The number of protons in the nucleus is known as the atomic number Z, and determines the chemical properties of the element.

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Oswin Leonard Crawford
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1 I. NUCLEAR PHYSICS I.1 Atomic Nucleus Very briefly, an atom is formed by a nucleus made up of nucleons (neutrons and protons) and electrons in external orbits. The number of electrons and protons is equal to assure neutrality of atomic nuclei. While the size of an atom is of magnitude m, the size of nuclei is in the order of m. There is experimental evidence that the shape of both may be considered approximately as spheres with diffuse boundaries. The masses of the most important building stones of atomic nuclei are about kg for nucleons and about kg for electrons (Nuclear Masses). There are other elementary particles such as neutrinos (ν), photons (γ), α-particles ( 2 He 4 ), etc. released from nuclei under certain conditions which do not play a role in reactor physics. The number of protons in the nucleus is known as the atomic number Z, and determines the chemical properties of the element. The number of neutrons is represented by the letter N. The total number of nucleons in the nucleus of an atom is known as the mass number A = Z + N. A nuclide X is characterized as follows: All elements with the same nuclear charge Z but different A are known as isotopes such as uranium (U): 92 U 234, 92 U 235, 92 U 238 representing the isotopes of natural uranium. Isobares are elements with the same mass number A but different Z such as 92 U 239, 93 Np 239 (neptunium), 94 Pu 239 (plutonium), etc. Correspondingly, isotones are elements with the same number of neutrons N which are relatively rare. The neutron is not stable unless it is bound to a nucleus. A free neutron decays to a proton with the emission of a β - (fast electron) and a ν (antineutrino). This process has a lifetime of about 12 minutes. The average lifetime of a free neutron in a reactor is a matter of milliseconds (10-3 s), thus neutron instability is of no consequence in reactor physics (Module 12, Section 1). EQUIVALENCE BETWEEN MASS AND ENERGY According to Einstein's theory of relativity, mass and energy are equivalent and convertible, one into the other according to: E = mc 2 (1-1) Whereby: E = rest energy 1

2 m = rest mass c = m/s, speed of light The electron volt (ev) is the unit of energy mainly used in nuclear physics. It is defined as follows: The electron volt is the energy gained by an electron when it passes through an electric field, the potential difference of which is 1 volt. Its Joule (J) equivalence is as follows: 1 ev = J 1 kev = 10 3 ev 1 MeV = 10 6 ev NUCLEAR MASSES The masses of atoms are expressed in terms of atomic mass units (amu's). An amu is defined as being one twelfth of the mass of one neutral atom of the isotope 6 C 12 (1 amu = 1/12 m( 6 C 12 )). The equivalence of 1 amu = kg is deduced as follows: The number of atoms or molecules in a mole (mass in grams equal to the atomic or molecular weight of the substance) is called Avogadro's number L, i.e atoms = 12 g C 12 The mass of one atom 6 C 12 is m( 6 C 12 ) = 12 g/ = g Thus: 1 amu = 1/ kg = kg and its energy equivalent is 1 amu = MeV. The most important element utilized in the conversion of nuclear energy is uranium (U). The elements used for this purpose are divided into two major groups, namely: fissionable or fertile materials, which require high-energy neutrons to achieve fission; fissile materials, which are easily fissionable, even with low-energy neutrons. Fissionable or Fertile Materials Fissile Percentage of Isotopic Mass Nuclides Weight (amu) Thorium Uranium Uranium Materials Uranium Table 1 illustrates the nuclides employed for the conversion of nuclear energy. 2

3 Particle/Atom Mass (amu) I.2. Binding Energy Electron Neutron Proton a-particle H H H He Li Be Be B Sr Xe Th Table 2 illustrates masses of selected particles/atoms. Source: Environmental Chemistry: Periodic Table of Elements The mass of a nucleus is always less than the sum of the masses of its constituent nucleons. The difference is known as mass defect: m = Zm p + Nm n - m(z,a) where m p and m n are the masses of an individual proton and neutron, respectively, and m(z,a) is the mass of the nucleus concerned. m is the mass that would be transformed into energy, if a nucleus is to be constructed by the necessary number of protons and neutrons. This same amount of energy would be needed to split a nucleus into its components. This quantity is taken as the measure of the energy needed to bind the nuclei. The energy equivalent of the mass defect is called the binding energy E B of the nucleus: 2 E B = ( Zm p + Nm n - m(z,a))c (1-1) Experience shows that the binding energy per nucleon in nuclei grows to about A = 60 (except in the case of some light nuclei) and then gradually decreases; i.e., the middle nuclei are more strongly bound than the light or heavy nuclei. 3

4 Figure 1. Source: John R. Lamarsh: Introduction to Nuclear Engineering, 2nd Edition, Addison-Wesley Publishing Company, 1983 Binding energy can be released either from light nuclei by fusion or from heavy nuclei by fission. When light elements fuse into larger groups, they lose mass, and heavy nuclei lose mass when they divide. Let s look at examples: 1 1) 0n + 1p 1 1H 2 + γ (1-2) (a γ ray of 2.23 MeV is emitted) Since this energy escapes when the deuteron 1 H 2 is formed, we say that the mass of the deuteron, expressed in units of energy, is 2.23 MeV less than the sum of the masses of the neutron and the proton. Separation between n and p can again be achieved if the system (deuteron) receives the binding energy via, for example, γ bombardment of the deuteron. γ + 1H 2 0n 1 + 1p 1 (E γ > 2.23 MeV) (1-3) Binding energy is the energy required to separate the nuclide into its individual nucleons. 2 2) 1H + H E B : 2.23 MeV 2.23 MeV E B : [8.23-2(2.23)] MeV = 4.02 MeV H + 1H 8.23 MeV (Binding energy for 1 H 1 is 0) (1-4) 4

