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
Magnetic field lines for a current loop
Current I flowing in circle in x-y plane
nAA ˆ I I Magnetic dipole moment
A circulating charge q,T
qI (T: period of motion)
A current loop =
Orbital Magnetism and the Zeeman Effect
An electron orbiting the nucleus of an atom should give rise to magnetic effects. Atoms are small magnets
2
q q q
2 r r A|L| | r p| r m 2m 2m
T T T
Orbital angular momentum
L2m
qˆA
T
qˆA I
q
nnMagnetic dipole
moment
Dipole moment vector is normal to orbit, with magnitude proportional to the angular momentum
For electrons, q = e and L2m
e
e
Magnetic dipole moment vector is anti-parallel to the angular momentum vector
Both L and are subject to space quantization !
v
I
(e: positive)
Magnetic dipole moment in an external B field
sincedt
LdB
Torque results in precession of the angular momentum vector
Larmor precession frequency:
e
eB
2mL
ˆ ˆˆdL and B
L̂
dL Lsin d
sin sin2 e
eLdt B dt B dt
m
L
d
dt
g
mr
Example: A spinning gyroscope(陀螺儀) in the gravity field
the rate at which the axle rotates about the vertical axis
p
d Mgh
dt I
Potential energy of the system
Change in orientation of relative to B produces change in potential energy
BdU dW d d
Defining orientation potential B B cosU
0.0 0.5 1.0
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
U (B
)
B // B //
For an orbiting electron in an atom:
B2
eU L
m
Quantum consideration for hydrogen-like atoms
Magnetic dipole moment for the rotating electron
12m
eL
2m
e
ee
m2m
eL
2m
e
ez
e
z
magnitude
z-component
(Note that electron has probability distribution, not classical orbit)
Quantization of L and Lz means that and z are also quantized !!
z Le e e
e e eBB L B L B m
2m 2m 2mU m
Total energy: L( ) n n BE B E m E Bm
Degeneracy partially broken: total energy depends on n and m
Bohr magneton:
e
24
e
2m
9.274 10 J/T
B
(magnetic quantum number)
0, 1, 2, ,lm l
Energy diagram for Z = 1 (hydrogen atom)
E
-13.6eV
-3.4eV
-1.5eV-0.85eV
=0 =1 =2
1s
2s
3s4s
2p
3p4p
3d4d
n=1
n=2 n=2
B=0 B0 B=0 B0
n=2, =1, m=1
n=2, =1, m=0
n=2, =1, m=1
B0
L
B
E
B
L0,0,112,1,
0,0,12,1,0
L0,0,12,1,1
o
o
o
o
A triplet spectral lines when B 0
Normal Zeeman Effect
Lorentz Zeeman
1902
1853~1928
1902
1865~1943
1896
First observation of spectral line splitting due to magnetic field
Requires Quantum Mechanics (1926) to explain
n=2, =1
n=1, =0
m=1m=0m=1
m=0
B = 0 B > 0
o Lo Lo
n=1
n=2
n=3
m21012
110
0
=1
=0
=2
2,3 1,2 3,1
Selection rules: 1
m 0, 1
The total angular momentum (atom + photon) in optical transitions should be conserved
(Cf. Serway,
Figure 9.5)
Le
eB B
2mB
Normal Zeeman effect – A triplet of equally spaced spectral lines when B 0 is expected
Selection ruleEnergy spacing = 5.810-5 [eV/T] B[T]
For B = 1 Tesla, 5 10L L5.79 10 eV, 8.78 10 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
L
2 1 2 1 2 1
56
B
5.8 10 10
a few tens eV
BE
E E E E E E
Leiden(08/2008)
homogeneous B field
Cf. Zeeman used Na atoms
Mysteries:
Other splitting patterns such as four, six or even more unequally spaced spectral lines when B 0 are observed
Anomalous Zeeman effect
existence of electron spin
(2/24/2009)
inhomogeneous magnetic field
Electron Spin Stern
1943
1888~1969
Gerlach
1889~1979
Direct observation of energy level splitting in an inhomogeneous magnetic field
ZF U r B B
Let the magnitude of B field depend only on z:
B(x,y,z) = B(z) ˆ ˆz z
dBF F z z
dz
Translational force in z-direction is proportional to z-component of magnetic dipole moment z
Quantum prediction:
ˆ ˆz s B
dB dBF z m g z
dz dz
g ≈ 2 for electrons
ˆ ˆ ˆ
ˆ ˆ( , , ) ( )
Z
Z Z Z
F U r B B
B B Bx y z
x y z
B zB x y z zB z
Bz = B(z)
Ag atom in ground state
Electronic configuration of Ag atom: [Kr]4d105s1
=0, m=0outermost electron
Stern and Gerlach (1922)
Expectation from normal Zeeman effect:
No splitting
Ex. expectations for = 1,three discrete lines
?
