficit is less than 1% of the mass of the nucleus. 2D 0 100 200 A, mass number Fission Fusion

Energy ladder of binding 10 Specific binding energy of nuclei in 4He-ban (7.07 MeV) 1 MeV Binding energy of electron in the inner shell (K) of the heaviest element Uran. 100 10 Coulomb-dam of the lightest elements in fusion reactor 1 keV 100 10 Binding energy of electrons in outermost shells of atoms Binding energy of atoms in molecules 1 eV Energy of photons, hν 100 Thermal energy, ½kBT, (25 meV at room temperature) 10 1 meV 1 eV = 1.6·10-19 J

Radioactivity: separation of radioactive radiations by static magnetic field (induction) neutral Original figure from Marie Curie’s dissertation (1904). positively charged particles Direction of external magnetic induction RS: radioaktive source (substance) negatively charged particles

Decay law of the radioactivity All radioactive decays follow the same statistical law of radioactivity: in a sample of particular radioisotope the number of atoms which disintegrate (-ΔN) in unit time (Δt) is directly proportional to the number of radioactive atoms (N) present in the sample at that time: Here λ is a constant, known as the decay rate, characteristic of the particular radioisotope concerned. The decay rate is the relative number of disintegration within unit time: If N0 is the number of radioactive atoms present at time t = 0and N is the number remaining after a time t, integrating gives: or where e (= 2.71828...) is the Euler’s number, the base of the natural logarithm. In many applications, the base is 2. In these cases, λ is replaced by ln2/TH, whereTH is the half-life (see later). Thus, a radioisotope decays exponentially and the rate of the decay (the initial slope of the curve) gives the decay constant λ.

Activity and half-life of the radioactivity If dN is the number of spontaneous nuclear transformations occuring in the time interval dt, then the activity a is defined by where a0 is the initial activity.The unit of activity is 1/s, otherwise known as the becquerel (Bq) The half-life THof a radioisotope is defined as the time taken for the activity to fall to half its initial value (or alternatively as the time taken for half the radioactive atoms to disintegrate): a (t = TH) = a0/2 or N (t = TH) = N0/2 The half-lifes of radionuclids in nature are between ~10-7 s and ~ 1017 years.

Radiometric dating methods 1)Potassium-argon dating method. abundance: 0,0117% 1/2 time TH = 1,23·109 years 2)Uranium-lead dating method. One of its great advantages is that any sample provides two clocks, one based on uranium-235's decay to lead-207 with a half-life of about 700 million years, and one based on uranium-238's decay to lead-206 with a half-life of about 4.5 billion years, providing a built-in crosscheck that allows accurate determination of the age of the sample even if some of the lead has been lost. 3)Radiocarbon dating method When an organism dies, it ceases to take in new carbon-14, and the existing isotope decays with a characteristic half-life TH = 5730 years. The carbon-14 dating limit lies around 58,000 to 62,000 years.

Example: Radiocarbon dating method Carbon dioxide in the atmosphere contains a small but readily detectable amount of the radioactive isotope 14C6. This isotope is produced by high energy neutrons (in cosmic rays) that transform nuclei of nitrogen atoms by the process 14N7 + 1n0→ 14C6 + 1H1 The 14C6 nucleus is unstable and decays by the first-order process 14C6 → 14N7 + 0β-1 with a half-life of TH = 5730 years. A sample of wood from the core of an ancient bristlecone pine in the White Mountains of California shows a 14C content that is 54.9% as great as that of atmospheric CO2. What is the approximate age of the tree? Solution. Since the disintegration of 14C after the death of the tree is a first order process (decay law of the radioactivity), we need know only the ratio of the present radioactivity to the original value: [14C]/[14C]0 = 0.549. where t is the age of the wood and ln 2 = 0.693. Numerically: t = 4990 years. We conclude that the carbon in the tree was taken from the atmosphere approximately 4990 years ago. It was assumed that the level of 14C in the atmosphere has not changed over the life of the tree. This is not strictly true, but corrections can be made, but they are small.

