Download Ch. 41 Atomic Structure

Chapter 41
Atomic Structure
PowerPoint® Lectures for
University Physics, Thirteenth Edition
– Hugh D. Young and Roger A. Freedman
Copyright © 2012 Pearson Education Inc.
Goals for Chapter 41
•
To see how to write the Schrödinger equation for a three-dimensional
problem
•
To learn how to find the wave functions and energies for a threedimensional box
•
To examine the full quantum-mechanical description of the hydrogen
atom
•
To study how an external magnetic field affects the orbital motion of
an atom’s electrons
•
To learn about the intrinsic angular momentum (spin) of the electron
•
To understand how the exclusion principle affects the structure of
many-electron atoms
•
To study how the x-ray spectra of atoms indicate the structure of these
atoms
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Introduction
• The Bohr model, in which an atom’s electrons orbit its nucleus like
planets around the sun, is inconsistent with the wave nature of
matter. A correct treatment uses quantum mechanics and the threedimensional Schrödinger equation.
• To describe atoms with more than one electron, we also need to
understand electron spin and the Pauli exclusion principle. These
ideas explain why atoms
that differ by just one
electron (like lithium with
three electrons per atom
and helium with two
electrons per atom) can
be dramatically different
in their chemistry.
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The Schrödinger equation in 3-D
• Electrons in an atom can move in all three dimensions of space. If
a particle of mass m moves in the presence of a potential energy
function U(x, y, z), the Schrödinger equation for the particle’s
wave function (x, y, z, t) is
h2  2  x, y, z,t  2  x, y, z,t  2  x, y, z,t 



2
2


2m 
x
y
z 2
 x, y, z,t 
 U x, y, z  x, y, z,t   ih
t
• This is a direct extension of the one-dimensional Schrödinger
equation from Chapter 40.
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The Schrödinger equation in 3-D: Stationary states
• If a particle of mass m has a definite energy E, its wave function
(x, y, z, t) is a product of a time-independent wave function
(x, y, z) and a factor that depends on time but not position. Then
the probability distribution function |(x, y, z, t)|2 = |(x, y, z)|2
does not depend on time (stationary states).
  x, y, z, t     x, y, z  eiEt /
• The function (x, y, z) obeys the time-independent Schrödinger
equation in three dimensions:
  2  x, y, z   2  x, y, z   2  x, y, z  





