Harris Chapter 7 - Atomic Structure

1
Harris Chapter 7- Atomic Structure

• 7.1
• Orbital Magnetic Moments, discovery of intrinsic
  spin
• 7.2 7.3
• Identical Particles (warning examples in book
  all inf-squ well)
• Exclusion Principle
• 7.4 7.5
• Multielectron Atoms, effective charges
• Hartree Treatment
• 7.6
• Spin-Orbit Effect
• 7.7
• Adding QM Angular Momenta
• 7.9 7.8
• Multielectron Spectroscopic Notation
• Zeeman Effect

2
Summary So Far
3
7.1 Orbital Magnetic Moments and Discovery of
Intrinsic Spin
4
Two kinds of Angular Momentum

• Classical Angular Momentum
• L r x p
• r vector, p vector ? L vector
• L obeys vector math
• Any L possible, no contraints on Lx Ly Lz
• Quantum
• Quantum Mechanical Angular Momentum
• L r x p
• r vector, p vector operator ? L 3 component
  operator
• L obeys got to be careful
• L described by two labels l , m
• L and Lz can be known, Lx and Ly cannot

5
Bohr Model of Ang Momentum
Classical or Semi-classical description
Note s-states (l0) have no Bohr model picture
Eisberg Resnick Fig 7-11
6
Vector Model of QM Ang. Momentum
quantum numbers
ER Fig 7-12
7
EdmondsA.M. in QM
pg 19 We might imagine the vector moving in an
unobservable way about the
z-axis...
pg 29 The QM probability density, not being
time dependent, gives us no
information about the motion of the
particle in its orbit.
Y(r,t) Y(r,t)
Y(r,t)Y(r) e-iwt
8
Morrison, Estle, Lane Understanding More QM,
Prentice-Hall, 1991
9
Otto Stern Walther Gerlach1922
Bohrs Q hypothesis
Sommerfelds Q hypothesis
3
2
1
Assigned by advisor Max Born to demonstrate
existence of the l, ml quantum numbers
10
Orbital Magnetic Moment
ER Fig 7-11
11
Orbital Magnetic Moment
ER Fig 7-11
12
ER Fig 7-11
Bohr magneton
13
B
m
14
B
m
Different ml states experience different forces
15
Use B as z-axis.
Different ml states experience different forces
16
Stern Gerlach1922
Harris Fig 7.3, 7.4
17
Stern Gerlach1922
Intended to demonstrate space quantization (l),
therefore expected odd number of spots, but
observed an even number.
http//upload.wikimedia.org/wikipedia/en/2/29/Ster
n-Gerlach_experiment.PNG
18
Despite Stern's careful design and feasibility
calculations, the experiment took more than a
year to accomplish. In the final form of the
apparatus, a beam of silver atoms (produced by
effusion of metallic vapor from an oven heated to
1000C) was collimated by two narrow slits (0.03
mm wide) and traversed a deflecting magnet 3.5 cm
long with field strength about 0.1 tesla and
gradient 10 tesla/cm. The splitting of the silver
beam achieved was only 0.2 mm. Accordingly,
misalignments of collimating slits or the magnet
by more than 0.01 mm were enough to spoil an
experimental run. The attainable operating time
was usually only a few hours between breakdowns
of the apparatus. Thus, only a meager film of
silver atoms, too thin to be visible to an
unaided eye, was deposited on the collector
plate. Stern described an early episode
http//www.physicstoday.org/pt/vol-56/iss-12/p53.h
tml
19
Stern described an early episode After
venting to release the vacuum, Gerlach removed
the detector flange. But he could see no trace of
the silver atom beam and handed the flange to me.
With Gerlach looking over my shoulder as I peered
closely at the plate, we were surprised to see
gradually emerge the trace of the beam. . . .
Finally we realized what had happened. I was
then the equivalent of an assistant professor. My
salary was too low to afford good cigars, so I
smoked bad cigars. These had a lot of sulfur in
them, so my breath on the plate turned the silver
into silver sulfide, which is jet black, so
easily visible. It was like developing a
photographic film.7
http//www.physicstoday.org/pt/vol-56/iss-12/p53.h
tml
20
Wolfgang Pauli 1924

• Pauli Exclusion Principle
• No two electrons can have the same quantum number
• Postulated an additional quantum number (i.e.
  label)
• Believed it came from the interaction between
  electrons.

