Chapter 2 Atoms and Nuclei PDF

Chapter 2 Atoms and Nuclei NUCLEAR ENGINEERING  Atomic Theory The most elementary concept is that matter is composed of individual particles—atoms Weight of gas is the sum of the weights of all gas atoms or molecules (e.g., He gas and CO2 gas) There are...

Chapter 2 Atoms and Nuclei NUCLEAR ENGINEERING  Atomic Theory The most elementary concept is that matter is composed of individual particles—atoms Weight of gas is the sum of the weights of all gas atoms or molecules (e.g., He gas and CO2 gas) There are more than 100 known elements NUCLEAR ENGINEERING  Most are found in nature; some are artificially produced. – Each is given a specific number in the periodic table of the elements; examples are hydrogen (H) 1, helium (He) 2, oxygen (O) 8, and uranium (U) 92. NUCLEAR ENGINEERING The symbol Z is assigned to the atomic number (number of protons), which is also the number of electrons in the atom and determines its chemical properties. NUCLEAR ENGINEERING NUCLEAR ENGINEERING  Generally, the further an element is in the periodic table, the heavier its atoms. The atomic weight M is the weight in grams of a definite number of atoms, 6.022 x 1023, which is Avogadro’s number, NA. NUCLEAR ENGINEERING  Although we often use the terms atomic weight and atomic mass interchangeably, atomic mass describes the mass of a single atom of a particular isotope, whereas atomic weight provides a weighted average mass for an element based on the abundance of its constituent isotopes. NUCLEAR ENGINEERING Hydrogen Isotopes NUCLEAR ENGINEERING NUCLEAR ENGINEERING  For the following elements, the values of M (atomic weight) are approximately – H, 1.008; – He, 4.003; – O, 16.00; and – U, 238.0. NUCLEAR ENGINEERING  Atomic weight is expressed using grams/mole or atomic mass units (u), and atomic mass is quantified using atomic mass units (u). Accurate values of atomic weights of all the elements are given in Table A.4 in Appendix A of the book. NUCLEAR ENGINEERING NUCLEAR ENGINEERING  If an element has a nonnatural abundance of its isotopes (i.e., the elemental material is either enriched or depleted), it is necessary to compute the atomic weight of the element (M) from the weighted sum of all the atomic masses of the isotopes (Mj) rather than to use the tabulated M value found in a reference. NUCLEAR ENGINEERING  In such cases, the isotopic abundance may be expressed either as – an atom abundance or fraction (γj), – or as a weight or mass fraction (ωj). This distinction leads to two formulas for determining the elemental atomic weight: NUCLEAR ENGINEERING γj = atom abundance or fraction ωj = weight or mass fraction NUCLEAR ENGINEERING  We can easily find the number of atoms per cubic centimeter in a substance if its density ρ (rho) in grams per cubic centimeter is known. This procedure can be expressed as a convenient formula for finding N, the atomic number density for any material The relationship holds for compounds as well, if M is taken as the molecular weight. NUCLEAR ENGINEERING NUCLEAR ENGINEERING NUCLEAR ENGINEERING  Gases Substances in the gaseous state are described approximately by the perfect or ideal gas law, relating pressure (p), volume (V), and absolute temperature (T), pV = n* RT where n is the number of particles and k is Boltzmann’s constant (k = 1.38 x 10-23 J/oK) R = K Na, R = Gas constant, NA = 6.022 x 1023 mol-1 n* = number of moles NUCLEAR ENGINEERING  An increase in the temperature of the gas as a result of heating causes greater molecular motion, which results in an increase of particle bombardment of a container wall and thus of pressure on the wall. P~T NUCLEAR ENGINEERING  Maxwell–Boltzmann Distribution The gas particles, each of mass m, have a variety of speeds v in accord with Maxwell’s gas theory, as shown in Figure 2.2. Maxwell’s formula for the number of molecules per unit speed is where no is the total number of molecules. NUCLEAR ENGINEERING  The most probable speed, at the peak of this Maxwellian distribution, depends on temperature according to the relation while the average speed is NUCLEAR ENGINEERING NUCLEAR ENGINEERING  The average energy of gas molecules is proportional to the temperature, 3 𝐸𝐸 = 𝑘𝑘𝑘𝑘 2 NUCLEAR ENGINEERING  The Atom and Light It is well known that the color of a heated solid or gas changes as the temperature is increased, tending to go from the red end of the visible region toward the blue end (i.e., from long wavelengths to short wavelengths). The measured distribution of light among the different wavelengths at a certain temperature can be explained by the assumption that light is in the form of photons. These are absorbed and emitted with definite amounts of energy E that are proportional to the frequency ν (Greek letter nu), according to where h is Planck’s constant, 6.6261 x 10–34 J s NUCLEAR ENGINEERING NUCLEAR ENGINEERING  Bohr (1913) first explained the emission and absorption of light from incandescent hydrogen gas with a novel model of the hydrogen atom. He assumed that the atom consists of a single electron moving at constant speed in a circular orbit about a nucleus—the proton— as shown in Figure 2.3. NUCLEAR ENGINEERING NUCLEAR ENGINEERING  