2
2
2m 
x
y
z 2

 U  x, y, z   x, y, z   E  x, y, z 
2
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Particle in a three-dimensional box
• For a particle enclosed in a cubical box with sides of length L
(see Figure 41.2 below), three quantum numbers nX, nY, and nZ
label the stationary states (states of definite energy).
• The three states shown here are degenerate: Although they have
different values of nX, nY, and nZ, they have the same energy E.
•
Follow Example 41.1.
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The hydrogen atom: Quantum numbers
• The Schrödinger equation for
the hydrogen atom is best
solved using coordinates (r, ,
) rather than (x, y, z) (see
Figure 41.5 at right).
• The stationary states are
labeled by three quantum
numbers: n (which describes
the energy), l (which
describes orbital angular
momentum), and ml (which
describes the z-component of
orbital angular momentum).
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The hydrogen atom: Degeneracy
• Hydrogen atom states
with the same value of n
but different values of l
and ml are degenerate
(have the same energy).
• Figure 41.8 (at right)
shows the five states with
l = 2 and different values
of ml. The orbital angular
momentum has the same
magnitude L for each
these five states, but has
different values of the zcomponent Lz.
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The hydrogen atom: Quantum states
• Table 41.1 (below) summarizes the quantum states of the
hydrogen atom. For each value of the quantum number n, there
are n possible values of the quantum number l. For each value of l,
there are 2l + 1 values of the quantum number ml.
• Read Problem-Solving Strategy 41.1, and follow Examples 41.2
and 41.3.
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The hydrogen atom: Probability distributions I
• States of the hydrogen atom with l = 0 (zero orbital angular
momentum) have spherically symmetric wave functions that
depend on r but not on  or . These are called s states. Figure 41.9
(below) shows the electron probability distributions for three of
these states.
• Follow Example 41.4.
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The hydrogen atom: Probability distributions II
• States of the hydrogen atom with nonzero orbital angular
momentum, such as p states (l = 1) and d states (l = 2), have wave
functions that are not spherically symmetric. Figure 41.10 (below)
shows the electron probability distributions for several of these
states, as well as for two spherically symmetric s states.
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Magnetic moments and the Zeeman effect
•
Electron states with nonzero orbital angular momentum (l = 1, 2, 3, …)
have a magnetic dipole moment due to the electron motion. Hence these
states are affected if the atom is placed in a magnetic field. The result,
called the Zeeman effect, is a shift in the energy of states with nonzero
ml. This is shown in Figure 41.13 (below).
•
Follow Example 41.5.
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The Zeeman effect and selection rules
•
Figure 41.14 (lower left) shows the shift in energy of the five l = 2 states
(each with a different value of ml) as the magnetic field strength is
increased.
•
An atom in a magnetic field can make transitions between different
states by emitting or absorbing a photon A transition is allowed if l
changes by 1 and ml changes by 0, 1, or –1. (This is because a photon
itself carries angular momentum.) A transition is forbidden if it violates
these selection rules. See Figure 41.15 (lower right).
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The anomalous Zeeman effect and electron spin
• For certain atoms the Zeeman effect does not follow the simple
pattern that we have described (see Figure 41.16 below). This is
because an electron also has an intrinsic angular momentum,
called spin angular momentum.
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Electron spin and the Stern-Gerlach experiment
•
The experiment of Stern and Gerlach demonstrated the existence of
electron spin (see Figure 41.17 below). The z-component of the spin
angular momentum has only two possible values (corresponding to
ms = +1/2 and ms = –1/2).
•
Follow Examples 41.6 and 41.7.
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Quantum states and the Pauli exclusion principle
•
The allowed quantum numbers for an atomic electron (see Table
41.2 below) are n ≥ 1; 0 ≤ l ≤ n – 1; –l ≤ ml ≤ l; and ms = ±1/2.
•
The Pauli exclusion principle states that if an atom has more than
one electron, no two electrons can have the same set of quantum
numbers.
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A multielectron atom
•
Figure 41.21 (at right) is a
sketch of a lithium atom, which
has 3 electrons. The allowed
electron states are naturally
arranged in shells of different
size centered on the nucleus.
The n = 1 states make up the K
shell, the n = 2 states make up
the L shell, and so on.
•
Due to the Pauli exclusion
principle, the 1s subshell of the
K shell (n = 1, l = 0, ml = 0) can
accommodate only two
electrons (one with ms = + 1/2,
one with ms = –1/2). Hence the
third electron goes into the 2s
subshell of the L shell (n = 2,
l = 0, ml = 0).
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Ground-state electron configurations
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Screening in multielectron atoms
• An atom of atomic number Z has a nucleus of charge +Ze and Z
electrons of charge –e each. Electrons in outer shells “see” a
nucleus of charge +Zeffe, where Zeff < Z, because the nuclear
charge is partially “screened” by electrons in the inner shells.
• Follow Examples 41.8 and 41.9.
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X-ray spectroscopy
•
When atoms are bombarded
with high-energy electrons,
x rays are emitted. There is a
continuous spectrum of x rays
(described in Chapter 38) as
well as strong characteristic
x-ray emission at certain
definite wavelengths (see the
peaks labeled K and K in
Figure 41.23 at right).
•
Atoms of different elements
emit characteristic x rays at
different frequencies and
wavelengths. Hence the
characteristic x-ray spectrum
of a sample can be used to
determine the atomic
composition of the sample.
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X-ray spectroscopy: Moseley’s law
•
Moseley showed that the square root of the x-ray frequency in K
emission is proportional to Z – 1, where Z is the atomic number of the
atom (see Figure 41.24 below). Larger Z means a higher frequency and
more energetic emitted x-ray photons. Follow Example 41.10.
•
This is consistent with our model of multielectron atoms. Bombarding
an atom with a high-energy electron can knock an atomic electron out
of the innermost K shell. K x rays are produced when an electron from
the L shell falls into the K-shell vacancy. The energy of an electron in
each shell depends on Z, so the x-ray energy released does as well.
•
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The same principle applies to
K emission (in which an
electron falls from the M shell
to the K shell) and K emission
(in which an electron falls
from the N shell to the K
shell).