Physical, biological and effective half-life Once a radioisotope has been administered to a patient, it is subject to biological removal from the body by processes such as respiration, urination and defaecation. This means that its effective half-life TH,eff is less than the physical half-life TH,physdue to pure radioactive decay. It is possible to define a biological half-life TH,biol as the time taken for biological processes to remove half the available material, assuming that no new material is arriving. If λbiol and λphys are the fraction of the radioactive isotope removed per second by biological process and radioactive decay, respectively, then the total fraction removed per second is: The kinetics of the activity of the radioisotope administered to the patient: As the decay constant relates to the half-life by λ = 0.693/TH, the effective half-life is The biological half-life of a given substance, reflecting its metabolic turnover, tends to vary from one individual to another and from one organ to another. It also depends on diet and disease. Accurate estimates of the biological (and hence effective) half-life are thus difficult to make and can cause serious problems regarding dosage assessments.

Frequently used radiopharmacons in human bodies

Examples: simultaneous metabolism and radioactive disintegration • What is the effective half-life of 131I in thyroid? The biological (metabolic) half-life is TH,biol = 15 days, and the radioactive half-life is TH,phys = 8.1 days. • Solution: After replacement of the numerical values: TH,eff = 5.3 days 2. The observed (effective) half-life of 18F isotope in the bone is TH,eff = 107 min, the physical half-life is TH,phys = 1.8 hours. Estimate the biological (metabolic) half-life! Solution: After replacement of the numerical values: TH,biol = 8.0 days

The stability of the nuclei (radioactive or not) depends on its number of protons (Z) and neutrons (N) Range of stable emitters β+ and α2+ emitters or K-capturers Generally, N = Z for the light elements. Above the element of 40Ca , the nucleus can be stable for N > Z only. Towards the heavier elements, the nuclei are more and more rich in neutrons (in-dependently whether they are stable or radioactive). The nuclei rich in neutrons (relative to the stable nuclei) disintegrate by emission of electrons and the nuclei poorer in neutrons (relative to the stable nuclei) are positron or alpha emitters or capture electron from the K electron shell.

Emissions of nuclear reactions α-emission. The alpha particle is identical to a very stable helium nucleus, or doubly-ionised helium atom (He2+) consisting of two protons and two neutrons. Decay by α-particle emission occurs mainly amongst nuclei of the heavier elements of Z > 60 and A > 144 and particularly Z > ZPb = 82. The decay of the mother nuclid (XM) results is a reduction of 2 in atomic number (Z) and 4 in mass number (A) of the daughter nuclid (XT): Penetration distance of α-particles in matter (R), the Geiger-Nuttal’s law. R: penetration depth, λ: decays constant,A and B constants. Although the α-particles have large speed (≈107 m/s), the penetration depth in air is small (≈ 6 cm). They are relatively harmless outside the body. However, if α-particle emitters are ingested, they can be extremely damaging, because of the dense ionisation they produce internally. Examples for α-emitters The α-emissions occur at certain discrete energies between 1.5 MeV and 10 MeV, determined by the emitter and the particular decay route. The decay of radium to radon is accompanied by emission of alpha particle having a kinetic energy of about 4.8 MeV.

α-decay, tunneling effect For nuclei of other radioisotopes, the energy of the emitted α-particles could be different but all particle have the same energy. All of the emitted alpha particles have exactly the same energy:Eα = 4.8 MeV. All α-particles should have the same energy in the nucleus before running against the potential wall. That reflects the shell structure of the nucleus, i.e. the nucleons are arranged according to srict energy levels in the nucleus. r (fm) The potential energyEP of the alpha particle in the nucleus of 238U as the function of distance r from the origin. The resulting (remaining) nucleus, 234Th has a radius of 9.2 fm. The α-particle has two possibilities to leave the nucleus: the potential wall can be either A) climbed (temperature-activated process) or B) tunneled (independent on temperature). The alpha particle selects pathway B), i.e. the tunneling.

Emissions of nuclear reactions β¯ -decay. The negative beta particle is simply a fast electron released from a nucleus during decay. It is termed a β—-particle mainly to distinguish it from orbital electrons. Radioisotopes having an excess of neutrons usually decay by this mode, during which a neutron (n) is converted into a proton (p): The neutrino and its antiparticle, the antineutrino, are particles of zero charge and approximately zero mass which carry away a certain amount of energy and momentum from such disintegration process. During β¯ -decay of the mother nuclid (XM), the daughter nuclid (XT) maintains A but Z increases by 1: The β-particles have a continuous energy spectrum extending from zero to a well-defined maximum energy, which depends on the particular radioisotope. This is because the energy available is shared between the emitted beta particle and its associated antineutrino. The β-energies can vary considerebly from several keV in the decay of tritium to a few MeV from potassium decay.