Each particle has an electric charge of 1.602 x10–19 coulombs (C), but the positively charged proton has a mass that is 1836 times that of the negatively charged electron. The radius of the orbit is set by the equality of electrostatic force, attracting the two charges toward each other, to centripetal force, required to keep the electron on a circular path. NUCLEAR ENGINEERING  If sufficient energy is supplied to the hydrogen atom from the outside, the electron is caused to jump to a larger orbit of definite radius. At some later time, the electron falls back spontaneously to the original orbit, and energy is released in the form of a photon of light. NUCLEAR ENGINEERING NUCLEAR ENGINEERING  The photon energy hν is equal to the difference between energies in the two orbits. The smallest orbit has a radius R1 = 0.53 x 10–10 m, whereas the others have radii increasing as the square of integers, n, which are called principal quantum numbers. Thus if n is 1, 2, 3,... ,7, the radius of the n-th orbit is NUCLEAR ENGINEERING  Figure 2.4 shows the allowed electron orbits in hydrogen. The energy of the atom system when the electron is in the first orbit is E = -13.6 eV, where the negative sign means that energy must be supplied to remove the electron to a great distance and leave the hydrogen as a positive ion. The energy when the electron is in the nth orbit is The various discrete levels are shown in Figure 2.5. -------------- 1 eV (electronvolt) = 1.602 x 10-19 J NUCLEAR ENGINEERING NUCLEAR ENGINEERING  The electronic structure of the other elements is described by the shell model, in which a limited number of electrons can occupy a given orbit or shell. The atomic number Z is unique for each chemical element and represents both the number of positive charges on the central massive nucleus of the atom and the number of electrons in orbits around the nucleus. NUCLEAR ENGINEERING  The maximum allowed numbers of electrons in orbits as Z increases for the first few shells are 2, 8, and 18. The number of electrons in the outermost, or valence, shell determines the chemical behavior of elements. NUCLEAR ENGINEERING NUCLEAR ENGINEERING  For example, oxygen with Z=8 has two electrons in the inner shell, six in the outer. Thus, oxygen has an affinity for elements with two electrons in the valence shell. The formation of molecules from atoms by electron sharing is illustrated by Figure 2.6, which shows the water molecule. NUCLEAR ENGINEERING NUCLEAR ENGINEERING  The Bohr model of atoms is useful for visualization, but quantum mechanics provides a more rigorous view. There, the location of the electron in the H atom is described by a probability expression. A key feature of quantum mechanics is Heisenberg’s uncertainty principle. It states that the precise values of both a particle’s position and momentum cannot be known at the same time. NUCLEAR ENGINEERING  The Heisenberg uncertainty principle is a rule in quantum mechanics. It states that there is a fundamental limit to how well you can simultaneously know the position and momentum (where momentum is classically mass times velocity) of a particle. This means if you know the position very precisely, you can only have limited information about its momentum and vice- versa. NUCLEAR ENGINEERING  Furthermore this is a mathematical inequality that does not depend on measurement technique or how you find the position and momentum. In fact, it has recently been shown experimentally that even if you have an instrument that is so precise that it should (theoretically) be capable of measuring the position and momentum of the particle below the Heisenberg uncertainty limit, the spread in measured values leads to an uncertainty that is still above this limit. NUCLEAR ENGINEERING ℏ = h/2𝜋𝜋 NUCLEAR ENGINEERING  Laser Beams Ordinary light in the visible range is a mixture of many frequencies, directions, and phases. In contrast, light from a laser (light amplified by stimulated emission of radiation) consists of a direct beam of one color and with the waves in synchronization. The device consists of a tube of material to which energy is supplied, exciting the atoms to higher-energy states. NUCLEAR ENGINEERING  A photon of a certain frequency is introduced. It strikes an excited atom, causing it to fall back to the ground state and in so doing emit another photon of the same frequency. The two photons strike other atoms, producing four identical photons, and so on. NUCLEAR ENGINEERING  The ends of the laser are partially reflecting, which causes the light to be trapped and to build up inside by a combination of reflection and stimulation. An avalanche of photons is produced that makes a very intense beam. Light moving in directions other than the long axis of the laser is lost through the sides, so that the beam that escapes from the end proceeds in only one direction. NUCLEAR ENGINEERING  The reflection between the two end mirrors assures a coherent beam (i.e., the waves are in phase). Nuclear application – isotope separation NUCLEAR ENGINEERING  Nuclear Structure Most elements are composed of atoms of different mass, called isotopes. For instance, hydrogen has three isotopes of weights