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)