Nuclear Physics PDF 2019

Nuclear physics The text under the slides was written by Peter Hajdu 2019 This instructional material was prepared for the biophysics lectures held by the Department of Biophysics and Cell Biology...

Nuclear physics The text under the slides was written by Peter Hajdu 2019 This instructional material was prepared for the biophysics lectures held by the Department of Biophysics and Cell Biology Faculty of Medicine University of Debrecen Hungary https://biophys.med.unideb.hu Related Chapters: I/1.1.1 pp 23, I/1.5 pp 40-44, II/3.2 pp 164-170 2 Outline of the lecture: - Discovery of radioactivity and parts of the nucleus (historical background, story- telling) - Major types of radioactive radiations and the changes in the nuclear structure upon decay - Energy scheme of decays - Law of radioactive decay - Composition of a nucleus: nuclei with different numbers of N and A (isotopes, isobars, isotones), isotope effect - Stability of a nucleus: the nuclear force and its properties, the relationship of the mass defect and binding energy - Spontaneous nuclear processes - Models describing the behavior of nuclei 3 In 1896 Becquerel discovered radioactivity while he wrapped photoplates in paper sheets and placed uranium salt over it: the silhouette of the uranium “crystal” could be detected on the negatives (“something” blackened the photoplate). Later Rutherford identified two radiations (alpha (α, positive) and beta (β, negative or positive)) stemming from the uranium. Then in 1900 Villard identified the third radiation coming from radium after penetration thru a thin lead plate, it was named gamma rays by Rutherford. In 1911 Rutherford performed his famous scattering experiment: he bombarded gold foil with alpha particles. As the majority of the alpha particles went thru the foil without deflection, he concluded that the radius of the atom is tiny as compared to the size of an atom. Also from the wide angle scatters (head- on collisions) he calculated that the nucleus has a charge equalto the total charge of the electrons in the atom (+Ze). In 1919 he published the discovery of the proton coming from a nuclear reaction (see in the slide). Approximately one and a half decades later Chadwick proved that uncharged particles coming from the decay of deuterium (see slide) is not gamma ray but a particle with a mass ca. of a proton and has no charge. It was named neutron (n). 4  ! Let’s go through the types of decays and particles ejected from a radioactive nucleus. Also, we introduce some basic concepts: parent element/nucleus: the starting radioactive nucleus, daughter nucleus: the mother nucleus transforms into this nucleus upon radioactive decay. Alpha- or α-particle/decay: the core of a 4 He atom is ejected from the nucleus, with a 2 double positive charge; meanwhile the parent nucleus’s mass decreases by 4 atomic mass unit (2 protons + 2 neutrons) and the number of protons by 2. Beta negative/positive - or β- / β+-decay: β- particle is an electron (the same as in the orbits of atoms) and during β- decay the parent nucleus’s mass does not change but the number of protons increases by 1 (new element is formed), and the number of neutrons decreases by 1. Also, another elementary particle is ejected together with the electron and it’s the (electron) antineutrino (𝜈̅ ). (it is required for the conservation of lepton (light elementary particles such as electron, muon) number and explanation of the decay energy scheme) For β+ decay the parent nucleus’s mass does not change but the number of protons decreases by 1 (new element is formed), and the number of neutrons increases by 1. The ejected particle is the so-called positron (an electron with positive charge), plus an (electron) neutrino. 5  ! The third type of radiation coming from a radioactive nucleus is the gamma/ particle that is a high energy photon. Two decay schemes - which are used in medicine (diagnosis (Mo into Tc (left)) and therapy (Co into Ni (right)) indicate that  particle is only emitted if it is preceded by an alpha or a beta decay and never alone (the excited nucleus loses energy via  particle emission). Gamma radiation is a tool for the excited nuclei to lose energy and get into a stable state. Finally, there is another possible way for nuclei with proton excess to reduce the atomic number: the nucleus captures an electron from the innermost K shell (K-capture). Here the electron fuses with a proton and becomes a neutron (p+ + e- no + ). Hence, the atomic number (number of protons) decrease by 1 while mass number is unaltered. Moreover, an X-ray photon or an Auger-electron is ejected from the atomic shells (here we refer to the production of characteristic X-ray). Also see the pic below. 