in proportion 1, 2, and 3—ordinary hydrogen, heavy hydrogen (deuterium), and tritium. NUCLEAR ENGINEERING  Each has atomic number Z=1 and the same chemical properties, but they differ in the composition of the central nucleus, where most of the mass resides. The nucleus of ordinary hydrogen is the positively charged proton; the deuteron consists of a proton plus a neutron; the triton contains a proton plus two neutrons. The neutron is a neutral particle of mass very close to that of the proton. NUCLEAR ENGINEERING  To distinguish isotopes, we identify the atomic mass number A as the total number of nucleons, which are the heavy particles in the nucleus. The atomic weight, a real number, is approximated by the mass number, which is an integer, M ≅ A. NUCLEAR ENGINEERING  The complete shorthand notation for an isotope is given by the chemical symbol X with leading superscript A and subscript Z values, that is, 𝐴𝐴𝑍𝑍𝑋𝑋 Figure 2.7 shows the nuclear and atomic structure of the three hydrogen isotopes NUCLEAR ENGINEERING NUCLEAR ENGINEERING NUCLEAR ENGINEERING  Nuclear Notation 235 235 235 U ~ U ~ U ~ U-235 92 143 Means: 235 nucleons (atomic mass # =protons + neutrons) 92 atomic number (number of protons) 143 number of neutrons NUCLEAR ENGINEERING For neutrons: 10𝑛𝑛 For electrons: −10𝑒𝑒 NUCLEAR ENGINEERING  Sizes and Masses of Nuclei Nucleus size ~ 10-15 m = 1 fentometer (fm) Atom size ~ 10-10 m = 1 angstrom (Å) The radii of nuclei can be calculated by NUCLEAR ENGINEERING  The masses of atoms, labeled M, are compared on a scale in which an isotope of 12 carbon 6𝐶𝐶 has a mass of exactly 12. The atomic mass of the proton is 1.007276, of the neutron 1.008665, the difference being only about 0.1%. The mass of the electron on this scale is 0.000549. A list of atomic masses appears in Table A.5. NUCLEAR ENGINEERING NUCLEAR ENGINEERING NUCLEAR ENGINEERING Mass-Energy Equivalence 1 eV (electron volt) = 1.602 x 10-19 J 1 amu = 1 u = 1.660539 x 10-27 kg = 931.49 MeV E = m c2 leads to Energy Mass NUCLEAR ENGINEERING  Binding Energy The force of electrostatic repulsion between like charges, which varies inversely as the square of their separation, would be expected to be so large that nuclei could not be formed. The fact that nuclei do exist is evidence that there is an even larger force of attraction. The nuclear force is of very short range NUCLEAR ENGINEERING  The radius of a nucleon is approximately 1.25 x 10–11 m; the distance of separation of centers is about twice that. The nuclear force acts only when the nucleons are very close to each other and binds them into a compact structure. Associated with the net force is a potential energy of binding. NUCLEAR ENGINEERING Mass Defect NUCLEAR ENGINEERING  To disrupt a nucleus and separate it into its component nucleons, energy must be supplied from the outside. Recalling Einstein’s relation between mass and energy, this is the same as saying that a given nucleus is lighter than the sum of its separate nucleons, the difference being the mass defect. NUCLEAR ENGINEERING  Let the mass of an atom including nucleus and external electrons be M, and let mn and MH be the respective masses of the neutron and the proton plus matching electron. Then the mass defect is NUCLEAR ENGINEERING  In nuclear physics, a magic number is a number of nucleons (either protons or neutrons) such that they are arranged into complete shells within the atomic nucleus. The seven most widely recognized magic numbers as of 2019 are 2, 8, 20, 28, 50, 82, and 126. Atomic nuclei consisting of such a magic number of nucleons have a high binding energy per nucleon NUCLEAR ENGINEERING Binding Energy per Nucleon (BE/A) NUCLEAR ENGINEERING http://csma31.csm.jmu.edu/physics/courses/163/older/bepn2.jpg NUCLEAR ENGINEERING http://www.schoolphysics.co.uk/ NUCLEAR ENGINEERING Binding energies per nucleon for some common elements: http://www.schoolphysics.co.uk/ NUCLEAR ENGINEERING NUCLEAR ENGINEERING NUCLEAR ENGINEERING  Example 2.6 shows the necessity of utilizing masses known to 6 or more significant digits in the computation of the mass defect. Calculations such as these are required for several purposes: to compare the stability of one nucleus with that of another, to find the energy release in a nuclear reaction, and to predict the possibility of fission of a nucleus. NUCLEAR ENGINEERING  We can speak of the binding energy associated with one particle such as a neutron. Suppose that M1 is the mass of an atom and M2 is its mass after absorbing a neutron. The binding energy of the additional neutron of mass mn is then NUCLEAR ENGINEERING  Explanations of binding energy effects by means of physical logic and measured atomic masses have led to what are called semiempirical formulas for binding energy. The BE for any nuclide may be approximated using a liquid drop model that accounts for (1) attraction of nucleons for each other due to strong nuclear force, (2) electrostatic (Coulombic) repulsion, (3) surface tension effects, and (4) the imbalance of neutrons and protons in the nucleus. NUCLEAR ENGINEERING The Bethe-Weizsacker formula is one such expression to calculate the binding energy: NUCLEAR ENGINEERING NUCLEAR ENGINEERING