21
Ralph Kronig 1925

• Spinning Electron Idea

22
Goudsmit Ulhenbeck 1925

• Studied high resolution spectra of alkali elements

23
Ocean Optics - Helium
24
Ocean Optics - Neon
25
Giancoli fig 36.21
26
The old and the new term scheme of hydrogen 5.
The scheme shows the multiplet splitting of the
excited states of the hydrogen atom with
principal quantum number n3, presented by
Goudsmit in the form in which it appeared in the
original publications of1926. The assignment in
the current notation has been added at the right.
With the development of quantum mechanics the
notation changed. The quantum numbers L and J now
usedfor the orbital and total angular momentum,
respectively, correspond to K-1/2 and J-1/2 in
the figure. The "forbidden component" referred to
by Goudsmit is of the type 3 2P1/2 --gt 2 2S in
which the total angular momentum is conserved and
L changes by plus or minus 1.
5 S. Goudsmit and G.E. Uhlenbeck, Physica 6
(1926) 273.
27
Uhlenbeck Goudsmit 1925
The discovery note in Naturwissenschaften is
dated 17 October 1925. One day earlier Ehrenfest
had written to Lorentz to make an appointment and
discuss a "very witty idea" of two of his
graduate students. When Lorentz pointed out that
the idea of a spinning electron would be
incompatible with classical electrodynamics,
Uhlenbeck asked Ehrenfest not to submit the
paper. Ehrenfest replied that he had already sent
off their note, and he added "You are both young
enough to be able to afford a stupidity!"
http//www.lorentz.leidenuniv.nl/history/spin/spin
.html
28
Uhlenbeck Goudsmit 1925
Ehrenfest's encouraging response to his students
ideas contrasted sharply with that of Wolfgang
Pauli. As it turned out, Ralph Kronig, a young
Columbia University PhD who had spent two years
studying in Europe, had come up with the idea of
electron spin several months before Uhlenbeck and
Goudsmit. He had put it before Pauli for his
reactions, who had ridiculed it, saying that "it
is indeed very clever but of course has nothing
to do with reality". Kronig did not publish his
ideas on spin. No wonder that Uhlenbeck would
later refer to the "luck and privilege to be
students of Paul Ehrenfest".
http//www.lorentz.leidenuniv.nl/history/spin/spin
.html
29
This isn't right. This isn't even wrong.
http//www-groups.dcs.st-and.ac.uk/history/Mathem
aticians/Pauli.html
There were some people thinking about electron
spin in those days, but there was a lot of basic
opposition to such an idea. One of the first was
Ralph de Laer Kronig. He got the idea that the
electron should have a spin in addition to its
orbital motion. He was working with Wolfgang
Pauli at the time, and he told his idea to Pauli.
Pauli said, "No, it's quite impossible." Pauli
completely crushed Kronig. Then the idea
occurred quite independently to two Young Dutch
physicists, George Uhlenbeck and Samuel Goudsmit.
They were working in Leiden with Professor Paul
Ehrenfest, and they wrote up a little paper about
it and took it to Ehrenfest. Ehrenfest liked the
idea very much. He suggested to Uhlenbeck and
Goudsmit that they should go and talk it over
with Hendrik Lorentz, who lived close by in
Haarlem.
His ability to make experiments self destruct
simply by being in the same room was legendary,
and has been dubbed the "Pauli effect" (Frisch
1991, p. 48 Gamow 1985).
"The Birth of Particle Physics," edited by Laurie
M. Brown and Lillian Hoddeson. The essay by Paul
A.M. Dirac is entitled "Origin of Quantum Field
Theory."
30
This isn't right. This isn't even wrong.
http//www-groups.dcs.st-and.ac.uk/history/Mathem
aticians/Pauli.html
They did go and talk it over with Lorentz.
Lorentz said, "No, it's quite impossible for the
electron to have a spin. I have thought of that
myself, and if the electron did have a spin, the
speed of the surface of the electron would be
greater than the velocity of light. So, it's
quite impossible." Uhlenbeck and Goudsmit went
back to Ehrenfest and said they would like to
withdraw the paper that they had given to him.
Ehrenfest said, "No, it's too late I have
already sent it in for publication "
His ability to make experiments self destruct
simply by being in the same room was legendary,
and has been dubbed the "Pauli effect" (Frisch
1991, p. 48 Gamow 1985).
"The Birth of Particle Physics," edited by Laurie
M. Brown and Lillian Hoddeson. The essay by Paul
A.M. Dirac is entitled "Origin of Quantum Field
Theory."
31
The calculation(using current values)
r lt 2.8 E-19 m
b gt 3 10 6
value from Bhabha scattering at CERN
32
This isn't right. This isn't even wrong.
http//www-groups.dcs.st-and.ac.uk/history/Mathem
aticians/Pauli.html
That is how the idea of electron spin got
publicized to the world. We really owe it to
Ehrenfest's impetuosity and to his not allowing
the younger people to be put off by the older
ones. The idea of the electron having two states
of spin provided a perfect answer to the
duplexity.
His ability to make experiments self destruct
simply by being in the same room was legendary,
and has been dubbed the "Pauli effect" (Frisch
1991, p. 48 Gamow 1985).
"The Birth of Particle Physics," edited by Laurie
M. Brown and Lillian Hoddeson. The essay by Paul
A.M. Dirac is entitled "Origin of Quantum Field
Theory."
33
Letter fm Thomas to Goudsmit
Part of a letter by L.H. Thomas to Goudsmit (25
March 1926). Reproduced from a transparency shown
by Goudsmit during his 1971 lecture. The original
is presumably in the Goudsmit archive kept by the
AIP Center for History of Physics.
http//www.lorentz.leidenuniv.nl/history/spin/goud
smit.html
34
intrinsic spin

