Download Moments

Restless electron(s) in an atom
Hydrogen-like atoms: H, He+, Li2+
Multielectron atoms:
many electrons are confined to a small space
 strong Coulomb ‘electron-electron’ interactions
Magnetic properties
(e.g., electrons interact with external magnetic fields)
Electron spin
Pauli exclusion principle
The periodic table
Orbital Magnetism and the Zeeman Effect
An electron orbiting the nucleus of an atom should give
rise to magnetic effects. Atoms are small magnets


Magnetic field lines
for a current loop
Current I flowing in circle in x-y plane

Magnetic dipole moment   I A  I Anˆ

A current loop = A circulating charge q,
q
I
T
(T: period of motion)
2 r
 r2
A
|L| | r  p|  r mq
 2mq
 2mq
T
T
T


q
q
Magnetic dipole   I Anˆ  Anˆ 
L
T
2m q
moment
Orbital angular
momentum
Dipole moment vector is normal to orbit, with magnitude
proportional to the angular momentum
(e: positive)
e 
L
For electrons, q = e and   
2m e

I
Magnetic dipole moment vector is anti-parallel to
the angular momentum vector
v
Both L and  are subject to space quantization !
Magnetic dipole moment in an external B field

   dL
since     B 
dt
L̂

ˆ
dLˆ  ˆ and B

Torque results in precession of the angular
momentum vector
Larmor precession frequency:
eB
L 
2me
dL  Lsin   d
d
 L
dt
 eL

 dt   B sin   dt  
B sin   dt
 2me

Example: A spinning gyroscope（陀螺儀）in the gravity field
the rate at which the axle rotates
about the vertical axis
d Mgh
p 

dt
I


  r  mg

Potential energy of the system
Change in orientation of  relative
to B produces change in potential
energy
Defining orientation potential

dU  dW   d  d   B
U     B    B cos
1.0
0.8
0.6
0.4
For an orbiting
electron in an atom:
U (B)
0.2
0.0
-0.2
e
U
LB
2m
-0.4
-0.6
-0.8
-1.0
0.0

 // B

0.5

1.0

  // B


Quantum consideration for hydrogen-like atoms
Magnetic dipole moment for the rotating electron
magnitude

e 
e
 
L
  1
2m e
2m e
z-component
e
e
z 
Lz 
m
2m e
2m e
Bohr magneton:
B 
e
2me
 9.274  1024 J/T
Quantization of L and Lz means that  and z are also quantized !!
(Note that electron has probability distribution, not classical orbit)
U    B 
Total energy:
e
e
eB
LB 
Lz B 
m
2me
2me
2me
 Lm
ml  0, 1, 2, , l
E ( B)  En  Lm  En  B Bm
Degeneracy partially broken: total energy depends on n and m
(magnetic quantum number)
Energy diagram for Z = 1 (hydrogen atom)
=0
=1
E
B=0
4s
3s
-0.85eV
-1.5eV
-3.4eV
n=2
2s
n=2
=2
B0
4d
3d
2p
B0
-13.6eV
1s
n=2, =1, m=1
n=2, =1, m=0
n=2, =1, m=1
 B B
2,1,11,0,0  o  L
2,1,01,0,0  o
n=1
B0
4p
3p
E  L
o
B=0
2,1,11,0,0  o  L
A triplet spectral
lines when B  0
Normal Zeeman Effect
1896
Lorentz
Zeeman
1853~1928
1865~1943
First observation of spectral line
splitting due to magnetic field
Requires Quantum Mechanics (1926)
to explain
n=2, =1
o
n=1, =0
B>0
B=0
o  L
1902
1902
m=1
m=0
m=1
o  L
m=0
m
2
1
0
1
2
1
0
1
n=3
n=2
=1
The total angular
momentum (atom
+ photon) in optical
transitions should
be conserved
Selection rules:
  1
m  0, 1
0
n=1
 3, 2
=2
 2,1
3,1
=0
(Cf. Serway,
Figure 9.5)
Normal Zeeman effect – A triplet of equally spaced spectral lines
when B  0 is expected
homogeneous B field
Selection rule
Energy spacing
e
L  B B 
B = 5.810-5 [eV/T]  B[T]
2me
For B = 1 Tesla,
L  5.79  105 eV, L  8.78  1010 rad/s
Ex. Relative energy change in the Zeeman
splitting. Consider the optical transitions from 2P
to 1S states in an external magnetic field of 1 T
E
L
B B


