The spin-orbit interactionIn the absence of an external magnetic field, internal field generated by electron motion (proportional to orbital angular momentum) will interact with spin dipole moment

Frame of nucleus

e

L

S

Ze

Frame of electron

e

L B

S

Ze

Nucleus circulates around electron a B-field due to the nuclear motion

s

Orbital dipole moment is anti-parallel to spin dipole moment

0sU B

higher energy state

when is parallel to L S

internal magnetic field自旋—軌道作用

(spin up)

Frame of nucleus

eZe

Frame of electron

e

S

L B

Ze

Nucleus circulates around electron a B-field due to the nuclear motion

s

Orbital dipole moment is parallel to spin dipole moment

0sU B lower energy

state

when is anti-parallel to L S

2p

1s

2p3/2

2p1/2

L

S

S

s

s The spin-orbit interaction

causes fine structure doubling of atomic spectral lines

S

(l = 0, |L| = 0)

(3/3/2010)

(l = 1, |L| > 0)

(spin down)

Total angular momentum (as a result of spin-orbit interaction)

J L S

Permissible values for the total angular momentum quantum number j

, 1, , | |j l s l s l s

For an atomic electron s = 1/2

1 1,

2 2j l l

Example,

for the 2p state l = 1, j = 3/2 or 1/2

for the 3d state l = 2, j = 5/2 or 3/2

for the s state l = 0, j = 1/2

magnitude: 1J j j

z-component: z jJ m , 1, , 1,jm j j j j

(maximum value)

(minimum value)

Neither nor is conserved separately!L S

(a) A vector model for determining the total angular momentum J = L + S of a single electron

(b) The allowed orientations of the total angular momentum J for the states j = 3/2 and j = 1/2. Notice that there are now an even number of orientations possible, not the odd number familiar from the space quantization of L alone

1 1,

2 2jm

j = 1/2

j = 3/2

3 1 1 3, , ,

2 2 2 2jm

mj has an even number of values

3| |

2J

15| |

2J

l = 1

Square of total angular momentum

22

2 2 2

J L S L S L S

L S L S

2 2 2

2

J L SL S

2

1 1 ( 1)2LSU L S j j s s

Spin-orbit interaction energy (a relativistic effect):

Energy shift due to spin-orbit interaction

2

2 2 3

1

4 2 4e

LSo e

Ze gU L S

m c r

r: radius of the orbiting electronc: speed of light

internal

internal ?

LS sU B

B

31 1

4LS oU U j j Write

Spin-orbit energy shift 31 1

4LS oU U j j

2P3/2

2P1/2

2S1/2

3/ 2 2 2 0

3 3 32P 1 1 1 1

2 2 4oE E U E U

1/ 2 2 2 0

1 1 32P 1 1 1 1 2

2 2 4oE E U E U

1/ 2 2 2

1 1 32S 1 0 0 1

2 2 4oE E U E

e.g., For a hydrogen atomUo = 1.510-5 eV

2S, 2P(n = 2)

2P3/2

2P1/2

ms=1/2 2S1/2

(8)

(2)

(2)

(4)

B=0B0

(but small) mj

3/21/21/23/2

1/21/2

1/21/2

3 1, 1,

2 2j l s

1 1, 1,

2 2j l s

1 1, 0,

2 2j l s

1

0

2 2 1n

Remarks:

In a small external B field, the spin-orbit interaction is dominant and the total angular momentum J as a whole precesses around the B field. The no. of split lines = the no. of mj values

In a large external B field, both the orbital angular momentum L and the spin angular momentum S precess independently around the B field. The no. of splitting lines = 2 (2l + 1)

Quantum numbers – in the absence of spin-orbit effect, a state of an atomic electron is specified by (n, l, ml, ms). If the spin-orbit interaction is taken into account, the state may be specified by (n, l, j, mj) 2P3/2, 2P1/2, 2S1/2

Example: electronic states associated with the principle quantum number n = 2

1 12,0,0, , 2,0,0,

2 2

1 12,1,0, , 2,1,0,

2 2

1 12,1,1, , 2,1,1,

2 2

1 12,1, 1, , 2,1, 1,

2 2

( , , , )sln l m m

1 1 1 12,0, , , 2,0, ,

2 2 2 2

3 3 3 12,1, , , 2,1, ,

2 2 2 2

3 1 3 32,1, , , 2,1, ,

2 2 2 2

1 1 1 12,1, , , 2,1, ,

2 2 2 2

( , , , )jn l j m

(in the absence of spin-orbit interaction)

