Atomic and Nuclear Radii

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Hilary Hawkins
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1 Atomic and Nuclear Radii By first approx. the nucleus can be considered a sphere with radius given by R 1.25 x A (1/3) {fm} A atomic mass number, fm m Since the volume of a sphere is proportional to R 3 Mass and Energy One of the striking results of Einstein s theory of relativity is that mass and energy are equivalent and convertible to one another. The complete annihilation of a particle or body of rest mass m o releases an amount of energy E rest m o c 2. For 1 g of matter E rest m o c kg (3 x 10 8 ) 2 9 x J 25 x 10 6 Kw-Hr Note: 1 Kw-Hr 3.6 x 10 6 J Burning 1 g of coal ~ 1 x 10-3 Kw -Hr Mass and Energy Electron- volt (ev): unit of energy equal to the increase in kinetic energy of an electron when it falls through and electric potential of 1 volt. So 1 ev x coulombs x 1 volt x joules (tiny amount) Note: 1 MeV 10 6 evand 1 KeV 10 3 ev Ex.. Compute the rest mass energy of 1u? Since 1u 1.66 x g Then E rest m o c 2 (1.66 x kg)(3x10 8 ) 2 / x MeV Kinetic Energy When a body is in motion its rest mass increases relative to an observer at rest Note: 1. for v 0, m m o 2. for v c, m E total E rest + E kinectic m c 2 E kinectic E total -E rest m c 2 -m o c 2 m o c 2 [ 1] Eq.1 1

2 Kinetic Energy binomial expansion (1+) 1+ So (1 ) 1+! Thus E kinectic m o c 2 [ E kinectic m ov 2 Eq. 2! +()! 1+ We can use Eq. 2 instead of Eq. 1 when; 1] m o c 2 (1+ + v 2 << c 2 or v c or v < 0.1 c ***Use Eq. 1 for electrons, but Eq. 2 for neutrons.*** -1) Photon Energy Photons and neutrinos have no rest mass and thus no KE. We compute the energy associated with these particles differently. E hν h planks constant (4.13 x ev-sec) ν frequency of EM wave (Hz) -Particle wavelength for particle with zero rest mass. λ. Photon Emission In a neutral atom (not ionized) it is possible for the electrons to be in a variety of orbits or states. The lowest state is the ground state. When at atom possesses more energy than it s ground state, it is said to be in an excited state or energy level. The highest energy state corresponds to the situation where the electron has been completely removed from the atom and the atom is ionized. An atom can not remain in an excited state indefinitely, it eventually decays to a state of lower energy, and a photon of equal energy will be released. Ex An ionized atom returns to ground state after releasing an 88 KeVphoton. What is the wavelength of the photon? λ x 10 m Nucleons are also thought to have orbits and energy levels. These energy levels are much larger than that of electrons. The photons emitted during Nuclei decay have much greater energy and are called gamma rays-γ. 2

3 Nuclear Stability and Radioactive Decay Chart of Nuclides B + Decay {Z } If too few neutronsa proton is transformed into a neutron and positron and a neutrino are emitted. Nuclides with Z > 20 (beyond calcium) have more neutrons than protons. The excess neutrons are required to balance the strong repulsive force between protons. If there are too many or two few neutrons the resulting nucleus is unstable, and it will undergo radioactive decay. Ex.. O-15 with 7 neutrons and 8 protons decays to nitrogen. + ν B - Decay {Z } If too many neutronsa neutron is transformed into a proton and an electron and a neutrino are emitted. B Decay Note: In both B+ and B-decay the atomic mass number (A) remains the same. Typical decay chain: Ex.. O-19 with 11 neutrons and 8 protons decays to fluorine, with 10 neutrons and 9 protons. + ν () daughter nucleus 3

4 Electron Capture In this process an atomic electron interacts with one of the protons in the nucleus and a neutron is formed. The electron is later replaced. e Alpha Decay (α) Large atomic nuclei may undergo radioactive α decay by the emission of an alpha particle. An αparticle is the highly stable nucleus of the isotope { 2 neutrons and 2 protons} P P P N This reduces Z by 2 and A by 4 Example + + α Radioactive Decay notes 1. The nucleus formed as the result of B+, B-, electron capture, or αdecay are often in an excited state. This excited daughter often emits subsequent ϒ rays. 2. Strictly speaking the term decay should not be used to describe the emission of ϒrays from excited nuclei, since only energy and not the character of the nucleus changes. However, it is the custom. 3. Radioactive nuclei decay via B, α, and ϒ emissions and NOT by emitting neutrons and protons. RADIOACTIVITY CALCULATIONS The probability per unit time that a nucleus will decay is constant. λ decay constant. This lead to an expression for the number of atoms yet to decay n(t) in terms of the number atoms at time 0 (n o ). n o t 4

5 Radioactive Decay Activity α(t) : the decay rate of a sample measured in disintegrations per unit time. α λ λ {units curies Ci} 1 Ci 3.7 x disintegrations/sec Note: λ λ Half-Life { } The time during which the activity decreases by a factor of 2. Solving for 2 2 (). Production and Loss Assume a radioactive nuclide is produced in a Rx at the constant rate R ( ) The time rate of change of the nuclide the rate of production the rate of loss + (1 ) ** Multiply by λgives α α + (1 ) ** see next slide Production and Loss The time rate of change of the nuclide the rate of production the rate of loss is found from> () Solve: () λ() () λ() λ, λ, λ 5

6 > R R < R Production and Loss α α + (1 ) t Nuclear Reactions When two nucleon combine to form two or more nuclear particles + + Laws 1. Conservation of nucleons 2. Conservation of charge 3. Conservation of energy 4. Conservation of momentum Conservation of Energy Rearranging ( + ) + [ + + ] Define the Q of the reaction as the change in the rest mass energy in MeV [ + + ] Q of the reaction Recall: 1u 1.66 x g 932 MeV So If Q>0 : Net increase in KE reaction is exothermic If Q<0 : Net decrease in KE reaction is endothermic 6

7 Ex Complete the following reaction +?+ 1. Z (atomic#) for Nitrogen 7 2. Z for n 0 3. Z for H 1 4. Z for? 7+0?+1 >? 6C 5. A for N A for n 1 7.A for H 1 8. A for? 14+1?+1 >? Ex Find the Q of the reaction Solution.. M(N14) M(C14) M(n) M(H1) u u Q ( )932MeV +3.9 MeV exothermic Mass is converted to KE 7

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