A neutral phosphorus atom has protons and electrons

PHYSICS · Multi-part · Fermi University Physics · Atomic, Molecular & Optical Physics · Many-electron atoms and the periodic table · Many-electron atoms and the periodic table

Deriving the Spectrum, Ch The central field and the periodic table

Problem

A neutral phosphorus atom has

Z=15

protons and

15

electrons. In the central-field approximation, one electron at distance

r

from the nucleus moves in a spherically averaged potential

U(r)

produced by the nucleus and the other

14

electrons. ((a)) Using Gauss's law on a sphere of radius

r

centred on the nucleus, find the effective charge

Z_{\rm eff}(r)

(in units of

e

) that the electron sees in the limit

r\to0

. ((b)) Find

Z_{\rm eff}(r)

in the limit

r\to\infty

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