MANY -ELECTRON ATOMS AND THE EXCLUSION PRINCIPLE

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MANY -ELECTRON ATOMS AND THE EXCLUSION PRINCIPLE

So far, our analysis of atomic structure has concentrated on the hydrogen atom. That’s natural; neutral hydrogen, with only one electron, is the simplest atom. If we can’t understand hydrogen, we certainly can’t understand anything more complex. But now let’s move to many-electron atoms. In general, an atom in its normal (electrically neutral) state has Z electrons and Z protons.

Recall from Section 43-2 that we call Z the atomic number. The total electric charge of such an atom is exactly zero because the neutron has no charge while the proton and electron charges have the same magnitude but opposite sign. We can apply the Schrodinger equation to this general atom. However, the complexity of the analysis increases very rapidly with increasing Z. Each of the Z electrons interacts not only with the nucleus but also with every other electron. The wave functions and the potential energy are functions of 3Z coordinates, and the equation contains second derivatives with respect to all of them. The  athematical problem of finding solutions of such equations is so complex that it has not been solved exactly even for the neutral helium atom, which has only two electrons.

Fortunately, various approximation schemes are available. The simplest approximation is to ignore all interactions between electrons and consider each electron as moving under the action only of the nucleus (considered to be a point charge). In this approximation the wave function for each electron is a function like those for the hydrogen atom, specified by four quantum numbers (n, 1,m ” m,); the nuclear charge is Ze instead of e. This requires replacement of every factor of iin the wave functions and the energy levels by Ze2.1n particular, the energy levels are given by Eq. (43-3) with e4 replaced by this approximation is fairly drastic; when there are many electrons, their interactions with each other are as important as the interaction of each with the nucleus. So this model isn’t very useful for quantitative predictions.

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MANY -ELECTRON ATOMS AND THE EXCLUSION PRINCIPLE