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Chapter 40
All About Atoms
In this chapter we continue with a primary goal of physics―discovering and
understanding the properties of atoms. 100 years ago researchers struggled
to find experiments that would prove the existence of atoms. Today, thanks to
scientific and technological progress, we can manipulate atoms in amazing
ways: we can image individual atoms using scanning tunneling microscopy;
we can drag them on surfaces to make quantum corrals, and even hold an
individual atom indefinitely in a trap in order to study its properties when
isolated.
(40-1)
40-2 Some Properties of Atoms
Basic Properties
Atoms are stable. Essentially all atoms have remained unchanged for billions
of years.
Atoms combine with each other. Atoms stick together to form molecules and
stack up to form rigid solids. Even though atoms are mostly empty space, their
interactions allow you to stand on a floor without falling through!
These basic properties can be explained by quantum mechanics.
(40-2)
Some Properties of Atoms
Subtler Properties
Atoms Are Put Together Systematically. There are repetitive (periodic)
patterns in the properties of different atoms that allow them to be organized into
a periodic table.
Six periods with 2, 8, 8, 18, 18, and 32 atoms
in each period, respectively. These numbers
are predicted by quantum mechanics.
Fig. 40-2
Ionization energy
vs. atomic number
(number of
protons in
nucleus)
(40-3)
Some Properties of Atoms
Subtler Properties, cont’d
Atoms Emit and Absorb Light:
hf  Ehigh  Elow
Atoms Have Angular Momentum and Magnetism:
“Orbit” of each electron (more
correct to think in terms of angular
momentum of electronic state) can
produce a magnetic moment.
Fig. 40-3
(40-4)
Some Properties of Atoms
Subtler Properties, cont’d
Einstein-de Haas Experiment:
Angular momentum and
magnetic moment of atoms
are coupled.
Aligning magnetic moments
of iron atoms using an
external magnetic field
causes the iron cylinder to
rotate in a direction opposite
to the now-aligned angular
momenta of the iron atoms
(conservation of angular
momentum).
Fig. 40-4
(40-5)
40-3 Electron Spin
Trapped or free, electrons have intrinsic spin angular momentum S (spin). This
is a basic characteristic like the electron’s mass or charge. This leads to two
additional quantum numbers that are required to fully specify the electronic
state: s (magnitude of the spin, which is always ½ for electrons) and ms (the
component of spin along the z-axis).
Table 40-1 Electron States for an Atom
Quantum Number
Symbol
Allowed Value
Related to
Principal
n
1, 2, 3, …
Distance from nucleus
Orbital
l
0, 1, 2, …, (n-1)
Orbital angular momentum
Orbital magnetic
ml
-l, -(l-1), …+(l-1), +l
Orb. ang. mom. (z-component)
Spin
s
½
Spin angular momentum
Spin magnetic
ms
±½
Spin ang. mom. (z-component)
States with same n form a shell.
States with same value for n and l form a subshell.
(40-6)
40-4 Angular Momenta and Magnetic Dipole
Moments
Orbital Angular Momentum and Magnetism
Orbital Angular Momentum:
L
Orbital Magnetic Dipole Moment:
 orb

 1
e
orb  
L
2m
e

  1
2m
Neither  orb nor L can be measured experimentally, but their
components along a given axis can be measured. Applying
a magnetic field B along the z axis allows the z components of
orb and L to be measured.
(40-7)
Orbital Angular Momentum and Magnetic Dipole Moments
orb,z  m B
Bohr magneton
eh
e
B 

4 m 2m
 9.274 1024 J/T (Bohr magneton)

Lz  m
 : semi-classical angle
between L and the z axis Fig. 40-5
Lz
cos 
L
(40-8)
Spin Angular Momentum and Spin Magnetic Dipole Moment
S, the magnitude of the spin angular momentum, has a single value for any
electron, whether free or trapped:
Fig. 40-6
S  s  s  1 
 12  12  1
 0.866
where s (=½) is spin quantum number of the electron.
The spin magnetic dipole moment s is related to S
and is given by:
e
e
s 
s  s  1
s   S
m
m
Neither S nor  s can be measured, but their
components along a given axis (say the z -axis) can be measured.
S z  ms
where ms   12 (spin up) or - 12 (spin down)
 s , z  2ms B
(40-9)
Orbital and Spin Angular Momentum Combined
J represents the total angular momentum of atoms containing more
than one electron. J is the vector sum of all the orbital and spin
angular momenta of all the electrons.
A neutrally charged atom with atomic number Z
will have Z electrons and Z protons.