E2  E1 E2  E1 E2  E1
5.8  105
a few tens eV
Cf. Zeeman used Na atoms
106
Leiden
(08/2008)
Mysteries:
Other splitting patterns such as four, six or even more
unequally spaced spectral lines when B  0 are observed
Anomalous Zeeman effect
inhomogeneous
magnetic field
existence of electron spin
(2/24/2009)
Electron Spin
Stern
Gerlach
Direct observation of energy level splitting
in an inhomogeneous magnetic field


F  U  r       B    Z B 
Let the magnitude of B field depend only on z:
B(x,y,z) = B(z)
F  Fz zˆ   z
dB
zˆ
dz
Translational force in z-direction is proportional to
z-component of magnetic dipole moment z
Quantum prediction:
F  z
dB
dB
zˆ  ms g B
zˆ
dz
dz
g ≈ 2 for electrons
1888~1969
1943
1889~1979


F  U  r       B    Z B 
B
B
B
 Z
xˆ  Z
yˆ  Z
zˆ
x
y
z
ˆ ( x, y , z )  zB
ˆ ( z)
B  zB
Ag atom in ground state
Electronic configuration
of Ag atom: [Kr]4d105s1
outermost
electron
Bz = B(z)
=0, m=0
Expectation from normal
Zeeman effect:
No splitting
?
Stern and Gerlach (1922)
Ex.
expectations for  = 1,
three discrete lines
Experimental confirmation
of space quantization !!
Experimental results
B
Not zero, but two lines
No B field
With B field on
Two lines were observed
Total magnetic moment is not zero. Something more than
the orbital magnetic moment
Orbital angular momentum cannot be the source of the
responsible quantized magnetic moment  = 0
Similar result for hydrogen atom (1927): two lines were observed
by Phipps and Taylor
Gerlach's postcard, dated 8 February 1922, to Niels Bohr. It shows
a photograph of the beam splitting, with the message, in translation:
“Attached [is] the experimental proof of directional quantization. We
congratulate [you] on the confirmation of your theory.”
1925, Goudsmit and Uhlenbeck
Goudsmit
Uhlenbeck
1902~1978
1900~1988
proposed that electron carries intrinsic
angular momentum called “spin”
Experimental result requires
1
s
2
2s  1  2
s: spin quantum number
a half integer !!
New angular momentum operator S
Sz    ms
1


2

 S    s  s  1 
2
2
 1  3 
   
 2  2 
Both cannot be changed in any way
2

Intrinsic property
Electron Spin

The new kind of angular momentum is called the electron SPIN
Why call it spin?

If the electron were spinning on its axis, it would have angular momentum and a
magnetic moment regardless of its spatial motion
However, this “spinning” ball picture is not realistic, because it would require that
the tiny electron be spinning so fast that parts would travel faster than c !
So we cannot picture the spin in any simple way … the electron’s spin is simply
another degree-of-freedom available to electron
A spin magnetic moment is associated with the spin angular momentum
s   S
U s   s  B
B=0
B0
sB
sB
E 2s|B|
B
s  9.2848  1024 J/T -- electron magnetic moment
 B = 9.2741  1024 J/T (the "Bohr magneton")
Note: All particles possess spin (e.g., protons, neutrons, quarks, photons)
Picturing a Spinning Electron
We may picture electron spin as the result of
spinning charge distribution
Spin is a quantum property
Electron is a point-like object with no
internal coordinates
Magnetic dipole moment
e
s   g e
S
2me
ge: electron gyromagnetic ratio = 2.00232 from measurement
(Agree with prediction from Quantum Electrodynamics)
Only two allowed orientations of spin vector S
So, we need FOUR quantum numbers to specify the
electronic state of a hydrogen atom