• Fundamental objects
• electron spin ½
• neutrino spin ½ , but LH only
• photon spin 1
• Composite objects
• proton spin ½
• neutron spin ½
• D delta spin 3/2

35
How to Denote Wavefunctions(version 1)
the spinor has no functional form because
spin is not a spatial feature
36
Two types of Magnetic Moments
L
S
37
interesting fundamental constants
-2.002 319 304 3622 (15)
1.602 176 487 (40) x 10-19 C   
38
7.2 7.3 Complications from having Identical
Particles
39
Exchange Symmetry
40
7.4 7.5 Multielectron Atoms
41
r n2 ao / Z
En ( -13.6 eV ) (Z2/n2)
ao 0.529 Å
42
Prob r2 R R
43
orbitals get sucked down the
most Crossings occur for the upper orbitals
4p
3d
4s
3p
3s
2p
2s
1s
1s sucked off bottom of page
44
Note This shows how the orbitals shift as
viewed from the perspective of an s-orbital.
45
Hartree-Fock Method
46
Hartree-Fock Methods
Choose initial shape For Coulomb Potl V(r)
Solve Schro Eqn for En Yn
Insert fine structure corrections
Build atom according to This set of orbital
energies En
Loop until V(r) doesnt change much
Use the collection of YnYn to Get new electron
charge distrib
Use Gauss Law to get new V(r) shape
47
(No Transcript)
48
Using effective charge is a very crude
approximation.
49
Hartree-FockEffective Charge Effects
r2 n2 ao / Zeff
En (Zeff2/n2) ( -13.6 eV )
50
(No Transcript)
51
(No Transcript)
52
7.6 Spin-Orbit Effect
53
Corrections to the Coulomb Potlfor H-atom

• Central Potential
• Spin-Orbit (electron viewpoint)
• Relativistic Spin (Thomas precession)
• Relativistic Kinetic Energy
• Spin-Orbit (nucleus viewpoint)
• Spin-Spin
• Impact of External Fields
• Zeeman Effect (applied B-field)
• Stark Effect (applied E-field)