J  L1  L2  L3 
 
LZ  S1  S 2  S3 
SZ

Total magnetic dipole moment eff is:
eff  1  2  3 

Z
Fig. 40-7



e
e

L1  L2  L3  LZ 
S1  S 2  S3  S Z
2m
m
 since S is weighted more than L, eff is not parallel to  J . (40-10)
40-5 Stern-Gerlach Experiment
Magnetic Deflecting Force on Silver Atom
U    B
U   z B
dU
dB
Fz  
 z
dz
dz
dB
is the magnetic field
dz
gradient along the z -direction.
Fig. 40-8
Stronger B
z
 Fz
Weaker B
The z -projection of  determines
the direction and magnitude of
the deflecting force.
(40-11)
Stern-Gerlach Experiment, cont’d
Experimental surprise
Silver atoms
Meaning of Experiment:
 z is quantized with two possible values
with opposite signs
B when
magnet ON
 Lz is also quantized the same way.
The dipole moments of all the electrons
in a silver atom vectorially cancel out
except for the moment of a single
electron.
Fig. 40-9
 s , z  2   12  B   B and  s , z  2   12  B   B
 dB 
 dB 
Fz   B 
 and Fz   B 

dz
dz




(40-12)
40-6 Magnetic Resonance
A proton has a spin magnetic moment 
that is associated with the proton's
intrinsic spin angular momentum S .
In a magnetic field B, the two spin states of
the proton will lead to two orientations of
 , which in turn will have two different
energies since U     B.
Fig. 40-10
E   z B    z B   2  z B
 photon absorption (nuclear magnetic
resonance) at hf  2  z B  spin flip
f
radio frequency
(40-13)
Magnetic Resonance, cont’d
The net magnetic field that a proton experiences consists of the vector
sum of the externally applied magnetic field Bext and internal fields Bint
B  Bext  Bint absorption when hf  2 z  Bext  Bint 
magnetic dipole moments of atoms and nuclei near the proton→ Bint
For fixed radio frequency light, when
Bext = hf/2mz - Bint→ absorption occurs.
Bint is different for protons in different
molecules, so the resonance Bext will be
different for protons in different
molecules (local environment).
Fig. 40-11
Resonances provide a fingerprint of
what (and where in the case of
Magnetic Resonance imaging) different
proton-containing molecules are present
in the material studied.
(40-14)
40-7
Pauli Exclusion Principle
No two electrons confined to the same trap (or atom) can
have the same set of values for their quantum numbers.
40-8
Multiple Electrons in Rectangular Traps
1. One-dimensional trap. Two quantum numbers n=1, 2, 3… (wavefunction
state along L) and ms= +½ or -½.
2. Rectangular corral. Three quantum numbers nx = 1, 2, 3… (wavefunction
state along Lx) , ny = 1, 2, 3… (wavefunction state along Ly), and ms= +½ or
-½.
3. Rectangular box. Four quantum numbers nx = 1, 2, 3… (wavefunction state
along Lx) , ny = 1, 2, 3… (wavefunction state along Ly), nz = 1, 2, 3…
(wavefunction state along Lz), and ms= +½ or -½.
(40-15)
Finding the Total Energy
Adding electrons to a rectangular trap:
Use energy level diagram.
Start at lowest energy level and move up as lower
levels become filled.
Empty (unoccupied) level
Partially filled level
Filled levels
Fig. 40-13
(40-16)
40-9
Building the Periodic Table
Four quantum numbers n, l, ml, and ms identify the quantum states of
individual electrons in a multi-electron atom.
Subshells are labeled by letters:
l=
0
1
2
3
4
5
. . .
s
p
d
f
g
h
. . .
Example: n = 3, l = 2→ 3d subshell
Neon: Z = 10→10 electrons
Energy
n
3
2
1
l = 0 (s) l = 1 (p)
ml = 0
-1 0 +1
__
__
__
__ __ __
__ __ __
l = 2 (d)
-2 -1 0
+1 +2
__ __ __ __ __
1s2 2s2 2p6
(40-17)
Building the Periodic Table, cont’d
Sodium: Z = 11→11 electrons
Energy
n
3
2
1
l =0 (s)
ml = 0
__
__
__
l =1 (p)l =2 (d)
-1 0 +1
-2 -1 0
__ __ __
__ __ __
+1 +2
__ __ __ __ __
1s2 2s2 2p6 3s1
degenerate
Chlorine: Z = 17→17 electrons
Energy
n
3
2
1
l =0 (s)
ml = 0
__
__
__
l =1 (p)l =2 (d)
-1 0 +1
-2 -1 0
__ __ __
__ __ __
+1 +2
__ __ __ __ __
1s2 2s2 2p6 3s2 3p6
For smaller atoms such as these, one can assume that the energy
only depends on n.
(40-18)
Building the Periodic Table , cont’d
Iron: Z = 26→26 electrons
For atoms with a larger number of electrons, the interactions among the
electrons causes shells with the same n but different l to have different
energies (degeneracy lifted).
1s2 2s2 2p6 3s2 3p6 3d6 4s2
Due to interactions, it takes less energy to start filling the 4s subshell before
completing the filling of the 3d subshell, which can accommodate 10
electrons.
(40-19)
40-10
X Rays and Ordering of Elements
X rays are short-wavelength (10-10 m), high-energy (~keV ) photons. Photons in
the visible range: ~ 10-6 m; ~eV.
Useful for probing atoms
Fig. 40-15
Fig. 40-14
K 0  hf 
Independent of target material
hc
min
min
hc

