Mass and Energy

1. Mass and Energy • Mass and energy are two different forms of the same thing. • Einstein’s theory of relativity demonstrated that the conversion from one to other was propositional to the square of the speed of light, c. • “E” is usually interpreted as rest energy and “m” is the rest mass • Rest mass is the mass of an object that is stationary relative to the observer.

2. Mass and Energy If one gram of material (any material) could be completely converted to energy it would provide the equivalent of all the electrical needs for 10,000 homes for 1 month, or be equivalent to the energy released by the nuclear weapon dropped on Nagasaki in 1945.

3. Mass and Energy • When a body is in motion its mass increases relative to a stationary observer. • The total energy of a particle in motion is the sum of its rest energy and its kinetic energy. • When “v “is much smaller than “c”, as is usually the case, then the total energy is just the usual expression for kinetic energy.

4. Mass and Energy • Electrons have a rest mass energy of 0.511 MeV and relativistic energy must be considered since kinetic energies can exceed this value. • Neutrons are much larger than electrons, with a rest mass equivalent of nearly 1000 MeV and for most problems of interest relativistic corrections are not required, and E=½mv2 may be used.

5. Particle Wavelength • Since photons (gamma rays) travel only at the speed of light and have no rest mass, kinetic energy has no meaning. Rather the total energy is given by • Where “h” is Planck’s constant and “ν” (lower case Greek Nu) is the frequency of the electromagnetic wave associated with the photon.

6. Particle Wavelength • All particles have an associated wavelength: • For particles with non-zero rest mass • For non-relativistic energies, where E is the kinetic energy: m0 is the rest mass.

7. Particle Wavelength • The non-relativistic equations are appropriate for neutrons in reactor calculations since they have such a large rest mass and kinetic energies do not typically exceed that value. • Substituting the value for neutron mass yields the following equation for neutron wavelength in centimeters, where energy is expressed in eV. ME 461

8. Particle Wavelength • For particles with a zero rest mass, the momentum is given by the relativistic equation • Where “E” is the energy of the particle and “c” is the speed of light. • The wave length for the relativistic case is then: ME 461

9. Excited States Typical visualizations of the atom with the nucleus (neutron & protons) surrounded by electrons in well defined orbits. ME 461

10. Excited States • Removing an electron from its orbital is called ionization. • Inner orbitals are more tightly bound to the nucleus than outer orbits and require higher ionization energies. • Atoms always tend to the lowest possible energy state, the ground state. • If an atom has more energy than the ground state because of high energy electrons in higher orbits the atom will emit an x-ray as the higher energy electron “drops” to a lower energy inner orbit. ME 461

11. Excited States • A similar process can occur in the nucleus. The neutrons and protons exist in “orbits” in the nucleus and will always tend to the lowest possible energy configuration. • If a nucleon is in a higher energy orbit, the nucleus will emit a gamma-ray ( ) as the nucleon transitions to a lower energy orbit. • The transition energies for nuclei are typically much higher than transition energies for atoms (electrons). • X-rays…. Electron transitions • …. Nucleon transitions ME 461

12. Radiation • Neutrons act as “glue” to maintain the stability of heavier nuclei where the repulsive forces of the large number positive protons would tend to push the nucleus apart. • In many cases there either too many or too few neutrons in the nucleus and stability is achieved by emitting various forms of radiation. • The nucleus will achieve stability by emitting α-rays or β-rays or undergo electron capture. • All of these processes may be followed by the further emission of γ-rays. ME 461

13. Radiation Alpha particles are “heavy”, high energy, positively charged but not very penetrating. Can be stopped by a piece of paper Beta particles (positive or negative) can be high energy , but can be easily stopped with plastic or light metals. Gamma-rays have an energy range from a few keV to a few MeV and are very penetrating. Typically travel 500 feet in air without an interaction. ME 461

14. Radiation Generalized decay scheme Co-60 decay scheme ME 461

15. Radioactivity Calculations • One fundamental law that governs all radioactive decay processes: • That constant is called the decay constant , λ, with units of inverse time. The probability per unit time that a nucleus will decay is a constant independent of time ME 461

16. Radioactivity Calculations ME 461

17. Radioactivity Calculations • Activity is defined as the decay rate of the sample, disintegrations per second. This is the fundamental law of radioactive decay ME 461

18. Radioactivity Calculations • Activity units; • Curie = 3.7x1010 disintegrations per second • Millicurie = 10-3 curie • Microcurie = 10-6 curie • In SI units, the Becquerel, Bq = 1.0 disintegration per second. • The time it takes for the activity to decrease by a factor of two is called the half-life, ,of the isotope in question. ME 461

19. Radioactivity Calculations • There are many situations where a radioactive isotope is produced in a reactor, or by some other means, that then decays to another radioactive isotope, etc., creating a decay chain. See Fig. 2.7 in the text and note the exponential behavior of radioactive decay ME 461

20. Radioactivity Calculations • The time rate of change of activity, or atoms, of any member of the chain is always described by a simple law of conservation. • Consider the first member of a chain that is produced at the rate, R, atoms/sec, then ME 461

21. Radioactivity Calculations There are two possible initial conditions If n(0) = 0 If n(0)=n0 No matter what the initial condition is, after a long time both situations achieve the same equilibrium value, R/λ. ME 461

22. Radioactivity Calculations • Decay Chains “A” decays to “B”. “ B” decays to “C”. “C” is stable. Assume there that “C” doesn’t exist at time zero, but there is nA0 and nB0 at time zero. There is no external production, except for the decay of the previous member of the chain itself. Time rate of change equals production minus loss For “A” ME 461

23. Radioactivity Calculations For “B” ME 461

24. Radioactivity Calculations For “C” ME 461

25. Decay Chains ME 461

26. Similar half lives – linear Plot ME 461

27. Similar half lives - semi log Plot ME 461

28. T½(B)<< T½(A) Linear Plot ME 461

29. T½(B)<< T½(A) Semi Log Plot ME 461

30. T½(B)>> T½(A) Linear Plot ME 461

31. T½(B)>> T½(A) Semi Log Plot ME 461

32. Decay Chains ME 461