54
Spin-Orbit Interaction
L
ms
ms
L
Note L.H. Thomas showed that in the x-form
between non-inertial
reference frames a factor of ½ appears.
55
Goal find expression for the orientational
potential energy of electron
intrinsic mag moment (ms) in terms of
orbital motion (L) and forces ( dV/dr).
ms
L
Note L.H. Thomas showed that in the x-form
between non-inertial
reference frames a factor of ½ appears.
56
ms
L
57
E
58
-2
L
S
NRG shift depends on relative orientation of L
and S
59
How to evaluate DE and SL
L
S
involved in radial integrations
depends on A.M. qu. no.s
60
(No Transcript)
61
electronSpin-Orbit locks the angle between L
S ? J is now a well-defined
direction.
J
S
NOTE Lz is no longer well-defined ml not a
good q. no.
L
S
L
62
Revised H-atom Level Scheme
add in spin-orbit correction
nlj
not required to specify NRG mj ml s ms
not required to specify NRG j mj l ml s ms
absolutely worthless
63
electron Spin-Orbit is more important in
higher-Z atoms
fnl expression only for H-atom, for all
others, must come from Hartree procedure
Li Na K Rb Cs
Splitting (eV) 0.42E-4 21.E-4 72.E-4 295.E-4 687.E-4
64
Bigger atoms larger Z (central charge)
same size
larger
65
7.7 QM Angular Momentum
66
Bohr Model of Ang Momentum
Note s-states (l0) have no Bohr model picture
Eisberg Resnick Fig 7-11
67
Vector Model of Ang. Momentum
quantum numbers
ER Fig 7-12
68
EdmondsA.M. in QM
pg 19 We might imagine the vector moving in an
unobservable way about the
z-axis...
pg 29 The QM probability density, not being
time dependent, gives us no
information about the motion of the
particle in its orbit.
69
Morrison, Estle, Lane Understanding More QM,
Prentice-Hall, 1991
70
ADDITION OFANGULAR MOMENTUM
L2
Ltot L1 L2
L1
71
Ltot L1 L2
72
Ltot L1 L2
73
Addition of Angular Momentum
aligned configuration
www.bokerusa.com
aligned does not mean straight
jack-knife configuration
www.cartowning.co.za/DBNRECGC.htm
jack-knife does not mean antiparallel
74
(No Transcript)
75
Detailed Example
L2
L1
Problem Two objects each travel in a p-orbit (
l1 ). The total energy of each object is
degenerate wrt ml, so we have no detailed
knowledge of ml. What are the allowed values of
ltot, mtot ?
76
l11, l21, ms degenerate
m1 m2 mtot
1 1 2
0 1
-1 0
0 1 1
0 0
-1 -1
-1 1 0
0 -1
-1 -2
mtot Possibilities (m1,m2)
2
1
0
-1
-2
77
Allowed Values of ltot mtot
78
Basic A.M. Math
J L S
J
S
L
79
Vector Representation of J
80
Annoying Pictures 1
Jeffs Qs i) what am I supposed to think about
the S L cones as drawn? ii) I
thought I was told earlier that L S were about
z ??
81
Annoying Pictures 2
Jeff Pictures such as this confuse the vector
symbols L and S with the
quantum numbers l and s . For instance, how could
L and S ever point in the same direction?
82
TOTAL ANGULAR MOMEMTUM
J L S
83
More Detailed H-atom Level Scheme
Energies Spectra not sensitive to j mj l
ml s ms till next page
Energies Spectra not sensitive to l ml
84
Ocean Optics - Helium
Because of the doublets, the states cannot be
completely degenerate
? spin-orbit effect
85
Ocean Optics - Neon
Because of the doublets, the states cannot be
completely degenerate
? spin-orbit effect
86
7.9 Multi-electron Spectra
87
Multi-e Spectroscopic Notation
QUANTUM NUMBERS principal n ltot ,
stot jtot .
stot 1, ltot0, jtot1 2S1
Stot S1 S2
Ltot L1 L2
Jtot Ltot Stot
88
Two Kinds of Notation

• Where an individ electron is at
• n l j
• 1s1/2
• 2s1/2
• 2p1/2
• 2p3/2

• A.M. for whole atom
• 2Stot1 ltot jtot
• 1S0
• 3S1
• 3P0 , 3P1, 3P2

89
(No Transcript)
90
Curious Things That HappenGround State of Helium
Ltot L1 L2
asym
1s
Stot S1 S2
sym
Ysystem (spatial wfn) (spin wfn)
1s
1S0
3S1
91
(No Transcript)
92
7.8 Atoms in External Magnetic Fields-- the
Zeeman Effect
93
Corrections to the Coulomb Potlfor H-atom

• Central Potential
• Spin-Orbit (electron viewpoint)
• Relativistic Spin (Thomas precession)
• Relativistic Kinetic Energy
• Spin-Orbit (nucleus viewpoint)
• Spin-Spin
• Impact of External Fields
• Zeeman Effect (applied B-field)
• Stark Effect (applied E-field)

94
Weak-Field Zeeman

• Hartree-Fock Coulomb related Procedures
• Fine Structure
• spin-orbit ( jtot becomes important )
• relativistic
• Zeeman

HZeeman - mtot Bext
Bext lt few 0.1s Tesla
95
Weak Field Zeeman
mtot
96
electronSpin-Orbit locks the angle between L
S ? J is now a well-defined
direction.
J
S
NOTE Lz is no longer well-defined ml not a
good q. no.
L
S
L
97
Weak-Field Zeeman
eSO makes jtot good quantum number, mltot
mstot become confused (near worthless).
Jtot
mtot
Jtot is well-defined direction jtot mjtot
g
project average mtot onto Jtot
98
Weak Field Zeeman
Jtot
projection of mtot onto J onto B
monto J
a
Bext
DEZeeman - mtot Bext
99
(No Transcript)
100
stot0
101
Strong-Field Zeeman

• Hartree-Fock Coulomb related procedures
• Zeeman
• Fine Structure
• spin-orbit
• relativistic

HZeeman - mtot Bext
102
Strong Field Zeeman
Ltot
Stot
Bext
HZeeman - mtot Bext
103
(No Transcript)