n, , m, ms (where ms = 1/2 and +1/2)
Complete wavefunction: product of spatial wave function 
and spin wave function 
   (r )  e
+ : spin-up wavefunction
- : spin-down wavefunction
Spin wave functions  : eigenfunctions of [Sz] and [S2]
Sz 
1
   
2
3
 S 2    
4
Sz 
1
   
2
3
 S    
4
2
2

2

 it
Wavefunction:

 e ( r ,)  Rn r Ym  ,   
states
Eigenvalues
n = 1, 2, 3,….
En = 13.6(Z/n)2 eV
 = 0, 1, 2,…, n-1
L    1
Lz  m
Sz  ms   / 2
m = 0, 1, 2,…, 
ms = 1/2
Degeneracy in the absence of a magnetic field:
Each state  has 2(2+1) degenerate states
Each state n has
2n2
degenerate states
two spin orientations
n 1
2  2  1
 0
In strong magnetic fields, the torques are large
Total magnetic
moment:
Both the L and S angular momenta
precess independently around the B field
   L  s
eB
e
U  (  L   s )  B 
(ml  g ems ) B
 Lz  ge S z  
2me
2me
For an electron: ge = 2
spin up
spin down
ms = 1/2
e
U 
 m  1 B
2me
ms = 1/2
U 
e
 m  1 B
2me
B=0
For a given m,
U  U   U   2 
e
B  2 L  2 B B
2me
Contribution to
energy shifts
B0
sB
sB
(orientation of s)
Magnetic field B  0
m=1, ms=1/2
m=0, ms=1/2
m=1
2P m=0
m=1
m=1, ms=1/2
m=0, ms=1/2
m=1, ms=1/2
o  L
o  L
m=0, ms=1/2
n=1,m=0
m=0, ms=1/2
1S
Selection rules:
  1, (m  ms )  0, 1
Otto Stern: “one of the finest experimental physicists of the
20th century” (Serway)
Specific heat of solids, a theoretical work under Einstein
“The method of molecular rays” – the properties of isolated atoms
and molecules may be investigated with macroscopic tools
Molecules move in a straight line (between collisions)
The Maxwell speed distribution of atoms/molecules
Space quantization – the Stern-Gerlach experiment
the de Broglie wavelengths of helium atoms
the magnetic moments of various atoms
the very small magnetic moment of proton !
(The experimental value is 2.8 times larger
than the theoretical value – still a mystery)
electron
spin
Otto Stern –
the Nobel Lecture, December 12, 1946
“The most distinctive characteristic property of the
molecular ray method is its simplicity and directness.
It enables us to make measurements on isolated
neutral atoms or molecules with macroscopic tools.
For this reason it is especially valuable for testing and
demonstrating directly fundamental assumptions of
the theory.”
Stern –Gerlach experiment with ballistic electrons in solids
Franck-Hertz Experiment
Direct confirmation that the internal
energy states of an atom are
quantized (proof of the Bohr model
of the atom)
4.9V
As a tool for measuring the
energy changes of the
mercury atom Franck and
Hertz used electrons, that
means an atomic tool
a
Recent Breakthrough – Detection of a single electron spin!

IBM scientists achieved a breakthrough
in nanoscale magnetic resonance
imaging (MRI) by directly detecting the
faint magnetic signal from a single
electron buried inside a solid sample
Nature 430, 329 (2004)
Next step – detection of single nuclear spin (660x smaller)
Dutt et al., Science 316, 1312 (2007)