Chapter 4 Atomic and nuclear physics INTRODUCTION: The technologies used in nuclear medicine for diagnostic imaging have evolved over the last century, starting with Röntgen s discovery of X rays and Becquerel s discovery of natural radioactivity. Each decade has brought innovation in the form of new equipment, techniques, radiopharmaceuticals, advances in radionuclide production and, ultimately, better patient care. All such technologies have been developed and can only be practised safely with a clear understanding of the behaviour and principles of radiation sources and radiation detection. Discoveries in nuclear physics have led to applications in many fields, such as nuclear power, nuclear weapons, nuclear medicine and magnetic resonance imaging, industrial and agricultural isotopes, ion implantation in materials engineering, and radiocarbon dating in geology and archaeology. Basic definitions for atomic structure: The constituent particles forming an atom are: protons neutrons electrons ( 54 )
Size of electron orbit is 5x10-11 m Nucleus is 5,000 times smaller than the atom! Nucleus size ~10-14 m Spacing between nucleons 10-15 m ( 1 fermi = 10-15 m ) Protons and neutrons are known as nucleons and form the nucleus of the atom. Protons have a positive charge, neutrons are neutral and electrons have a negative charge mirroring that of a proton. In comparison to electrons, protons and neutrons have a relatively large mass exceeding the electron mass by a factor of almost 2000 (note: m p /m e = 1836). ( 55 )
If the atom were the size of your classroom, the nucleus would be the size of a single grain of sand in the center of the room. Most of an atom s mass is concentrated in the nucleus. Electrons are outside the nucleus in the electron cloud. Because electrons are so fast and light, physicists tend to speak of the "electron cloud" rather than talk about the exact location of each electron. Figure 7.1: Excitation vs ionization processes ( 56 )
Nuclear forces: 1 - electromagnetic forces: Electrons are bound to the nucleus by electromagnetic forces. The force is the attraction between protons (positive) and electrons (negative). The momentum of the electron causes it to move around the nucleus rather than falling straight in. 2 - strong nuclear force: The strong nuclear force attracts neutrons and protons to each other, otherwise the positively charged protons would repel each other. However we do not understand the nature of this force. But we do know that enormous amounts of energy are needed to separate the nucleus into separate protons and neutrons. 3 - weak force: The weak force is weaker than both the electric force and the strong nuclear force. 4 - gravitation force: The force of gravity inside the atom is much weaker even than the weak force. ( 57 )
Every process we know in the universe can be explained in terms of these fundamental forces. The following general definitions apply to atomic structure: Atomic number Z: is the number of protons and number of electrons in an atom. Atomic mass number A: is the number of nucleons in an atom, i.e. the number of protons Z plus the number of neutrons N in an atom: A = Z + N Atomic mass: (sometimes mistakenly called atomic weight): The mass of a neutral atom. Is value in atomic mass units (u) is approximately equal to the sum of the number of protons and neutrons in the nucleus of the atom. 1 u is equal to one twelfth of the mass of the 12 C atom (unbound, at rest and in the ground state) or 931.5 MeV/c 2 Nucleus: The core of the atom, where most of its mass and all of its positive charge is concentrated. Neutron number ( N ): The total number of neutrons in the nucleus. ( 58 )
Isotope: Isotopes of a given element have the same atomic number (same number of protons in their nuclei) but different mass numbers (different number of neutrons in their nuclei). 238 U and 235 U are isotopes of uranium.( Same Z, different N ). Ex: 8 8, 1 1 1 6 6 6 Isobars: Atoms (nuclides) of different chemical elements that have the same number of nucleons. Correspondingly, isobars differ in atomic number (or number of protons) but have the same mass number( Same A and different Z and N ). An example of a series of isobars would be 40 S, 40 Cl, 40 Ar, 40 K, and 40 Ca. 26 27 ( 59 )