Emissions of nuclear reactions β+ -emission (positron-decay) Neutron-deficient (proton-rich) radioisotopes often decay by converting a proton into a neutron and emitting a positron having the same rest mass as an electron but carrying an equal and opposite charge of +e: In such a decay of the mother nuclid (XM), the atomic number A of the daughter nuclid (XT) remains unchanged and Z decreases by 1: The nuclear transition cannot occur spontaneously because additional energy is needed: the mass of the proton is smaller than the sum of the masses of the neutron and the positron. The difference (mass deficiency) is the mass (or energy-equivalent) of two electrons that should be offered by the other nucleons of the radiactive nucleus. Therefore, the energy of the initial (mother) nucleus should be at least 1.02 MeV larger than that of the product (daughter) nucleus.

Emissions of nuclear reactions K-capture. A proton in the nucleus can capture an extra-nuclear electron from one of the inner electron shells, usually the K-shell, and combine with it to form a neutron: The electron or K-capture (like β+-decay) leaves A unchanged but decreases Z by 1: Isotopes commonly decaying by electron capture include 51Cr24, 55Fe26 and 64Cu29. Since an electron generally moves in from an outer orbit to fill the K-vacancy, a subsequent emission of characteristic X-rays accompanies K-capture. γ-ray emission. It is an electromagnetic radiation from a nucleus during transitions from an excited nuclear state to a lower-energy nuclear state. Gamma ray emission often follows another decay process such as alpha or beta emission, which has left the daughter nucleus in an excited state. The mass and atomic numbers remain unchanged: The γ-quanta have discrete energies between 0.1 MeV and 20 MeV. Like the beta particles, they have no precise range in matter (see Beer’s law of absorption).

discrete spectrum Distribution of the number of particles (quanta) according to their energies in different radiations: spectra Energy (MeV) average energy continuous spectrum Energy (MeV) discrete spectrum Energy (MeV)

Soddy-Fajans displacement rule A: mass number = number of nucleons (neutrons and protons) in the nucleus; Z:atomic number = number of protons in the nucleus. AX*Z→ (A+ΔA)X(Z+ΔZ)

Families (series) of radioactive elements After disintegration of a (mother) nucleus, the produced (daughter) nucleus can be also radioactive which disintegrates to a new element and so on. The group of elements is called radioactive decay series that represents various stages of radioactive decay in which the heavier members of the group are transformed into successively lighter ones, the lightest being stable. There are 4 similar families in the nature among them the two uran radioactive families are the best known. • Among the α-, β- and γ-decays, only the α-emission changes the atomic mass A of the nucleus. Because the atomic mass changes by ΔA= 4 after the α-decay, theoretically four decay families can be distinguished depending on the remainder (0, 1, 2 or 3) of the division of the atomic mass by 4: • (4k) family: 232Th90 (TH = 1.8·1010 years) →...→...→ 208Pb82 • (4k + 1) family: 237Np90 (TH= 2.14·106 years) →...→...→ 209Bi83. As the half-life is very small (in geological terms), the members of the series disappeared (they are not available anymore) in the nature. They can be produced artificially only. • (4k + 2) family: 238U92 (TH = 4.51·109 years) →...→...→ 206Pb82 • (4k + 3) family: 235U92 (TH= 7.04·109 years) →...→...→ 207Pb82

A = 4k 232Th90 → 208Pb82 Thorium decay series Z Np 93 U 92 Pa 91 Th 90 Ac 89 Ra 88 Fr 87 Rn 86 β-decay At 85 Po 84 Bi 83 α-decay Pb 82 Tl 81 A 206 210 214 218 222 226 230 234 238

A = 4k + 1 237Np93 → 209Bi83 Neptunium decay series Z Np 93 U 92 Pa 91 Th 90 Ac 89 Ra 88 Fr 87 Rn 86 β-decay At 85 Po 84 Bi 83 α-decay Pb 82 Tl 81 A 206 210 214 218 222 226 230 234 238

A = 4·k +2 238U92 → 206Pb82 Uran-Radium decay series Z Np 93 U 92 Pa 91 Th 90 Ac 89 Ra 88 Fr 87 Rn 86 β-decay At 85 Po 84 Bi 83 α-decay Pb 82 Tl 81 A 206 210 214 218 222 226 230 234 238

