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Corso di Fisica Moderna a.a. 2011/12 2nd semestre The Stability of the nuclear atom At the center of the atom is a nucleus whose mass is approximately that of the enAre atom and whose charge is equal to the atomic number Z Ames e; around this nucleus there exist Z electrons, neutralizing the atom as a whole. 1)  We have an atom which would rapidly collapse (for a diameter of 10-­‐10 m the Ame of collapse is ≈ 10-­‐12 sec) 2)  The conAnuous spectrum of radiaAon that would be emiNed in this process is not in agreement with the discrete spectrum which is known to be emiNed by atoms OpAcal spectrum of hydrogen atom With the use of spectroscopy in the late 19th century, it was found that the radiaAon from hydrogen, as well as other atoms, was emiNed at specific quanAzed frequencies. It was the effort to explain this radiaAon that led to the first successful quantum theory of atomic structure, developed by Niels Bohr in 1913. He developed his theory of the hydrogenic (one-­‐
electron) atom from four postulates: 1)  An electron in an atom moves in a circular orbit about the nucleus under the influence of the Coulomb aNracAon between the electron and the nucleus, obeying the laws of classical mechanics. 2)  Instead of the infinity of orbits which would be possible in classical mechanics, it is only possible for an electron to move in an orbit for which its orbital angular momentum L is and integral mulAple of h, Planck’s constant. 3)  Despite the fact that it is constantly acceleraAng, an electron moving in such an allowed orbit does not radiate electromagneAc energy. Thus, its total energy E remains constant. 4)  ElectromagneAc radiaAon is emiNed if an electron, iniAally moving in an orbit of total energy Ei , disconAnuously changes its moAon so that it moves in an orbit of total energy Ef . The frequency of the emiNed radiaAon is equal to the quanAty (Ei-­‐Ef) divided by h. 2nd postulate: Angular Momentum QuanAzaAon In the Bohr model, the wavelength associated with the electron is given by the DeBroglie relaAonship and the standing wave condiAon that circumference = whole number of wavelengths. In the hydrogenic case, the number n is the principal quantum number. These can be combined to get an expression for the angular momentum of the electron in orbit. (Note that this assumes a circular orbit, a generally unwarranted assumpAon.) Thus L is not only conserved, but constrained to discrete values by the quantum number n. This quanAzaAon of angular momentum is a crucial result and can be used in determining the Bohr orbit radii and Bohr energies. Combining the energy of the classical electron orbit with the quanAzaAon of angular momentum, the Bohr approach yields expressions for the electron orbit radii and energies: SubsAtuAon for r gives the Bohr energies and radii: Electron TransiAons The Bohr model for an electron transiAon in hydrogen between quanAzed energy levels with different quantum numbers n yields a photon by emission with quantum energy: This is ocen expressed in terms of the inverse wavelength or "wave number" as follows: MoAon of nucleus The measured RH differs from it’s theoreAcal value R∞. The difference is 60 cm-­‐1 (1eV=8066 cm-­‐1) MoAon of nucleus Hydrogen-­‐like atoms Hydrogen-­‐like atoms Lenard’s experiment Franck-­‐Hertz experiment Lenard’s experiment Franck-­‐Hertz experiment Franck-­‐Hertz experiment: improved Rydberg atoms A Rydberg atom is an excited atom with one or more electrons that have a very high principal quantum number. These atoms have a number of peculiar properAes including 1)  an exaggerated response to electric and magneAc fields, 2)  long decay periods 3)  and electron wavefuncAons that approximate, under some condiAons, classical orbits of electrons about the nuclei. The core electrons shield the outer electron from the electric field of the nucleus such that, from a distance, the electric potenAal looks idenAcal to that experienced by the electron in a hydrogen atom. It is now apparent why Rydberg atoms have such peculiar properAes: the radius of the orbit scales as n2 (the n = 137 state of hydrogen has an atomic radius ~1 µm) and the geometric cross-­‐secAon as n4. Thus Rydberg atoms are extremely large with loosely bound valence electrons, easily perturbed or ionized by collisions or external fields. Rydberg atoms Rydberg atoms Schrodinger equaAon in a central potenAal The hydrogen atom soluAon to the Schrodinger equaAon produces three quantum numbers which can be seen to arise naturally from geometrical constraints on the wavefuncAon. the wavefuncAon takes the form which gives three equaAons. The equaAon for each of the three variables gives rise to a quantum number and the quanAzed energy states of the atom can be specified in terms of these quantum numbers. Schrodinger equaAon in a central potenAal The Angular wavefuncAon P=Legendre polynomial The Radial wavefuncAon of Hydrogen Laguerre polynomials The Radial wavefuncAon of Hydrogen Alkali atoms Alkali atoms: screening Alkali atoms: licing of orbital degeneracy