6  ! The electrons of atoms are able to absorb or emit energy in certain portions/quanta because the energy levels of a bound electron are quantized. The energy quantization is also held for the radioactive decay and the energy of three decays is characteristic for the parent nucleus; hence it can be used to identify the decay process and the element (just like characteristic X- ray). The highest energy loss of a nucleus is achieved by alpha radiation, then less with beta and the lowest with gamma. The energy of alpha and gamma particles coming from a nucleus is well-defined (quantized), typical for the emitting nucleus. Though the energy of the beta decay is quantized, the energy spectrum of beta particles is not. Why? Because the energy of the beta decay is randomly distributed/departed between the beta particle and the other particle, i.e., the neutrino or antineutrino. 7  ! As the time elapses and radioactive nuclei disintegrate, the number of parent nuclei decreases. The time course of this decay can be described with a negative exponential function: N(t) = N0  e-t = N0  e-t/ (e: Euler/natural number, app. 2.71828…), where N defines the number of the undecayed parent nuclei at time point “t”, N0 is the initial number of parent nuclei, and  (lambda) is the decay constant (a rate constant, unit: [time] -1) of the disintegration process,  is the average lifetime of the nuclei.  can be determined from the decay curve: it is the reciprocal of the average lifetime, that is, the time point is read where N equals to N0/e (app. 37% percent of the initial value). Also the decay formula can be rewritten in the following form: N(t) = N0  2-t/T½, here T½ is the half-life of the radioactive nuclei (relationship between T½ and  is shown in the top right corner). The example on the right shows that at the time point equal the half-life, app. half of the nuclei remain intact, at the double of the half-life one-fourth of the nuclei are undecayed. When the elapsed time is triple of the half-life, the remaining fraction of the undecayed nuclei is one-eighth (check it!!!). 8  ! To determine the value of the average lifetime from the graph shown on the left (linear-linear plot) is not accurate (single-point reading), and also the plotting and fitting a curve to the data points could be difficult mainly for free-hand graphing. Instead, using the semi-logarithmic coordinate system (axis y is logarithmic, axis x is linear) the plot of a negative exponential function turns into a line. Or with the derivation in the mid below one can easily perform 𝑵 linearization of the exponential equation/function. By plotting (𝐥𝐧 𝑵 ) vs the elapsed time 𝟎 “t” one could obtain the graph illustrated on the right, which is a straight line with a negative slope (slope = -). To determine  (lambda), the slope of the line with the triangle method should be determined, and then we get that  = (-1)  slope. This way the evaluation of parameters of radioactive decay can be done easier. 9  ! After injection of radioactive pharmacons into the patient’s body (for PET, SPECT etc. imaging) the amount of radioactive nuclei can decrease in two ways: 1) the parent nucleus decays into a daughter nucleus and emits beta or gamma ray (physical decay); 2) the radiopharmacon is removed from the body via excretion (urine through kidneys) or exhalation etc. (biological decay). This means that the “decay process” of the radioactive pharmacons is accelerated in the patient’s body: the effective decay, which defines how fast the radioactivity vanishes from a living organism, is characterized by the effective decay constant (eff), which is the sum of the physical (phys) and biological (biol) decay constants (the rate constant describes how fast a process occurs). Consequently, the effective half-life is the lowest out of the three (see bottommost equation on the right), and the curve for the effective decay runs under both the physical and biological decay curves. 10  * 5 The radioactive elements occurring in nature can be classified into three groups that form radioactive decay chains/series. The parent element of each series decays with alpha decay and the subsequent daughter elements can decay with either alpha or beta decay. The mass number of each member of any series can be given with the formula: A = 4n + C, where C= 0, 1, 2, 3 (remainders for the division by four). E.g. for C=0 we can have the members of the thorium series. 