(in the presence of spin-orbit interaction)

The angular momentum vectors L, S, and J for a typical case of a state with l = 2, j = 5/2, mj = 3/2. The vectors L and S precess uniformly about their sum J, as J precesses randomly about the z axis

Strong B fieldWeak B field

In a strong B field, the orbital angular momentum L precesses about the z axis.(Similarly for the spin angular momentum S )

(3/8/2010)

Thus, electrons do not pile up in the lowest energy state, i.e, the (1,0,0) orbital They are distributed among the higher energy levels according to

the Pauli Principle Particles that obey Pauli exclusion principle are called “fermions”

Pauli Exclusion PrincipleFrom spectra of complex atoms, Pauli (1925) deduced a new rule:

In a given atom, no two electrons can be in the same quantum state, i.e. they cannot have the same set of quantum numbers n,,m,ms

Every “atomic orbital with n,,m” can hold two electrons:

(n,,m,) and (n,,m,)

More generally, no two identical fermions (any particle with spin of ħ/2, 3ħ/2, etc.) can be in the same quantum statestate

Pauli (1900–1958)

Two-particle problems

– Consider two particles in a one-dimensional potential well U = U(x)

There exist a series of eigenstates

Quantum state labels: a, b, c, ……

Eigenfunctions: a(x), b(x), c(x), ……

Eigenvalues: Ea, Eb, Ec, ……

e.g., one-particle problem – if there is only one particle in the potential well and the particle is in the state a

2 2

2( ) Write:

2 a a a a a a a

dx U x x E x H x E x

m dx

H: Hamiltonian

Now consider two particles, 1 and 2, in the potential well. Assume particle 1 be in the state a and particle 2 be in the state b

2 2

1 1 1 1 1 1 121

( ) ( )2 a a a a a a a

dx U x x E x H x x E x

m dx

2 2

2 2 2 2 2 2 222

( ) ( )2 b b b b b b b

dx U x x E x H x x E x

m dx

Guess: the total wavefunction for this two-particle system

2 2 2 2

1 2 1 22 21 2

1 2 1 2

1 2 1 2

1 2 1 2

1 2

1 2

( ) ( ) ,2 2

( ) ( ) ,

( ) ( )

( )

( ) ,

a b

a a b a b b

a b a b

a b

d dU x U x x x

m dx m dx

H x H x x x

H x H x x x

E x x x E x

E E x x

E E x x

(x1,x2) = a(x1)b(x2) i

s a solution, with the total

energy E = Ea + Eb

The wavefunction for particle 1 in state a and particle 2 in state b

Now let the positions of the two particles be exchanged. Let particle 2 be in the state a and particle 1 be in the state b

2 2

1 1 1 1 1 1 121

( ) ( )2 b b b b b b b

dx U x x E x H x x E x

m dx

2 2

2 2 2 2 2 2 222

( ) ( )2 a a a a a a a

dx U x x E x H x x E x

m dx

Guess: the total wavefunction for this two-particle system

2 1 2 1

2 1 2 1

2 1 2 1

2 1

2 1

( ) ( ) ,

( ) ( )

( )

( ) ,

a b

a a b a b b

a b a b

a b

H x H x x x

H x H x x x

E x x x E x

E E x x

E E x x

(x2,x1) = a(x2)b(x1) i

s a solution, with the total

energy E = Ea + Eb

The wavefunction for particle 2 in state a and particle 1 in state b

Assume for a given potential energy U(x),

state a – wavefunction,

state b – wavefunction,

Two-particle wavefunction for particle 1 in state a, particle 2 in state b

Two-particle wavefunction for particle 2 in state a, particle 1 in state b

sina ax A k x

cosb bx A k x

1 2 1 2 1 2, ( ) ( ) sin( )cos( )a b a bx x x x A k x k x

2 1 2 1 2 1, ( ) ( ) sin( )cos( )a b a bx x x x A k x k x

“Probability density” for particle 1 in state a and particle 2 in state b

“Probability density” for particle 2 in state a and particle 1 in state b

The guess solutions (x1,x2) = a(x1)b(x2) and (x2,x1) = a(x2)b(x1) imply t

hat the “probability density” for finding the two particles is not the same under exchange of the two particle coordinates

2 2 2 21 2 1 2 1 2, | , | | | sin ( )cos ( )a bP x x x x A k x k x

2 2 2 22 1 2 1 2 1, | , | | | sin ( )cos ( )a bP x x x x A k x k x

2 1 1 2, ,P x x P x x

Quantum particles are “indistinguishable” – a purely quantum-mechanical concept; no classical analogy !

