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Physics 228 Spring 2008
ManyElectron Atoms
Due at 11:59pm on Monday, April 14, 2008
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ManyElectron Atoms
Learning Goal:
To understand how electrons are placed into subshells as atomic number increases and how this
leads to the trends of the periodic table.
The Schrödinger equation was highly successful at describing the hydrogen atom. Unfortunately, for any other atom
the equation becomes too complex to solve, because of the interactions of the electrons with one another. Luckily,
approximations can be made to get some useful results, including an explanation of the periodic table of elements!
The most useful approximation is called the
central field approximation
. In the central field approximation, the
potential energy is described by a spherically symmetric function (i.e., a function dependent only on distance from
the center of the atom). In this approximation, it is assumed that the electrons closest to the nucleus (
)
experience a force from the full charge on the nucleus, but electrons further out experience force from only a fraction
of that charge, because the inner electrons "screen" the nucleus.
To understand this, think of the probability distribution of the electrons as a spherically symmetric charge
distribution. From Gauss's law, you know that the electric field at a distance
from the nucleus will be proportional
to the net charge contained within a sphere of radius
. Since such a sphere will always contain the positive charge
on the nucleus, increasing
leads to a decreased net charge, as more of the charge from the electrons is included to
cancel the nucleus's charge.
This approximation tells us several important things. First, since the potential is spherically symmetric, the electrons
must have the same angular wave functions
and
as the hydrogen atom. This means that the same
quantum numbers
and
will be used to describe electrons in manyelectron atoms, and that the same rules that
apply to these numbers in hydrogen will apply in manyelectron atoms.
Part A
Which of the following gives the correct permitted values of
for
?
ANSWER:
Page 1 of 14
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5/8/2008
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A second important result is that electrons will fill the lowest energy states available. This would seem to indicate
that every electron in an atom should be in the
state. This is not the case, because of
Pauli's exclusion
principle
. The exclusion principle says that no two electrons can occupy the same state. A state is completely
characterized by the four numbers
, ,
, and
, where
is the spin of the electron.
An important question is, How many states are possible for a given set of quantum numbers? For instance,
means that
with
are the only possible values for those variables. Thus, there are two possible states:
and
. How many states are possible for
?
Express your answer as an integer.
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This note was uploaded on 11/25/2010 for the course PHYSICS 228 taught by Professor Staff during the Spring '08 term at Rutgers.
 Spring '08
 Staff
 Physics

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