5 This energy appears as kinetic energy of 1 H 3 and 1 H 1. Example of mass defect in a fusion reaction The currently most important fusion reaction is: 1H H 2 2 He n 1 or H 3 (d,n) He 4 (1-5) Balance of masses: Masses before the reaction: = amu (H 3 ) amu (H 2 ) = amu Masses after the reaction: = (He 4 ) ( 0 n 1 ) = amu The mass defect m= amu is equivalent to 17.6 MeV. If the kinetic energy of the 1 H 2 is 1 MeV and the 1 H 3 nucleus is stationary, then the sum of the energies of the emergent neutron and the a-particle ( 2 He 4 ) will be 18.6 MeV. Example of mass defect in a fission reaction The binding energy per nucleon of the U 238 is about 7.5 MeV, while it is about 8.4 MeV for a nucleus with A=119 (238/2). Thus, if a uranium nucleus splits into two lighter nuclei each with about half the uranium mass, there is a gain in the binding energy of the system. Binding energy before reaction 7.5 MeV per nucleon Binding energy after reaction 8.4 MeV per nucleon Mass defect = 0.9 MeV per nucleon Total mass defect = MeV = 214 MeV. This process is called nuclear fission and it is the source of energy in nuclear reactors. I.3. Nuclear Forces Between particles equally charged there are repulsive Coulomb forces, and as the nucleus of the atom contains a large number of protons, each repels the others in accordance with Coulomb's Law. Clearly, there should also be other forces in the nucleus that attract. These are referred to as nuclear forces. They act between nucleons and drop rapidly to zero when separated from each other. When the number of protons increases, the long-term Coulomb forces grow faster than the attractive short-term nuclear forces. Heavy nuclei to remain stable require more neutrons, so that the attracting forces of all particles are superior to the repulsive Coulomb forces. For this reason, the n/p ratio grows gradually from 1 to 1½. 5

6 The nature of these forces, which bind the protons and neutrons in the nucleus is short-term, strong and charge independent. It is generally thought that the nucleons are bound in the nucleus by means of the continuous exchange of particles called mesons. It may be observed that for stable nuclides with a low mass number, the neutron/proton ratio is near one. For heavy nuclides this n/p ratio rises progressively, maintaining stability up to a limit level, at which point they are no longer stable and may be formed artificially. This occurs in elements with mass number A larger than 238. It is said that a nucleus is in the ground state when the nucleons in the nucleus are at their lowest potential energy. Otherwise, the nucleus is excited within discrete energy states, as long as all constituents of the nucleus are bound, which are referred to as energy levels. I.4. Radioactivity LAWS OF RADIOACTIVE DECAY All nuclides which are heavier than Pb (Z=82), and a few light nuclei as well, are unstable and naturally radioactive. They decay, emitting either g or b - particles. In most cases, the resulting nucleus, or daughter, is produced in an excited state, which decays to the ground state by emission of one or more photons. Usually, but not always, this will happen instantaneously within s of the formation of the daughter. A radioactive nuclide, or radionuclide, can also decay, by means of capturing an orbital electron (k capture). After abandoning the nucleus, the photons may be absorbed, emitting an electron from the orbit of the same atom. This emission of a secondary particle is known as internal conversion. In the following section, we will review the laws of radioactive disintegration. ACTIVITY Radioactive decay is governed by the laws of probability and is independent of the external environment such as pressure, temperature, chemical treatment, etc.. The number of atoms in a radioactive substance that decay N within a certain interval of time, is proportional to the number of present atoms N and the interval of time t considered: - N = λ N t (1-6) Whereby: λ constant, known as the radioactive decay constant. It is characteristic of each nuclide and its dimension is time -1 (s -1 ; min -1 ). If in (1-6) we apply a small t and pass it to the first member (1-7) 6

7 N(t) = N o e -λt with N 0 = N(t=0) (1-8) The value A, which measures the decay speed of an active nuclide (the minus sign is due to the fact that there are atoms that disappear), is called activity. Thus, activity is the number of atoms that disintegrate within a unit of time. HALF-LIFE Half-life is defined as the time necessary for a significant number of atoms to reduce to half, and is represented by T ½. Mathematically speaking: (1-9) Connecting this equation with (1-7): (1-10) Half-life indicates the average lifetime of atoms. Following the corresponding mathematical analyses, and once a certain time lapse t has passed, we have the following: (1-11) Whereby: A 0 = source activity at the initial time (t=0) e = natural logarithmic base Activity is expressed in disintegrations per unit of time (disintegrations per minute; disintegrations per second). The unit of activity is called a Curie, [A] = Ci and it is equivalent to disintegrations per second. The SI unit of activity is Bequerel, [A] = Bq which is equivalent to 1 disintegration per second. 1 Ci = Bq. 7

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Nuclear Theory - Course 227 NUCLEAR STRUCTURE

Nuclear Theory - Course 227 NUCLEAR STRUCTURE The Nucleus, Nuclear Particles The atomic nucleus consists of Z protons and N neutrons, where Z and N are the atomic number and neutron number respectively.