Can measurable physical quantities be differing under particle exchange ?

The scattering of two electrons as a result of their mutual repulsion. The events depicted in (a) and (b) produce the same outcome for identical electrons but are nonetheless distinguishable classically because the path taken by each electron is different in the two cases. In this way, the electrons retain their separate identities during collision. (c) According to quantum mechanics, the paths taken by the electrons are blurred by the wave properties of matter. In consequence, once they have interacted, the electrons cannot be told apart in any way!

Electrons are identical particles. It is impossible to distinguish one electron from another ! Also, wavefunctions can overlap !!

a(x1)b(x2) and a(x2)b(x1) are not the correct wavefunctions

Need to search for the right wavefunctions ?!

Exchange Symmetry

The simplest case: Two identical (“indistinguishable”) particle system

Consider two identical particles satisfying 3D Schrödinger equations with coordinates r1 and r2. Assume no interaction between them

2

21 1 1 1 1 1 12 a a a a ar U r r E r H r r

m

2

22 2 2 2 2 2 22 b b b b br U r r E r H r r

m

1 2 1 2 1 2, ,H r H r r r E r r

exchange particles

1 2 1 2, a br r r r

Particle 1 has energy Ea and particle 2 has energy Eb. E = Ea + Eb

r1 r2

2 1 2 1, a br r r r

Particle 1 has energy Eb and particle 2 has energy Ea

A distinguishable solution !

Try linear combinations of the two wavefunctions:

1 2 1 2 2 1

1,

2S a b a br r r r r r S is a solutions

of the (linear) Schrödinger eq. with E = Ea + Eb

1 2 1 2

1 2 1 2 2 1

,

1

2

S

a b a b

H r H r r r

H r H r r r r r

1 1 2 1 2 1

2 1 2 2 2 1

1

21

2

a b a b

a b a b

H r r r H r r r

H r r r H r r r

1 2 2 1

1 2 2 1

1

21

2

a a b b a b

b a b a a b

E r r E r r

E r r E r r

1 2 2 1 1 2

1( ) ( , )

2a b a b a b a b SE E r r r r E E r r

Try linear combinations of the two wavefunctions:

1 2 1 2 2 1

1,

2A a b a br r r r r r

A is a solutions of the (linear) Schrödinger eq. with E = Ea + Eb

1 2 1 2

1 2 1 2 2 1

,

1

2

A

a b a b

H r H r r r

H r H r r r r r

1 2 2 1

1 2 2 1

1

21

2

a a b b a b

b a b a a b

E r r E r r

E r r E r r

1 2 2 1

1 2

1

2( ) ( , )

a b a b a b

a b A

E E r r r r

E E r r

Under exchange of particle coordinates r1 r2

1 2 a 1 b 2 a 2 b 1

1r ,r r r r r

2S

2 1 12 22 1 1

1, ,

2aS b Sb ar r rr r r rr

1 2 1 2 2 1

1,

2A a b a br r r r r r

2 1 12 1 1 22, ,1

2a b a bA Ar r r rr r r r

symmetric

anti-symmetric

Both satisfy the Exchange Symmetry Principles

2 2

1 2 2 1, , ,S Sr r r r No observable difference !!