Isotones: Two nuclides are isotones if they have the same neutron number N, but different in Z and A. For example, boron-12 and carbon-13 nuclei both contain 7 neutrons. 5 6 or: 6 7 8 See http://www.nndc.bnl.gov/wallet/ Meson(1934 Hideki Yukawa): It is a carrier of the nuclear force that holds atomic nuclei together. Its mass is between the electron m 0 mass and the proton mass m p, and has electric charge -1e, 0e, +1e. All mesons( about 140 types) are bosons and unstable. photon: A packet of electromagnetic energy ( x-ray or gamma ray ). Photons have momentum and energy, but no rest mass or electrical charge. Positron: or antielectron is the antiparticle of the electron. The positron has an electric charge of +1 e, and has the same mass as an electron. When a positron collides with an electron, annihilation occurs. If this collision occurs at low energies, it results in the production of two or more gamma ray photons. ( 60 )
Nuclear Properties 1 - Nucleus charge ( Ze ): The charge of nucleus is equal to the proton charge, therefore equal to Ze, where Z is the number of proton in the nucleus( atomic number ) and e is elementary charge: e = 1.602 x 10-19 coulombs 2 - Radius of the nucleus: Assuming spherical shape, nuclear radii can be calculated according to following formula: R = R 0 A 1/3 Where R: the nuclear radius, A: mass number and R 0 : radius constant( nuclear unit radius ) R 0 = 1.4 Fermi ( for nuclear particles scattering on nuclei ) = 1.2 Fermi ( for electron scattering on nuclei ) ( 1 Fermi = 10-15 meter = 1 femtometer (fm) in SI unit; one ). ( 61 )
By assuming the nucleus to be a sphere we can calculate the density of the nucleus where the volume V: V = V = The mass of the whole nucleus is the mass of a nucleon (m) multiplied by the number of nucleons (A) Mass M = Am, then the Density is: The nuclear density of a nucleus is = 2.4 x 10 17 kg/m 3, this is the density of the earth if it were compressed to a ball 200 m in diameter. With Avogadro's number ( N a =6.022 x 10 23 ), the mass of an individual atom is ( 62 )
( ) Since the atomic radius of about 2 x 10-8 cm is 10 5 times greater than the nuclear radius, the nucleus occupies only about 10-15 of the volume of the atom. Example: Estimate the mass of an atom of 238 U. M( 238 U ) = We have the mass of 238 U is found to be 238.050782 u which is the atomic weight, therefore: M( 238 U ) = ( ) = 3.952925 In many calculations, we will need to know the number of atoms in 1 cm 3 of a substance. The atom density N is: N atoms cm 3 ρ A N a N = atom density ( atoms/cm 3 ) = density ( g/cm 3 ) N = Avogadro's number (6.022 x 10 atoms/mole) M = gram atomic weight ( 63 )
Example: Calculate the atomic density of 13Al 27 if the gram atomic weight of aluminum is 26.9815 g and its density is 2.699 g/cm 3. N = Example: What is the hydrogen atom density in water? For water, the molecular weight A is A(H 2 O) = 2A H + A O = 18 and the molecular density is: N ( H 2 O ) = = = 3.35 x 10 22 molecules/cm 3 The hydrogen density N ( H ) = 2N( for water ) = 2 x 3.35 x 10 22 = 6.69 x 10 22 atoms/cm 3 ( 64 )
Radiation types: Alpha radioactivity. Alha particles consist of two protons and two neutrons bound together into a particle identical to a helium nucleus. Because of its very large mass (more than 7000 times the mass of the beta particle) and its charge, it heavy ionizes material and has a very short range. Beta radioactivity. Beta particles are high-energy, high-speed electrons or positrons emitted by certain types of radioactive nuclei such as potassium-40. The beta particles have greater range of penetration than alpha particles, but still much less than gamma rays. Gamma radioactivity. Gamma rays are electromagnetic radiation of an very high frequency and are therefore high energy photons. They are produced by the decay of nuclei as they transition from a high energy state to a lower state known as gamma decay. Neutron emission. Neutron emission is a type of radioactive decay of nuclei containing excess neutrons (especially fission products), in which a neutron is simply ejected from the nucleus. This type of radiation plays key role in nuclear reactor control, because these neutrons are delayed neutrons. ( 65 )
( 66 )