A = 4k + 3 235U92 → 207Pb82 Uran-Actinium decay series Z Np 93 U 92 Pa 91 Th 90 Ac 89 Ra 88 Fr 87 Rn 86 β-decay At 85 Po 84 Bi 83 α-decay Pb 82 Tl 81 A 206 210 214 218 222 226 230 234 238

Equilibrium in the radioactive decay series:mother-daughter system We are dealing witht the most simple case: the radioactive mother nuclid converts to a radioactive daughter nuclid. Let’s denote the half-life (decay constant) of the mother nuclid by TM (= (ln2)/ λM) and those for the daughter nuclid by TT (= (ln2)/ λT). The scheme of chain reactions is where NM and NT mean the number of mother- and daughter nuclids at time t. The change (drop) of the number of mother nuclids within an infinitesimal time rangedt is The change of the number of daughter nuclids within the same time range: where the first term on the right-hand-side offers the number of nuclids originated from the disintegration of the mother nuclids within time range dt, and the second term includes all nuclids that disappear via disintegration of the daughter nuclids within the same time rang dt.

Mother-daughter system in equilibrium The kinetics (time variation) of the number of mother and daughter nuclids can be obtained by integration of the two equations. As initially only mother nuclids are available, the initial conditions are NM = NM,0 andNT = 0, at t = 0: Dominant, if λM < λT , and t →∞ The number of daughter nuclids runs parallel with the number of the mother nuclids after the transient period (if t→∞). The decay of the number of mother nuclids follows a simple exponential function. The kinetics of the number of the daughter nuclids, however, is biexponential: initially it increases very fast, but later decreases as it is limited by the available (and slowly disintegrating) mother nuclids. After tmax when it reaches the maximum, it will decay with the same rate as does the mother nuclid. Daughter nuclid Mother nuclid

Kinetics of the activities (special cases) Steady state (very slowly changing) equilibrium a = - dN/dt= λ·N „running (time-dependent) equilibrium” If the decay constant of the mother nuclid is not significantly smaller than that of the daughter nuclid (the decay of the mother nuclid becomes faster) then the activity of the daughter nuclid will be larger when the equilibrium is established. After a short transient time, the activities run parallel but a bit shifted. If the decay constant of the mother nuclid is significantly smaller than that of the daughter nuclid then the activities of the mother and daughter nuclids in equilibrium are identical:

Generalization: equilibrium in radioactive series As the transitions are unidirectional in the radioactive series, the members (with the exemption of the first one) are produced from the previous (neighboring) member of the chain. The isotopes will gradually disappear and convert finally to the last and the only stable member of the series. Let’s consider an intermedier member! Within the short time range Δt, the number of disintegrated i-th nuclid is ΔNi. If this loss is compensated by the increase due to the decay of the previous (i-1)-th member of the chain, and this will be true for every (with the exemption of the first) members of the always shorter chain, then the remaining members of the radioactive decay series are in equilibrium: ΔN1 = ΔN2 = ΔN3 = ... The last (and stable) member of the series will be gradually accumulated in expense of the first member while the number (and activity) of the intermediate members will not change in the radioactive equilibrium. This will occur after appropriate time and last for appropriate long. The loss due to decay: which, however, will be replaced from decay of the previous (mother) nuclid (with the exemption of the first member of the series). The kinetic and equilibrium equations of mother-daughter systems play important roles in several topics of nuclear medicine (molybdenum→technetium-99m transition, tracer studies, etc.).

Example: What was the minimum quantity of uranium ore (pitchblende) the Curie couple had to start with to obtain 1 gramm of radium after purifications? The radium is an intermediate member of the 238U radioactive decay series. The half-life of the 238U mother nuclid is TU = 4.51·109 years, and that of the radium (daughter nuclid) is TRa = 1602 years. As first approximation, let’s suppose that the uranium ore contains the nuclei of these two elements (238U92 and 226Ra88) only. The final product should be 1 g radium, which means that it should contain NRa = (1/226) ·6·1023 = 2.65· 1021 nuclei of radium. As the members of the chain are in radioactive equilibrium, the number of uran nuclei should be: NU = TU/TRa·NRa = 7.46·1027. The mass of so many uran nuclei (in individual mass unit) should be: mU = 7.46·1027/6·1023 mol = 12.4 kmol, which corresponds to a mass of 12.4·103· 238 g = 2960 kg. In a somewhat more refined approximation, the other members of the radioactive decay series should be taken also into account. Their effect is, however, negligible: the value calculated above will be modified in the fourth digit only. The first approximation was a reasonable approach. Marie and Pierre Curie had to work with much more uranium ore because it contained also other elements which did not belong to the radioactive decay family. The quantity and the effect of these impurities were not considered in our calculations.