K0
(cutoff wavelength)
(40-20)
Characteristic X-Ray Spectrum
1. Energetic electron strikes atom in target, knocks
out deep-lying (low n value). If deep-lying electron
in n = 1 (K-shell), it leaves a vacancy (hole)
behind.
2. Another electron from a higher energy shell in the
atom jumps down to the K-shell to fill this hole,
emitting an x-ray photon in the process.
If the electron that jumps into the hole starts from
the n = 2 (L-shell), the emitted radiation is the Ka
line. If it jumps from the n = 3 (M-shell), the emitted
radiation is the Kb line. The hole left in the n = 2 or
n = 3 shells is filled by still higher lying electrons,
which relax by emitting lower energy photons
(higher lying energy levels are more closely
spaced).
Fig. 40-16
(40-21)
Ordering Elements
Moseley (1913) bombarded different elements with x rays. Nuclear charge, not
mass, is the critical parameter for ordering elements.
Fig. 40-17
(40-22)
Ordering Elements, cont’d
Accounting for the Moseley Plot
me4 1 13.60 eV
Energy levels in hydrogen: En   2 2 2 =
, for n  1, 2,3,
2
8 0 h n
n
Approximate effective energy levels in multi-electron atom with Z protons
(replace e2 x e2 with e2 x (e(Z - 1))2:
2
En
13.60 eV  Z  1


n2
 13.60 eV  Z  1  13.60 eV  Z  1
Ka energy: E  E2  E1 

2
2
12
2
 (10.2 eV)  Z  1
2
2
(10.2 eV)  Z  1
2
E
2
15
Ka frequency: f 

  2.46  10 Hz   Z  1
15
h
 4.14 10 eV  s 
f  CZ  C where constant C  4.96  10 Hz
7
1
2
(40-23)
40-11 Lasers and Laser Light
1. Laser light is highly monochromatic: Its spread in wavelength is as small
as 1 part in 1015.
2. Laser light is highly coherent: Single uninterrupted wave train up to 100
km long. Can interfere one part of beam, with another part that is very far
away.
3. Laser light is highly directional: Beam spreads very little. Beam from
Earth to Moon only spreads a few meters after traveling 4 x 108 m.
4. Laser light can be sharply focused: Can be focused into very small spot
so that all the power is concentrated into a tiny area. Can reach intensities of
1017 W/cm2, compared to 103 W/cm2 for oxyacetylene torch.
Lasers have many uses:
Small: voice/data transmission over optic fibers, CDs, DVDs, scanners
Medium: medical, cutting (from cloth to steel), welding
Large: nuclear fusion research, astronomical measurements, military
applications
(40-24)
40-12
How Lasers Work
hf  Ex  E0
Thermal distribution
(Boltzmann):
N x  N0 e
 Ex  E0  kT
Fig. 40-19
To get more stimulated emission than absorption, Nx > N0 → population
inversion → not in thermal equilibrium
(40-25)
Helium-Neon Gas Laser
Thermal Equilibrium Population Inversion
Fig. 40-20
Fig. 40-21
Fig. 40-22
(40-26)