Types of reactions: 1 - Elastic scattering: In this case, the incident particle strikes the target nucleus and leaves without losing energy,but its direction may change. The elastic collision involves conservation of momentum and kinetic energy without dissipative force. 2 - Inelastic collision: In this type of nuclear reaction,the kinetic energy is not conserved but a part of the energy of the incident particle is taken up by the target nucleus which is excited to a higher quantum state and then later it decays to the ground state radiating the excess energy in the form of γ photon. ( 67 )
Binding Energy: The difference between the actual mass and the mass of all the individual nucleons is called the total binding energy B.E. It represents the work necessary to dissociate the nucleus into separate nucleons, or the energy which would be released if the separated nucleons were assembled into a nucleus. ` B E tot A, Z [ Zm p + Nm n z M A ] c 2 The average binding energy is given by: B E ave A, Z B E B E tot A,Z A ( 68 )
Example: Calculate the total and average binding energy for deuterons ( 1 H 2 or 1 d 2 ) and α particles. Then calculate the separation energy of α particles from 92 U 238. The mass of deuterons 1 H 2 is 2.0140 u The mass of Neutron = 1.008664 u The mass of Proton = 1.007276 u 1.008664 + 1.007276 = 2.01594 u 2.0159-2.0140 = 0.00194 u ( mass defect ) = 0.00194 X 931.49 = 1.8070906 MeV, = = 0.9035453 MeV/nucleon The Bohr model of H atom: As described in chapter one, a simple definition of Bohr s atomic model is: electrons orbit the nucleus at set distances. When an electron changes orbits, the energy difference between the initial and final orbit is emitted by the atom in bundles of electromagnetic radiation called photons. Bohr model was proposed in 1913 by Niels Bohr and was really an expansion on the Rutherford model of 1911. The Rutherford model had several flaws that the Bohr model overcame. ( 69 )
2.2.1 - Bohr s Postulates: Various postulates of Bohr s atomic model are: 1- In an atom, the electrons revolve around the nucleus in certain definite circular paths called orbits, or shells. 2 - Each shell or orbit corresponds to a definite energy. Therefore, these circular orbits are also known as energy levels or energy shells. 3 - The orbits or energy levels are characterized by an integer n, where, n can have values 1, 2, 3, 4 etc. The orbits are numbered as 1, 2, 3, 4 etc., starting from the nucleus side. Thus, the orbit for which n=1 is the lowest energy level. The orbits corresponding to n = 1,2,3,4..etc. When the electron is in the lowest energy level, it is said to be in the ground state. 4 - The electrons present in an atom can move from a lower energy level (E lower ) to a level of higher energy (E higher ) by absorbing the appropriate energy. Similarly, an electron can jump from a higher energy level (E higher ) to a lower energy level (E lower ) by losing the appropriate energy. The energy absorbed or lost is equal to the difference between the energies of the two energy levels, i.e., ΔE= E higher - E lower Coulomb attraction force = centrifugal force ( 70 )
Centrifugal force F c and Coulomb force F e are respectively: We know that ω = v / r, therefore when the two forces are equal: Or v e πε m r The corresponding energy is the sum of the kinetic and the potential energies of the electrons: E = E k +E p Where the kinetic energy, as usual, is given by ( ½ ) m 0 v 2 or ( ½ ) m 0 r 2 ω 2. The potential energy is defined as the work which one obtains on allowing the electron to approach the nucleus under the influence of the Coulomb force from infinity to a distance r. Since the work is defined as the product of force and distance, and the Coulomb force changes continuously with the distance from the nucleus, we must integrate the contributions to the work along a differential path dr, this gives ( 71 )
The minus sign because E p = 0 at r = E p as a binding energy, may be seen to be negative, with the zero point being the state of complete ionization. The total energy is thus found to be: Substituting we get: E e πε r ( 72 )
Example: Experiments indicate that 13.6eV is required to separate a hydrogen atom into a proton and an electron, that is, its total energy is E = -13.6 ev. Find the orbital radius and velocity of the electron in a hydrogen atom. Solution: 13.6 ev = 13.6 x 1.6 x 10-19 = 2.2 x 10-18 J r = 5.3 x 10-11 m The electron velocity can be found from = 2.2 x 10 6 m/s ( 73 )