Harris Chapter 7- Atomic Structure

• 7.1– Orbital Magnetic Moments, discovery of intrinsic spin

• 7.2 & 7.3– Identical Particles (warning: examples in book all inf-squ well)– Exclusion Principle

• 7.4 & 7.5– Multielectron Atoms, effective charges– Hartree Treatment

• 7.6– Spin-Orbit Effect

• 7.7– Adding QM Angular Momenta

• 7.9 & 7.8– Multielectron Spectroscopic Notation– Zeeman Effect

Summary So Far

http://asd-www.larc.nasa.gov/cgi-bin/SCOOL_Clouds/Cumulus/list.cgi

RER

rmVr

rm rr

2

22

2

2 1

2

1

2

,mnl YrRr

EV

m2

2

2

2222

22

2

sin

1sin

sin

11

rr

rr rr

7.1 Orbital Magnetic Moments and Discovery of Intrinsic Spin

Two kinds of Angular Momentum

• Classical Angular Momentum– L = r x p– r vector, p vector L vector– L obeys vector math– Any L possible, no contraints on Lx Ly Lz

• Quantum– Quantum Mechanical Angular Momentum– L = r x p– r vector, p vector operator LL 3 component operator– LL obeys …… got to be careful– LL described by two labels l , m– L and Lz can be known, Lx and Ly cannot

Bohr Model of Ang Momentum

Note: s-states (l=0) have no Bohr model picture

Eisberg & Resnick: Fig 7-11

Classical orSemi-classical

description

Vector Model of QM Ang. Momentum

quantum numbers

m

E&R Fig 7-12

pg 19: “We might imagine the vector moving in an unobservable way about the z-axis...”

Edmonds“A.M. in QM”

pg 29: “The QM probability density, not being time dependent, gives us no information about the motion of the particle in it’s orbit.”

*(r,t) (r,t)

(r,t)=(r) eit

Morrison, Estle, Lane “Understanding More QM”, Prentice-Hall, 1991

Otto Stern & Walther Gerlach~1922

nprL

nqdp

Assigned by advisor Max Born to demonstrate existence of the l, ml quantum numbers

1

3

2

Bohr’s Q hypothesis

Sommerfeld’s Q hypothesis

22 1 L

mLz

Ai

vr

q

t

Qi

/2

mvrprL

Lm

q 2

Lm

eg

2

electron

neutron

proton

g

1

0

1

Orbital Magnetic Moment

E&R Fig 7-11

Ai

Lm

eg

2

12

m

eg

mm

egzz 2

electron

neutron

proton

g

1

0

1

Orbital Magnetic MomentE&R Fig 7-11

Lm

eg

2

22310927.02

Amm

ebohr

L

bohrg

Bohr magneton

E&R Fig 7-11

BU z

BUnOrientatioofEnergyPotential

0 BUF z

B

BU z

BUnOrientatioofEnergyPotential

BUF z

Different ml states experience different forces

B

dz

dBF zz

Lm

eg

2

BU z

BUnOrientatioofEnergyPotential

Use B as z-axis.

BUF z

dz

dBF zz

Different ml states experience different forces

Stern & Gerlach~1922

Harris Fig 7.3, 7.4

Stern & Gerlach~1922

http://upload.wikimedia.org/wikipedia/en/2/29/Stern-Gerlach_experiment.PNG

Intended to demonstrate space quantization (l), & therefore expected odd number of spots, but observed an even number.

Despite Stern's careful design and feasibility calculations, the experiment took more than a year to accomplish. In the final form of the apparatus, a beam of silver atoms (produced by effusion of metallic vapor from an oven heated to 1000°C) was collimated by two narrow slits (0.03 mm wide) and traversed a deflecting magnet 3.5 cm long with field strength about 0.1 tesla and gradient 10 tesla/cm. The splitting of the silver beam achieved was only 0.2 mm.

Accordingly, misalignments of collimating slits or the magnet by more than 0.01 mm were enough to spoil an experimental run. The attainable operating time was usually only a few hours between breakdowns of the apparatus. Thus, only a meager film of silver atoms, too thin to be visible to an unaided eye, was deposited on the collector plate.