Experimental results
B
No B field With B field onTwo lines were observed
Not zero, but two lines
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
Experimental confirmation of space quantization !!
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.”
Goudsmit
1902~1978
Uhlenbeck
1900~1988
1925, Goudsmit and Uhlenbeck
proposed that electron carries intrinsic angular momentum called “spin”
Experimental result requires
212 s2
1s
New angular momentum operator S
smzS
2 21S s s
1
2
21 3
2 2
s: spin quantum number
Both cannot be changed in any way Intrinsic property
a half integer !!
Electron Spin The new kind of angular momentum is called the electron The new kind of angular momentum is called the electron SPINSPIN
Why call it 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
24
24
9.2848 10 J/T -- electron magnetic moment
= 9.2741 10 J/T (the "Bohr magneton")
s
B
Note: All particles possess spin (e.g., protons, neutrons, quarks, photons)
E sB|
B
B=0B0
sB
sB
A spin magnetic moment is associated with the spin angular momentum
s S
s sU B
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
2s ee
eg S
m
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
1
2zS 2 23
4S
Spin wave functions : eigenfunctions of [Sz] and [S2]
+ : spin-up wavefunction- : spin-down wavefunction
1
2zS 2 23
4S
( ) e i tr
Wavefunction: ,),r( mne YrR
states
n = 1, 2, 3,….
= 0, 1, 2,…, n-1m = 0, 1, 2,…, ms = 1/2
Eigenvalues
En = 13.6(Z/n)2
eV 1L
zL m / 2z sS m
Degeneracy in the absence of a magnetic field:
Each state has degenerate states
2(2+1) Each state n has degenerate states2n2
1
0122
n
two spin orientations
In strong magnetic fields, the torques are large
( ) ( )2 2L s z e z l e s
e e
eB eU B L g S m g m B
m m
Both the angular momenta precess independently around the B field
and L S
For an electron: ge = 2
spin up ms = 1/2
spin down ms = 1/2
12 e
eU m B
m
12 e
eU m B
m
2 2 22 L B
e
eU U U B B
m
For a given m,
Contribution to energy shifts
Total magnetic moment:
L s
B=0B0
sB
sB
(orientation of s)
m=1m=0m=1
n=1,m=0
Magnetic field B 0
Lo
m=1, ms=1/2
m=1, ms=1/2
m=0, ms=1/2
m=1, ms=1/2
m=0, ms=1/2
m=0, ms=1/2
m=0, ms=1/2
Lo
1, ( ) 0, 1sm m Selection rules:
1S
2P
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)
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
4.9V
a
Franck-Hertz Experiment
Direct confirmation that the internal energy states of an atom are quantized (proof of the Bohr model of the atom)
As a tool for measuring the energy changes of the mercury atom Franck and Hertz used electrons, that means an atomic tool
Recent Breakthrough – Detection of a single electron spin!
IBM scientists achieved a breakthrough IBM scientists achieved a breakthrough in nanoscale magnetic resonance in nanoscale magnetic resonance imaging (MRI) by directly detecting the imaging (MRI) by directly detecting the faint magnetic signal from a faint magnetic signal from a singlesingle
electron buried inside a solid sampleelectron buried inside a solid sample
Next step – detection of single nuclear spin (660x smaller)
Nature 430, 329 (2004)
Dutt et al., Science 316, 1312 (2007)