Exchange symmetry: For both S and A, all the probability densities (observable effects) are unaffected by the interchange of particles

2 2

1 2 2 1, ,A Ar r r r

A two-electron system: the helium atom He

2e+

1a r

2b r

2 2

21 1 1 1

1

2

2 4a a a ae o

er r E r

m r

1 1 1a a aH r r E r

2 2

22 2 2 2

2

2

2 4b b b bo

er r E r

m r

2 2 2b b bH r r E r

1 2 1 2 2 1

1,

2S a b a br r r r r r

1 2 1 2 2 1

1,

2A a b a br r r r r r

symmetry space wavefunction

anti-symmetric space wavefunction

Total two-particle wavefunction:

1 2 space 1 2 spin( , ) , ( , )r r r r

A describes a class of particles called “fermions”

S describes a class of particles called “bosons”

Pauli exclusion principle states that no two electrons in an atom can have the same set of quantum numbers (no two electrons can occupy the same quantum state). The exclusion principle applies only to fermions

It is an experimental fact that integer spin particles are bosons, but half-integer spin particles are fermions

1 2 2 1( , ) ( , )A Ar r r r

1 2 2 1( , ) ( , )S Sr r r r

(Both the spatial coordinates and the spin orientations of the two particles are to be simultaneously interchanged)

The symmetry character of various particles

particleparticle symmetrysymmetry Spin (s)Spin (s)

Electron antisymmetric 1/2

Positron antisymmetric 1/2

Proton antisymmetric 1/2

Neutron antisymmetric 1/2

Muon antisymmetric 1/2

particle symmetric 0

He atom (G) symmetric 0

meson symmetric 0

Photon symmetric 1

Deuteron symmetric 1

Fermions（費米子）

Bosons（玻思子）

half-integer spin particles

integer spin particles

e.g., Helium isotope 3He (2 protons + 1 neutron) – a fermion

Example 9.5 Ground state of the helium atom

Construct explicitly the two-electron ground state wavefunction for the helium atom in the independent particle approximation

(Assume each helium electron sees only the doubly charged helium nucleus)

Hydrogen-like atom with the lowest energy, Z = 2

3/ 2

2100 10 00

1 2( ) ( , ) or a

o

r R r Y ea

E1 = 2213.6

= 54.4 eV

1 2 0

1 2 100 1 100 2 100 2 100 1

3

2( ) /

0

1, ( ) ( ) ( ) ( )

2

2 2

S

r r a

r r r r r r

ea

symmetric spatial

wavefunction

Spin eigenfunctions for two-electron systems

1 1| | | , | ,

2 2A

| | ,

1 1| | | , | ,

2 2| | ,

S

Singlet state (單重態 ), S = 0, ms = 0

Triplet state (三重態 ), S = 1, ms = 0, 1

ms=0

ms=1

ms=1

| , |

| , |

Linear combinations of

anti-symmetric spin wavefunction

symmetric spin wavefunction

1 2 0

1 2 1 2

3

2( ) /

0

( , ) ( , )

1 2 | , | ,

A S A

r r a

r r r r

ea

Total two-electron wavefunction for a helium atom

The ground state – only once choice: 1S2, spin singlet (total spin = 0)

The equal admixture of the spin states |+ and |+ means the spin of the first helium electrons is just as likely to be up as it is to be down

The spin of the second electron must always be opposite the first

If both electrons had the same spin orientations A = 0. The theory allows no solution no two electrons can occupy the same quantum state

Remarks:

In practice, it requires only 24.6 eV to remove the first electron. To remove the second electron requires 54.4 eV. The electron left behind screens the nuclear charge and make the ionization of the first electron easier

The other possible choice of wavefunction A(r1,r2) = A(r1,r2)triplet must comprise a higher energy eigenstate

E = 2 ( Z213.6 eV) = 108.8 eV, with Z = 2, n = 1

Energy for the two electrons in a helium atom

A naïve estimate:

symmetric under spin exchange

Example: The first excited state, 1S2S

-110

-90

-70

-50

Simple prediction by ignoring e-e interaction

Experimental spectrum

E (eV)

(ground state)

(1st excited state)

A naive estimate:

E = 13.6 eV [(4/1)+(4/4)]

= 68 eV

Electron-electron interaction plays an important role in multi-electron atoms and should not be ignored

the first four excited states

the ground state

The wavefunction (a) and probability density (b) for the approximate molecular wave + formed from the symmetric combination of atomic orbitals centered at r = 0 and r = R. (c) and (d): Wavefunction and probability density for the approximate molecular wave - formed from the anti-symmetric combination of these same orbitals

Hydrogen molecular ion: H2+

The bonding state (a), a spin singlet state, possesses a lower energy than the anti-bonding state (c), a spin triplet state

