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Muonic atoms and the size of the nucleus

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Muonir atoms and the size of the nucleus. The negative muon (symbol pf) is a particle with the same charge as the electron but with a larger mass (m = 207 m,). High-speed muons are produced in violent nuclear collisions. These muons can be slowed down in matter and captured into orbits around the nuclei of atoms in the material. The resulting system. with a negative muon in the inner orbit. is called a muonic atom. After about 2 microseconds (on the average) the muon decays into an electron and two neutrinos. destroying the muonic atom. Despite this brief existence. the muonic atom can be considered stable since its lifetime corresponds to very many periods of revolution of the muon in a Bohr orbit about the nucleus. Because of its large mass relative to the electron. the inner orbit of the muon lies very much closer to the nucleus than that of any electron. This makes it possible to use muonic atoms to probe the size and structure of nuclei.
(a) For a central charge Ze. obtain an expression for the radius r,, of the nth muonic orbit. Express this as a multiple of the radius 0,, of the first Bohr (electron) orbit in hydrogen.
(b) For the muonic orbit of (a). what is the energy E, expressed as a multiple of the energy of the lowest state of an electron in the hydrogen atom?
(c) For Z = 13 (aluminum) calculate the energy in keV and the wavelength in Angstroms of the n = 2 —> n = l muonic x-ray transition.

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Muonic atoms and the size of the nucleus are determined. The central charge obtained by an expression is provided.

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