11  ! Now let us go through again the basic properties of nuclei: they can only contain positively charged protons (proton/atomic number: Z) (charge magnitude equals to one electron charge) and electroneutral neutrons (neutron number: N), and their sum in a nucleus is the nucleon/mass number (A). The mass of a proton and the mass of a neutron are almost equal, the neutron being a bit heavier (see the text on the right). The graph on the left shows all the elements with their isotopes found in nature: if we pick a certain Z (proton) number and move along a vertical line, we can have the isotopes of a certain element with Z. When we fix the N number and move parallel to the Z axis, we get the isotones (elements with the same N, horizontal line). Finally, elements which have the same mass/nucleon (A) number are the isobars (elements positioned on a straight line with slope -1, blue line here). Further information we can obtain from the graph is about the stability of nuclei (check the legend in the left corner). This part we will discuss later, and we will refer back to this graph. 12  ! Isotopes of various elements can have very different chemical and physical properties, e.g. hydrogen and its isotope deuterium are non-radioactive, while the other isotope tritium is radioactive and ejects beta negative particles. Hence, the latest can be applied in detection of cell proliferation, protein secretion etc. The carbon 14 isotope (6 protons + 8 neutrons) is unstable with a relatively long half-life and can be utilized in determination of age of fossils (carbon dating method). The ratio of 12C and 14C is constant as long as the living organism is alive, but after its death the ratio increases due to the decay of 14C: this can be used to estimate the time of death. More on applications of isotopes comes in a subsequent lecture. 13  ! The protons of a nucleus due to the Coulomb/electrostatic force repel each other (like charges repel each other). Hence, another force should arise between the protons that compensates this repulsive interaction and keeps the nucleus together: this is called nuclear force. However, the nuclear force between the protons is not enough to keep the nucleus together, so the presence of neutrons in the nucleus can add more attractive force and stabilize it (no repulsion between two neutrons or a proton and a neutron). E.g. in a helium at least 1 neutron is necessary to preserve it stability, otherwise it falls apart. The nuclear force has the following properties: 1) very strong; 2) its range is short, only neighboring nucleons affect each other; 3) magnitude of the interaction is not influenced by the charge of the nucleons; 4) it is only attractive (just like gravitational force). 14  ! When the nucleons of a nucleus (protons and neutrons) are in a free state (just like the balls that can move freely on a frictionless horizontal surface), their energy is comprised of the kinetic and potential terms. Their potential energy is zero as there is no attractive interaction between them. As the nucleons form a nucleus (fusion happens, see later) and get into a bound state (attractive force between them), the total potential energy of the nucleons becomes negative. (the balls get into a well where they still move around but just inside the well). 15  ! When the nucleons are in a bound state in the nucleus, the nucleus can be characterized with the binding energy. The definition of the binding energy is the energy released by the free nucleons when they fuse into the nucleus or, alternatively, it is the energy needed to remove all nucleons from the nucleus (see the graph below, too). The binding energy can be calculated using the mass-energy equivalence equation by Einstein (E = mc2). First, the mass of the free nucleons should be determined (Z  mp + (A-Z)  mn), and from this the mass of the nucleus formed by these nucleons (Mnucleus) must be subtracted (as shown in the equation above). For every nucleus we should get a positive difference, which means the mass of the nucleus is lower than that of the free nucleons (and the potential energy of the nucleus is negative). This is called mass defect, and by using Einstein’s famous equation Ebinding can be calculated. 16  ! Plotting the binding energy per nucleon (total binding energy of the nucleus divided by the number of nucleons (A)) for each nucleus vs. the mass number (A) the curve in the pic above can be obtained. It clearly shows that the maximum is around the iron 56 (56Fe). Those elements, whose Z number is lower than 26 can reach the most stable nucleus configuration via fusion with other lighter nuclei (as it happens in the stars such as the Sun). While the nuclei heavier than iron try to get the stable configuration via fission process: they split into smaller nuclei (each is heavier than the alpha particle) with radioactive decay. Elements with the most stable isotopes can be found around the A = 56. 