Corpuscular Intensity - Report Example

Nuclear Energy Report Name: Abstract This report is a study of the particles involved in nuclear energy. The atomic and nuclear energy levels of the finite, discrete energy values of alpha particles and gamma photons involved in the decay of unstable elements have been discussed with regard to nuclear energy. The report also studies a special case of beta decay mechanism and continuous beta particle energy spectrum of elements. These beta decay mechanisms include; forbidden decay and double decay. Key words: atomic energy levels; nuclear energy levels; discrete energy; beta decay; double beta decay; forbidden decay; continuous beta particle energy spectrum. 1.0 Atomic and Nuclear Energy Levels, and origin of the finite and discrete energy values of alpha particles and gamma photons involved in the decay of unstable elements. Atoms and nucleus in physical systems exist with certain values of energy that are characteristic of each physical system. Atoms are made of protons, located in the nucleus at the center of the atom, and electrons that orbit around the protons. In atomic physics, a principal energy level, or an electron shell may be described as an orbit around the atom nucleus along which electrons follow (Turner, 2008). The shells are labeled with corresponding quantum numbers. Given that electrons are attracted to the nucleus by electrical forces, the electrons in an atom will generally occupy the inner shells, and only occupy the outer shells when the inner shells are already filled up. This is however, not a strict requirement since atoms can have incomplete outer shells. If the potential energy is zero at an infinite distance from the nucleus of an atom, then the bound electron states possess negative potential energy. An atom at the lowest energy level and its electrons is said to be at ground state (Tsoulfanidis, 2013). If the atom is at a higher energy level, then it is said to be excited. The energy levels are said to be degenerate if more than a single quantum mechanical state have the same energy. If an atom has more than one electron around it, the interactions between the electrons raise the energy level. However, these interactions are normally neglected if there is a low spatial overlap of the wave functions of the electrons (Smit, Lindsay, & Förtsch, 2012). Like the atom, the nucleus has discrete energy levels that are governed by quantum mechanics. A quantum mechanical system can only take on given discrete energy values. Each nucleus has a different location of excitation states and the excitation energy is dependent on the internal structure of a given nucleus of an atom (Sarkar, 2007). Each of the excited states is characterized by certain quantum numbers that sufficiently describe its parity, angular momentum, and isospin. According to a proposal by Niels Bohr, electrons in an atom occupy only given specific orbitals which have specific energy values. This means that an electron does not have continuous energy, but “quantized” energy. This is to say that the energy possessed by electron in an atom is never continuous, but rather quantized. These discrete energy values that correspond to each of the specifically allowed orbitals are referred to as energy levels (Rajput, 2009). The energy spectrum of such a system with discrete energy levels is termed as a quantized system. Quantized energy levels originate from the interaction between energy of a particle and its wavelength. Confined particles, like electrons in atoms, the wave function is in the form of standing waves. Each of the energy levels is labeled using a quantum number and it is possible to determine its energy by the following: ( Where: - The element’s Rydberg constant. For instance, in hydrogen atom, the energy levels are given by: A negative energy level indicate that the electron is trapped or bound in the atom. In case the electron escapes, an ion is forms with or. When an electron moves from a higher energy level to a lower energy level, a photon is emitted, which carries away the jump energy. The energy levels exhibited by a hydrogen atom explain why the light spectrum emitted by hydrogen is characterized by discrete lines. An energy level of an atom can consist of a number of quantum states that depend on angular moment and spin. As a result of Pauli Exclusion Principle, electrons in an atom must move into higher energy levels before occupying the lower energy levels. Electrons in atoms may undergo transitions in energy levels by emission or absorption of a photon whose energy is equal to that of the difference in energy between the two levels. Transition of energy levels can also occur non-radiatively, meaning that absorption or emission of a photon is not involved. 1.2 Special Case of Beta Decay Beta decay affects the highest number of nuclei. This decay mechanism is very important in a number of applications, including PET imaging and betavoltaics. There are two types of beta decay – beta minus (β-) and beta plus (β-). Beta minus leads to emission of an electron accompanied by emission of antineutrino, while beta plus leads to emission of a positron accompanied by emission of an electron neutrino (Civitarese, 2014). There are three kinds of particle decays; the Fermi decays, Gamow-Teller decays and Forbidden decays. Fermi decays and Gamow-Teller decays are referred to as “super-allowed” and “simple allowed” decays. Forbidden decays are substantially more improbable as a result of parity violation and change in orbital angular momentum, leading to longer decay times (Arfken, 2012). Beta decay processes may be classified depending on the L-value of the radiation it emits. The decay is classified as forbidden decay if the value of L > 0. Allowed transitions occur when the required conditions are full filled, i.e., retaining the 1st term of unity of the exponential wave function. However, in a case where these conditions are not full filled, the matrix element becomes zero (Kingery, 2011). For a beta transition to continue, higher order terms of the exponential wave function have to be included. These terms usually correspond to beta transitions where the lepton pair takes away a finite orbital angular momentum with L-value = 1, 2, 3…etc. The transitions are classified as 1st forbidden, second forbidden, etc. However, these subsequent higher beta transitions are hindered by allowed transition by a given factor that increases rapidly as the decrease in forbidden-ness increases. In some cases, a nuclei may undergo a double β decay, where the nucleus charge change by two units. Double decay mechanism is characterized by long half-life. In a nucleus where both beta and double beta decay are likely to occur, the double beta decay is not effectively possible to observe (Murray & Holbert, 2014). In a nuclei where beta decay is forbidden, but double beta decay is allowed, the process can be observed and half-life of the decay mechanism determined. Double beta decay process does not alter the mass number of an atom, thus, a given nuclide with a certain mass number has to be stable in both single and double beta decays. 1.3 Continuous Beta Particle Energy Spectrum Beta decay is a process that clearly follows 1st order kinetics, and therefore, a single constant can be used to describe the rate of decay. It has been experimentally observed that beta decay occurs within a big range of half-lives, from milliseconds to about 1016 years. The beta decay process involves creating two particles and is not continuous because the emission of electrons occur as a stream of discrete particles. However, the emitted electrons have a continuous kinetic energy spectrum. The kinetic energy ( ) range from zero to the maximum energy available, unlike in the case of predictable energy ofparticles. This continuous energy spectrum is due to the fact that is shared between the antineutrino and the β particle electron. The β takes part of the energy and an antineutrino also takes part of the energy (Murray & Holbert, 2014). Thus, the energy is divided between the two particles, and each of the particles has a continuum of energies. A typical value of energy is about 1 MeV, although it can range from KeV to tens of MeV. Most of beta particles have speeds that are closer to that of light. The two particles add up to equal discrete amount. Acknowledgements I wish to acknowledge my professor for the knowledge acquired on nuclear energy, specifically nuclear elements and decay mechanisms. I also acknowledge the effort by authors of materials listed in the reference list below. 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