B

kitefleaUrban and Civil

Nov 15, 2013 (3 years and 4 months ago)

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Chapter 8

Spin and Atomic Physics


Magnetic Dipoles

Magnetic Dipoles

Torque and Angular Momentum

B

q

m
L

L

dL

Force on e
-

by
B
-
Field

Force on e
-

by
B
-
Field

-

+

p

E

F
+

F
-

B

F
+

F
-

I

m

No field gradient, no force.

Force on e
-

by
B
-
Field

-

+

p

E

F
+

F
-

B

F
+

F
-

I

m

Field gradient, force.

Stern
-
Gerlach

Experiment

s
-
state electrons →

l
= 0

Classical prediction

Experimental outcome

S

N

B

z
B


S

N

Stern
-
Gerlach

Experiment

p
-
state electrons →

l
= 1

Classical prediction

Experimental outcome

N

S

B

Intermission

New Quantum Number


Intrinsic angular momentum (magnetic dipole)





s is particle dependent


½
-
integer for fermions (1/2, 3/2, 5/2, …)


Integer for bosons (0, 1, 2, …)


)
1
(


s
s
S
New Quantum Number


Intrinsic angular momentum is due to an
intrinsic dipole





g is the
gyromagnetic

ratio


g = 2 for the electron


g = 5.6 for the proton

S
m
e
g
S


2

m
New Quantum Number


Spin quantum number









½
-
integer
m
s

for fermions (1/2, 3/2, 5/2, …)


-
1/2, 1/2 (spin down, spin up)


-
3/2,
-
1/2, 1/2, 3/2


Integer
m
s
for bosons (0, 1, 2, …)


0


-
1, 0, 1


-
2,
-
1, 0, 1, 2

s
s
s
s
m
m
S
s
s
z
,
1
,...
1
,







NMR at Centenary

(Nuclear Magnetic Resonance)


For protons to be measured, radio frequency
radiation at the
Larmour

frequency is used to
interact with the intrinsic spin (resonance)


Our spectrometer has a magnetic field that makes
the proton
Larmour

frequency 300 MHz

hf
B
hf
U
s


m
NMR at Centenary

(Nuclear Magnetic Resonance)

hf
B
hf
U
s


m
NMR

aspirin

NMR

aspirin

Degeneracy

Energy of Spins in B
-
Fields


How does energy differ for spin up
(
m
s
=1/2) and spin down (
m
s
=
-
1/2) in a
magnetic field ?

N

S

N

S

m
s

S

B
z

m
s

S

B
z

Multi
-
particle Systems


Two particles bouncing
around in an infinite well





2

1

Multi
-
particle Systems


Since the SEQ is just energy,
the total energy is the sum of
the kinetic and potential for
both particles.




All functions are separable





2

1

Multi
-
particle Systems


The
wavefunctions

for equal
masses are





And the energies are





2

1

Multi
-
particle Systems


Single particle states





Multiparticle

states




No interactions between particles


Wavefunctions

are independent of one
another





2

1

Multi
-
particle Systems


Probability density





2

1

Multi
-
particle Systems





2

1

There’s a problem here. What is it?

Probability density MUST be unchanged if
indistinguishable particle labels are switched

Multi
-
particle Systems

Distinguishable

Indistinguishable,
antisymmetric

Indistinguishable, symmetric

Probability Densities

The Schrodinger Equation


Reminder


SEQ is a linear P.D.E


Sums and differences of solutions are also solutions

Exchange Symmetry


Changing particle labels


No change in
y


Symmetric


Change in
y

-

y


Antisymmetric

Exchange Symmetry


Changing particle labels


No change in
y


Symmetric

Exchange Symmetry


Changing particle labels


Change in
y

-

y


Antisymmetric

Exchange Symmetry

Pauli Exclusion Principle


Indistinguishable

fermions

may NOT
occupy the same individual
particle state


Wolfgang Pauli


1924


Nobel Prize
-

1945

Some interesting things…


Superconductivity






Bose
-
Einstein Condensate

Some interesting things…


Fermions can
combine to
fermion

or boson


Depends on the
“glue”


Bosons combine
to form bosons,
always

Intermission

Why did the chicken cross the road?

Wolfgang Pauli: There already was a chicken on this side of the
road.

More Complicated Atoms


Hydrogen is nice but rare


Hydrogen is a naturally diatomic, H
2


Pauli Exclusion has some pretty amazing
consequences

More Complicated Atoms


Deviations from ideal Coulomb potential


More energy dependence on
l



Electron screening









E ~ n +
l


This effect is larger for circular orbits (large
l
)


Electron stays further from nucleus

More Complicated Atoms

1s

2s

2p

Subshell

(
n
,
l
)

n

+
l

n

l

# electrons

possible

Calculating the number of
electrons per shell

More Complicated Atoms

1s

2s

2p

3s

3p

4s

3d

4p

5s

4d

5p

6s

4f

Subshell

(
n
,
l
)

n

+
l

n

l

# electrons

possible

1

2

2

3

3

4

3

4

5

4

5

6

4

0

0

1

0

1

0

2

1

0

2

1

0

3

1

2

3

3

4

4

5

5

5

6

6

6

7

2

2

6

2

6

2

10

6

2

10

6

2

14

2(2
l
+1)

More Complicated Atoms

More
Complicated
Atoms

Hund’s

Rules


The term(s) arising from the ground state
configuration with the maximum
multiplicity (2S+1) lies lowest in energy


Spins will fill with parallel
m
s

EOS

1
s

2
s

2
p

1
s

2
s

2
p

(correct)

Hund’s

Rules

1
s

2
s

2
p

1
s

2
s

2
p

(correct)

1
s

2
s

2
p

1
s

2
s

2
p

(incorrect)

Hund’s

Rules


The term(s) arising from the ground state
configuration with the maximum
multiplicity (2S+1) lies lowest in energy


Spins will fill with parallel
m
s

1
s

2
s

2
p

1
s

2
s

2
p

(correct)

This is occasionally called the "bus seat rule" since it is analogous to the behavior of
bus passengers who tend to occupy all double seats singly before double occupation
occurs.

Hund’s

Rules


Of several levels with the same
multiplicity, the one with the maximum
L

lies lowest in energy


L

is a vector coming from the sum of all
electron’s orbital momentum


Electrons orbiting in the same direction can
avoid one another (Pauli Exclusion)

z

z

Hund’s

Rules


Of several sublevels with the same
multiplicity and total quantum number
L:


The sublevel with the minimum
J

lies lowest
in energy if the configuration has a shell that
is less than half
-
filled;


The sublevel with the maximum J lies lowest
in energy if the configuration has a shell that
is more than half
-
filled.

The Aufbau Principle ...

[He]
2
p
2

[He]
2
p
3

[He]
2
p
4

[He]
2
p
5

EOS

[He]
2
p
6

More Complicated Atoms

Chemical Behavior

Chemical Behavior

Characteristic X
-
rays

Electron comes in and
knocks out inner shell
electrons

Energy Dispersive X
-
Ray Analysis

Scanning Electron Microscope

EDX spectrum of
Rimicaris

exoculata

Rules for Electron
Configurations

For
orbitals

of identical energy, electrons enter empty
orbitals

whenever possible


Hund’s

rule

Electrons in half
-
filled orbitals have parallel spins

EOS