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Transcript
D EPARTMENT OF P HYSICS , C OLORADO S CHOOL OF M INES PHGN 422: Nuclear Physics γ Decay Supplement Kyle G. Leach October 21, 2016 1 I NTRODUCTION Nuclei that are in an excited state generally decay via the emission of a γ ray or through internal electron conversion to decrease the energy of the nucleus. The transitions can occur between two excited states or an excited state and the ground state. This decrease in energy does not change the isotope, it merely reconfigures the nucleons within the nucleus. In the γ-decay process, electromagnetic radiation of a specific energy is released when the nucleus undergoes a transition from an excited state to a lower energy state. With internal conversion, the energy that would be emitted through electromagnetic radiation instead liberates an atomic electron, causing it to move into an unbound state. These two processes generally compete with each other. 1.1 γ R ADIATION E NERGETICS In order to understand the energy transfer in γ decay, we must consider the initial and final states of the nucleus as well as its recoil momentum. Using conservation of energy and momentum, we obtain: Conservation of Momentum: Conservation of Energy: p~R + p~γ = 0 E i = E f + E γ + TR , 1 where: TR = p R2 2M x = p γ2 2M x = E γ2 2M x c 2 . Therefore; ∆E = E γ + E γ2 2M x c 2 , (1.1) where ∆E = E i − E f is the difference in the energies of the nuclear states involved. This expression can be solved for the γ-ray energy, µ ¶ ∆E E γ ≈ ∆E 1 − . (1.2) 2m x c 2 We see that the energy released in the electromagnetic transition is slightly less than the energy difference between the initial and final nuclear states due to the recoil of the daughter nucleus. 1.2 γ-R AY T RANSITION M ULTIPOLARITY The emitted γ-ray photon can be understood in a simple model using classical electrodynamics, where a radiation field can be described in terms of a multipole expansion. This classical theory is then supplemented using a quantum mechanical description for the electric and magnetic multipoles in terms of the power radiated for each: λe (L) = P e (L) ~ω and λm (L) = P m (L) , ~ω (1.3) where the subscripts m and e represent magnetic and electric multipoles, respectively, L is the multipolarity of the respective transition (see Table 1.1), and E γ = ~ω, where ω is the angular frequency of the oscillating field. We can then expand the decay constants in multipoles by expanding the radiation powers. In order to obtain order of magnitude expressions for the expected transition rates we can make a number of approximations. These are referred to as the Weisskopf estimates. They are i) that the initial and final states are given by the single particle wave functions ψi = R i (r )Y`i mi (θ, φ) and ψ f = R f (r )Y` f m f (θ, φ), and ii) the radial terms in the wavefunctions are constant over the entire nuclear volume, and zero elsewhere. Once these approximations are made, the electric and magnetic multipole decay constants can be expressed as [1]: · ¸ µ ¶ 2e 2 (L + 1) 3 2 E γ 2L+1 2L λe (L) = R , 4π²0 ~L[(2L + 1)!!]2 L + 3 ~c (1.4) and λm (L) = 20e 2 ~(L + 1) 4π²0 c 2 m p2 L[(2L + 1)!!]2 · 3 L +3 ¸2 µ Eγ ~c ¶2L+1 R 2L−2 , (1.5) 2 Table 1.1: Properties and nomenclature for electromagnetic multipole radiation. [1] Radiation Nomenclature Symbol Multipolarity (L) Parity Electric Dipole Magnetic Dipole Electric Quadrupole Magnetic Quadrupole Electric Octopole Magnetic Octopole . . . E1 M1 E2 M2 E3 M3 . . . 1 1 2 2 3 3 . . . -1 +1 +1 -1 -1 +1 . . . where R is the nuclear radius, and E γ is expressed in MeV. The total angular momentum (L) of the photon is subject to selection rules, which are related to the angular momentum of the initial and final nuclear states by |~ Ji − ~ J f | ≤ L ≤ |~ Ji + ~ Jf | where L = 1, 2, 3... . (1.6) It is important to note that the angular selection rules do not include 0 → 0 transitions since they can only be satisfied with L = 0, and there are no L = 0 photons. There are also parity selection rules that are dependent on the angular momentum of the photon, For E L transitions, For M L transitions, πi = π f (−1)L πi = π f (−1) L+1 . (1.7) (1.8) For various initial and final nuclear spin and parity states there are in general a number of allowed γ-ray transitions that can occur. In the case where the lowest multipole permitted by the selection rules is electric, it will dominate the decay. If the lowest allowed multipole L is magnetic, there will, in general, be a competition between M L and E (L +1) multipole radiation. 1.3 I NTERNAL C ONVERSION The internal conversion decay constant is, in general, a sum of the decay constants for the conversion of electrons from the various atomic shells (K, L, M, etc.). As mentioned previously, this process competes with photon emission, which implies that the total decay constant for a transition between the initial and final nuclear states is a sum of the γ and internal conversion decay constants, λ = λe + λγ , where the γ-decay constant is given above for M L and E L transition. The internal conversion coefficient, α, is defined as the ratio of the decay constant for electron 3 conversion to the decay constant for γ emission, α= λe , λγ (1.9) which is then expressed in terms of the total decay constant λ = λγ (1 + α). (1.10) The internal conversion coefficients can be calculated theoretically for each atomic shell [2]. An internal conversion coefficient, combined with a measurement of the γ-decay constant, will therefore yield the total electromagnetic decay constant. R EFERENCES [1] Richard A. Dunlap. The Physics of Nuclei and Particles. Thomson Learning. Brooks/Cole, 2004. [2] T. Kibédi et. al. BRICC 2.0b. http://www.nndc.bnl.gov/bricc, 2003. 4