17  * 5 The fusion of light nuclei can happen spontaneously provided the nuclei are so close to each other that the nuclear force overcomes the repulsion due to the Coulomb interaction (e.g. the nuclei are condensed into a tiny volume or collide at a high speed with each other). The fusion of the light nuclei leads to an energetically favorable state: the newly formed nucleus has a lower potential energy than the total potential energy of the original nuclei. The energy diffence between two light nuclei and the fused nucleus is irradiated as an electromagnetic wave (photon(s)), that could be utilized in a future power plant. As the potential energy of the resultant nucleus is lower, its stability increases. 18  ! The stable nuclei for light elements contain app. the same number of neutrons and protons (the N/Z ratio is close to 1), while for heavy nuclei the neutrons outnumber the protons. The extra neutron number contributes to the stability of the nuclues as it increases the binding energy (no electrostatic repulsion occurs upon neutron addition, see slide on nuclear forces). The stable nuclei in the N-Z graph are located in the green band (it is above the 1:1 ratio line for large Z values). Those (mainly heavy) nuclei, which have N/Z ratio much higher than 1 (red band) have excess neutron number, which leads to instability. In these nuclei a a neutron is transformed into a proton via - decay and attain stable configuration. For nuclei in the blue strip the N/Z ration is closer to one than for the stable (too many protons), so they a proton is transformed into a neutron via + particle ejection; thus, the proton number is reduced by one. In the hatched elliptical area those nuclei are located, which are unstable and heavy enough to undergo  decay (or fission upon e.g. neutron impact). Please note that light nuclei are unable to emit  particles (energetically cannot happen, fusion is 5 * favourable energetically). The figure to the right of the text clearly shows the stability graph at “higher resolution” for different nuclei: it details that at higer Z values the N = Z line and the stability curve defintely separated. 19  ! To describe the stability of nuclei, first a semiempirical model was formulated. As the density of different nuclei was constant (density = mass / volume, mass  A  mnucleon, volume  A ) it was logical to regard it as an incompressible droplet. According to this model, the binding energy of a nucleus (Ebinding, Weizsäcker formula) is influenced by the following terms: volumetric term (first one), which is proportional to A (number of nucleons, nuclear force contribution), the only positive term. The second term accounts for the decrease in the nuclear potential energy: as there are more nucleons in the nucleus, there will be more on the surface of the spherical nucleus (which are not surrounded by nucleons from every direction, so the attractive interaction is weaker for these). The third term considers the repulsion between the protons that reduces the binding energy. Finally, the last term considers the symmetrical distribution of the nucleons: though we expect that the nucleus with lower Z and higher N has a higher Ebinding (due to the lower electrostatic repulsion, right nucleus in the figure), there is a term reducing the binding energy because of the unequal number of protons and neutrons. Explanation: neutrons due to the asymmetric nucleon 20 distribution are at a higher energy state than the protons for the symmetric case (Pauli’s exclusion principle dictates that protons can fill up only proton states and neutrons only neutron states, no mixing allowed, i.e., neutrons cannot be in a proton state and vice versa). 21  ! As the energy of radiations stemming from a nucleus is discrete (just as the spectrum of atoms), it was assumed that energy levels of the nucleons in a nucleus are also quantized. Also it was found that elements whose N, Z, or A number is a magic number, have many stable isotopes and are very stable (like closed electron shells for atoms, group VIII elements). Physicists using various potential well approcahes could solve Schrödinger’s eqation for different nuclei. The model had several handicaps, it only worked for the lighter nuclei, the model did not give back the magic numbers for heavy nuclei, etc. According to the shell model the neutrons and protons build up two separate shell systems, no tresspassing being allowed between these two, i.e., a neutron cannot jump onto a proton shell and vice versa. The shells are named similar as for electrons in the atom: 1s2+2, 1p6+6, 2s2+2, 1d10+10, 1f14+14, 2p, 3f, etc. For example on the 1s orbit altogether 4 nucleons reside: 2 protons and 2 neutrons. In summary, it was a nice approach just as the Bohr model for atoms, however, it could not explain all phenomena experienced for nuclei. 22  ! The scientist made a statistics on the stability of nuclei and the table above contains all data. It can be clearly seen that an even Z and N result in a stable nucleus. For light nuclei the key to the stability is N/Z  1, for heavy one N/Z is higher than 1. 23 24