Problems for seminar • The lifetime of 51Cr24 is TH = 27.7 days. How much is the activity of m = 1 g chromate? • The lifetime of 42K19 is TH = 12.36 hours. After how much time will drop the initial A0 = 1·108 1/s (Bq) activity of the potassium preparatum to At = 1·105 Bq? • To what portion of the initial activity of the radioactive sample of 5 years lifetime will decay after 25 years? • The radioactive decay time of 137Cs55 is TH = 30.17 year. What time is needed to drop the initial activity to 10%? 5. What layer of thickness from lead is needed to attenuate the intensity of the gamma radiation to 5%? The half-value thickness of the lead is 5 mm. 6. 14C is found in living organisms in the amount of about 100 atoms of 14C for every 1020 atoms of 12C. 14C has a half-life of 5760 years. It is found in a particular fossil, the amount of 14C has decreased to about 10 atoms of 14C for every 1020 atoms of 12C. Estimate the age of the fossil.

Problems for seminar • 7. The count rate from a radionuclide falls from 800 counts per minute to 100 counts per minute in 6 hours. What is the decay constant of the nuclide? • 8. A radon 222Rn86 nucleus of mass 3.6·10-25 kg decays by the emission of an alpha particle of mass 6.7·10-27 kg and energy 5.5 MeV. • What are the mass number (A) and atomic number (Z) of the resulting nuclide? • What is the momentum of the α-particle? • Find the velocity of recoil of the resulting nucleus (ignore relativistic effects). • 9. A small volume of a solution which contained the sodium radioisotope 24Na had an activity of 12000 disintegrations per minute when it was injected into the blood stream of a patient. After 30 hours, the activity of 1 cm3 of the blood was found to be 0.5 disintegration per minute. If 24Na has a half-life of 15 hours, estimate the volume of blood in the patient.

Problems for seminar 10. Two radioactive sources X and Y initially contain the same number of radioactive atoms. Source X has a half-life of 15 minutes and source Y a half-life of 30 minutes. What is the ratio of the activity of X to that of Y a) initially; b) after 30 minutes; c) after 2 hours? 11. Two samples, each of volume 1 mL, contained tritiated water, the activity of each sample being 4 MBq. One sample was injected into the blood stream of a patient and after a suitable period of time 4 mL of blood was withdrawn. The corrected count rate produced by this blood in a liquid scintillation counter was 207 counts per second. The other sample was diluted 10000 times with ordinary water and 4 mL of the diluted liquid produced 745 counts per second, after correction, in the same scintillation counter. Calculate the volume of water in the patient’s body. The half-life of tritium may be assumed to be very long compared with the duration of the experiment.

Problems for seminar 12. What is the specific binding energy (binding energy per nucleon) in a 12C6 nucleus? Use any of the following data: m12C = 12 au (exact) mH atom = 1.00782504 au mproton = 1.00727647 atomic unit mneutron = 1.00866501 au melectron = 5.4858026·10-4 au • 13. The binding energy per nucleon in 52Cr24 is 8.776 MeV/nucleon. Using the following masses • mproton = 1.0072765 atomic unit • mneutron = 1.0086650 au • melectron = 5.485802·10-4 au • what do you predict for the nuclear mass of 52Cr24? • What do you predict for the atomic mass?

Problems for seminar 14. The age of water or wine may be determined by measuring its radioactive tritium content. Tritium is present in a steady state in nature. It is formed primarily by cosmic irradiation of water vapor in the upper atmosphere, and it decays spontaneously with a half-life of 12.5 years. The formation reaction does not occur significantly inside a glass bottle at the surface of the earth. Calculate the age of a suspected vintage wine that is 20% as radioactive as a freshly bottled specimen. Would you recommend to a friend that he consider paying a premium price for the „vintage” wine?

Quantum Physical Phenomena in Life (and Medical) Sciences