(Serway, Fig. 11.7)

1921 – Stern-Gerlach experiment of anomalous Zeeman effect

1925 – Pauli exclusion principle (the fourth degree of freedom)

1925 – Goudsmit-Uhlenbeck theory of electron spin

當泡利處心積慮的鑽研反常塞曼效應時，當時在慕尼黑的朋友問他：「你為何總是愁眉苦臉的？」泡利馬上反問道：「要是你陷入了反常塞曼效應的思慮，難道還快樂的起來嗎？」根據當時的條件，泡利是無法圓滿的解釋這個效應的。而沿著這條路走下去，卻使他得以發現不相容原理。

《微觀絕唱》， p.143

Ehrenfest on Goudsmit-Uhlenbeck theory:

「你們還很年輕，做點蠢事也沒有什麼關係！」 Lorentz

Wolfgang Pauli Otto Stern

Kondo effect – a singlet ground state formed by the localized moment and the screening conduction electron spins

Electron-electron interactions and Screening effects

Helium atom: 1 2 1 20 1 2

( )( ),

4 | |

e eH r r H r H r

r r

Each electron sees not only the attractive helium nucleus but also the other electron

Electrons confined to the small space of an atom exert strong repulsive electrical forces on each other (especially in multi-electron atoms)

Is the charge of nucleus = +2e from electron’s viewpoint ?

Inner electrons screen the positive charge of nucleus, and hence reduce the effective Z and the attractive potential smaller negative potential energy (especially for multi-electron atoms)

(??)

The resulting effective potential energy seen by an electron in the atom can be notably smaller than the bare potential energy

1834~1907

MendeleevMendeleev’s table as published in 1869 – 63 elements

“All good theories should be able to make predictions and this was no exception”

GaGe

Sc

The Periodic Table

Build a multiple electron atom

Chemical properties of an atom are determined by the least tightly bound electrons

Occupancy of subshell Energy separation between the subshell and the next higher subshell

s shell =0

p shell =1

Helium and Neon and Argon are inert. Their outer subshell is closed

n

1

2

3

Hydrogen: (n, , mℓ, ms) = (1, 0, 0, ±½) in ground stateIn the absence of a magnetic field, the state ms = ½ is degenerate with the

ms = −½ state

Helium: (1, 0, 0, ½) for the first electron; (1, 0, 0, −½) for the second electron.

The two electrons have anti-aligned spins (ms = +½ and ms = −½)

The principle quantum number has letter codes.n = 1 2 3 4...Letter = K L M N…

n = shells (e.g., K shell, L shell, …)

n = subshells (e.g., 1s, 2p, 3d, …)

How many electrons may be placed in each subshell?

= 0 1 2 3 4 5 …

Letter = s p d f g h …

Total

For each mℓ: two values of ms 2

For each : (2 + 1) values of mℓ 2(2 + 1)

1s

1s2

1s22s22p63s2

1s22s22p63s

1s22s22p6

1s22s22p1

1s22s22p2

1s22s22p3

1s22s22p4

1s22s22p5

1s22s2

1s22s

The filling of electronic states must obey the Pauli exclusion principle and Hund’s rule (spin-spin correlation)

Energies of orbital subshells increase with increasing

Energies of orbital subshells are lower for lower . Lower angular momentum implies more eccentric classical orbits, with greater penetration into the nuclear regime

Fill electrons to states according to the lowest value of n+, with “preference” given to n

The lower values have more eccentric classical orbits than the higher values

Electrons with higher values are more shielded from the nuclear charge

Electrons lie higher in energy than those with lower values

4s fills before 3d

1

23

34

8

455

566

67

78

n+

1s2s 2p

7s 7p 7d 7f 7g 7h 7i 6s 6p 6d 6f 6g 6h

3s 3p 3d4s 4p 4d 4f5s 5p 5d 5f 5g

Periods: Horizontal rows Correspond to filling

of the subshells

Groups: Vertical columns Same number of electrons in an orbit Can form similar chemical bonds

Inert Gases: Last group of the periodic table Closed p subshell except helium Zero net spin and large ionization energy Their atoms interact weakly with each other (van der Waals forces)

Alkalis: Single s electron outside an inner core Easily form positive ions with a charge +1e Lowest ionization energies Electrical conductivity is relatively good

