1. Chapter IX Atomic Structure
Restless electron(s) in an atom

2. 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

3. 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
A current loop =
A circulating charge q,
(T: period of motion)

4. Orbital angular momentum
Magnetic dipole moment
Dipole moment vector is normal to orbit, with magnitude proportional to the angular momentum
(e: positive) v I
For electrons, q = e and
Magnetic dipole moment vector is anti-parallel to the angular momentum vector
Both L and  are subject to space quantization !

5. Magnetic dipole moment in an external B field
since
Torque results in precession of the angular momentum vector
Larmor precession frequency:

6. Example: A spinning gyroscope（陀螺儀）in the gravity field
the rate at which the axle rotates about the vertical axis

7. Potential energy of the system
Change in orientation of  relative to B produces change in potential energy
Defining orientation potential
For an orbiting electron in an atom:

8. Quantum consideration for hydrogen-like atoms
Magnetic dipole moment for the rotating electron
Bohr magneton:
magnitude
z-component
Quantization of L and Lz means that  and z are also quantized !!
(Note that electron has probability distribution, not classical orbit)
Total energy:
Degeneracy partially broken: total energy depends on n and m
(magnetic quantum number)

9. Energy diagram for Z = 1 (hydrogen atom)
=0
=1
=2 E
B=0
B0
B=0
B0
-0.85eV 4s 4p 4d
-1.5eV 3s 3p 3d
n=2
n=2
-3.4eV 2s 2p
B0
n=2, =1, m=1
n=2, =1, m=0
n=2, =1, m=1
A triplet spectral lines when B  0
n=1
-13.6eV 1s

10. Normal Zeeman Effect 1896 First observation of spectral line
Lorentz
Zeeman
1902
1853~1928
1865~1943
Normal Zeeman Effect
1896
First observation of spectral line
splitting due to magnetic field
Requires Quantum Mechanics (1926)
to explain
B > 0
B = 0
m=1
n=2, =1
m=0
m=1
n=1, =0
m=0

11. m =2 n=3 =1 n=2 Selection rules: n=1 =01 2 1
n=2
=1 1
The total angular momentum (atom + photon) in optical transitions should be conserved
Selection rules:
n=1
=0
(Cf. Serway, Figure 9.5)

12. Normal Zeeman effect – A triplet of equally spaced spectral lines when B  0 is expected
homogeneous B field
Selection rule
= 5.810-5 [eV/T]  B[T]
Energy spacing
For B = 1 Tesla,
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
11B
Leiden
(08/2008)
Cf. Zeeman used Na atoms

13. Anomalous Zeeman effect
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)

14. Stern
1943
1888~1969
Gerlach
1889~1979
Electron Spin
Direct observation of energy level splitting in an inhomogeneous magnetic field
Let the magnitude of B field depend only on z:
B(x,y,z) = B(z)
Translational force in z-direction is proportional to z-component of magnetic dipole moment z
Quantum prediction:
g ≈ 2 for electrons

16. ? =0, m=0 Ag atom in ground state
Electronic configuration of Ag atom: [Kr]4d105s1
outermost electron
=0, m=0
Bz = B(z)
Expectation from normal Zeeman effect:
No splitting ?
Ex.
expectations for  = 1,
three discrete lines
Stern and Gerlach (1922)

17. 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

18. Gerlach's postcard, dated 8 February 1922, to Niels Bohr
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.”

19. a half integer !! 1925, Goudsmit and Uhlenbeck
proposed that electron carries intrinsic angular momentum called “spin”
Experimental result requires
1902~1978
1900~1988
s: spin quantum number
a half integer !!
New angular momentum operator S
Both cannot be changed in any way
Intrinsic property

20. A spin magnetic moment is associated with the spin angular momentum
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
B0
B=0
+msB
DE = 2ms|B| B
-msB
Note: All particles possess spin (e.g., protons, neutrons, quarks, photons)

21. 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
ge: electron gyromagnetic ratio
= from measurement
(Agree with prediction from Quantum Electrodynamics)
Only two allowed orientations of spin vector

22. n, , m, ms (where ms = 1/2 and +1/2)
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 
+ : spin-up wavefunction
- : spin-down wavefunction
Spin wave functions  : eigenfunctions of [Sz] and [S2]

23. Wavefunction: states Eigenvalues n = 1, 2, 3,…. En = 13.6(Z/n)2 eV
m = 0, 1, 2,…, 
ms = 1/2
Degeneracy in the absence of a magnetic field:
Each state  has degenerate states
2(2+1)
Each state n has degenerate states
2n2
two spin orientations

24. In strong magnetic fields, the torques are large
Total magnetic moment:
Both the angular momenta precess independently around the B field
For an electron: ge = 2
spin up
ms = 1/2
Contribution to energy shifts
spin down
ms = 1/2
B=0
B0
+msB
-msB
For a given m,
(orientation of s)

25. Magnetic field B  0 m=1 2P m=0 m=1 n=1,m=0 1S Selection rules:
m=1, ms=1/2
m=0, ms=1/2
m=1 2P
m=0
m=1, ms=1/2
m=1
m=0, ms=1/2
m=1, ms=1/2
m=0, ms=1/2
n=1,m=0
m=0, ms=1/2 1S
Selection rules:

26. 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 !
electron spin
(The experimental value is 2.8 times larger than the theoretical value – still a mystery)

27. 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.”

28. Stern –Gerlach experiment with ballistic electrons in solids

29. 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

30. 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)