Alkaline Earths: Two s electrons in outer subshell Largest atomic radii High electrical conductivity

Halogens: Need one more electron to fill outermost subshellNeed one more electron to fill outermost subshell Form strong ionic bonds with the alkalisForm strong ionic bonds with the alkalis More stable configurations occur as the More stable configurations occur as the pp subshell is filled subshell is filled

Transition Metals: Three rows of elements in which the 3Three rows of elements in which the 3d, 4, 4d, and 5, and 5d are being filled are being filled Properties primarily determined by the Properties primarily determined by the ss electrons, rather than by the electrons, rather than by the

d subshell being filled subshell being filled Have Have d-shell electrons with unpaired spins-shell electrons with unpaired spins As the As the d subshell is filled, the magnetic moments, and the tendency subshell is filled, the magnetic moments, and the tendency

for neighboring atoms to align spins are reducedfor neighboring atoms to align spins are reduced

Lanthanides (rare earths): Z = 57 ~ Z = 70 Have the outside 6s2 subshell completed As occurs in the 3d subshell, the electrons in the 4f subshell have

unpaired electrons that align themselves The large orbital angular momentum contributes to the large

ferromagnetic effects

Actinides: Z = 89 ~ Z = 102 Inner subshells are being filled while the 7s2 subshell is complete Difficult to obtain chemical data because they are all radioactive Have longer half-lives

X-ray spectra and Moseley’s law Target :W

series of sharp lines: “characteristic spectrum” of the target material

1913–4, Moseley measured characteristic x-ray spectra of 40 elements, and observed “series” of x-ray energy levels, called K, L, M, … etc.

Analogous to optical series for hydrogen (e.g. Lyman, Balmer, Paschen…)

Electronic transitions within the inner shells of heavy atoms are accompanied by emission of high-energy photons (x rays)

scattered incident electron

incident electron

ejected K-shell electron

hole in K-shell (n = 1)

The characteristic spectrumdiscovered by Bragg systematized by Moseley

x-ray

hole in L-shell (n = 2)

The characteristic peak is created when a hole in the inner shell created by a collision event is filled by an electron from a higher energy shell

K

L

M

N

O

n=1

n=2

n=3

n=4

n=5

K K K K K

L L L L

M M M

N N

When a K-shell electron be knocked out, the vacancy can be filled by an electron from the L-shell (K radiation) or the M-shell (K radiation)

K series: n = 2, 3, … to n = 1

2L K 2

113.6 1 1 eV

hcE E Z

n

Consider the K line. An electron in the L shell is partially screened from the nucleus by the one remaining K shell electron

2

L 2

( 1)13.6 eV

ZE

n

2

K 2

( 1)13.6 eV

1

ZE

e.g., (K) = 0.723 Å for molybdenum (Z = 42)

Atomic number versus square root of frequency (Moseley,

1914)

L series: n = 3, 4, … to n = 2

22 2

1 113.6 7.4 eV

2L

hcE Z

n

Moseley’s law: the square root of photon frequency should vary linearly with the atomic number Z (A direct way to measure Z )

Moseley established the fact that the correct sequence of elements in the periodic table is based on atomic number rather than atomic mass

Nuclear Magnetic Resonance Imaging (NMRI or MRI)

MRI is a medical imaging technique used in radiology to visualize the internal structure and function of the body. It provides great contrast between the different soft tissues of the body, making it very useful in brain and cancer imaging. It uses no ionizing radiation, but uses a powerful magnetic field to align the nuclear magnetization of hydrogen atoms in water in the body

Proton magnetic moment: (very small compared with electron magnetic moment)

proton protonproton

12.79 , =

2

eS s

M

2proton 10 10 MHzU B

Resolution: 1 – 0.1 mm

An expensive, large-volume superconducting magnet of several Tesla

proton 2.79e

SM

proton

24

2.79

2.79 2.792 2

2.79

Bohr magneton: 9.274 10 J/T

z

B

B

eU B S B

Me m e

B BM M m

mB

M

B=0B 0

42 MHz, = 1 TE B

1

2zS

Summary:

Orbital magnetism and the normal Zeeman effect

Electron spin

The spin-orbit interaction

Exchange symmetry and the Exclusion principle

The periodic table

X-ray spectra and Moseley’s law