Stern described an early episode:

http://www.physicstoday.org/pt/vol-56/iss-12/p53.html

Stern described an early episode:

After venting to release the vacuum, Gerlach removed the detector flange. But he could see no trace of the silver atom beam and handed the flange to me. With Gerlach looking over my shoulder as I peered closely at the plate, we were surprised to see gradually emerge the trace of the beam. . . . Finally we realized what [had happened]. I was then the equivalent of an assistant professor. My salary was too low to afford good cigars, so I smoked bad cigars. These had a lot of sulfur in them, so my breath on the plate turned the silver into silver sulfide, which is jet black, so easily visible. It was like developing a photographic film.7

http://www.physicstoday.org/pt/vol-56/iss-12/p53.html

Wolfgang Pauli ~ 1924

• Pauli Exclusion Principle• No two electrons can have the

same quantum number

• Postulated an additional quantum number (i.e. label)

• Believed it came from the interaction between electrons.

Ralph Kronig ~1925

• Spinning Electron Idea

Goudsmit & Ulhenbeck ~ 1925

• Studied high resolution spectra of alkali elements

Ocean Optics - Helium

Ocean Optics - Neon

Giancoli – fig 36.21

The old and the new term scheme of hydrogen [5]. The scheme shows the multiplet splitting of the excited states of the hydrogen atom with principal quantum number n=3, presented by Goudsmit in the form in which it appeared in the original publications of1926. The assignment in the current notation has been added at the right. With the development of quantum mechanics the notation changed. The quantum numbers L and J now usedfor the orbital and total angular momentum, respectively, correspond to K-1/2 and J-1/2 in the figure. The "forbidden component" referred to by Goudsmit is of the type 3 2P1/2 --> 2 2S in which the total angular momentum is conserved

and L changes by plus or minus 1.

[5] S. Goudsmit and G.E. Uhlenbeck, Physica 6 (1926) 273.

Uhlenbeck & Goudsmit~ 1925

The discovery note in Naturwissenschaften is dated 17 October 1925. One day earlier Ehrenfest had written to Lorentz to make an appointment and discuss a "very witty idea" of two of his graduate students. When Lorentz pointed out that the idea of a spinning electron would be incompatible with classical electrodynamics, Uhlenbeck asked Ehrenfest not to submit the paper. Ehrenfest replied that he had already sent off their note, and he added: "You are both young enough to be able to afford a stupidity!"

http://www.lorentz.leidenuniv.nl/history/spin/spin.html

Uhlenbeck & Goudsmit~ 1925

Ehrenfest's encouraging response to his students ideas contrasted sharply with that of Wolfgang Pauli. As it turned out, Ralph Kronig, a young Columbia University PhD who had spent two years studying in Europe, had come up with the idea of electron spin several months before Uhlenbeck and Goudsmit. He had put it before Pauli for his reactions, who had ridiculed it, saying that "it is indeed very clever but of course has nothing to do with reality". Kronig did not publish his ideas on spin. No wonder that Uhlenbeck would later refer to the "luck and privilege to be students of Paul Ehrenfest".

http://www.lorentz.leidenuniv.nl/history/spin/spin.html

“This isn't right. This isn't even wrong.” There were some people thinking about

electron spin in those days, but there was a lot of basic opposition to such an idea. One of the first was Ralph de Laer Kronig. He got the idea that the electron should have a spin in addition to its orbital motion. He was working with Wolfgang Pauli at the time, and he told his idea to Pauli. Pauli said, "No, it's quite impossible." Pauli completely crushed Kronig.

Then the idea occurred quite independently to two Young Dutch physicists, George Uhlenbeck and Samuel Goudsmit. They were working in Leiden with Professor Paul Ehrenfest, and they wrote up a little paper about it and took it to Ehrenfest. Ehrenfest liked the idea very much. He suggested to Uhlenbeck and Goudsmit that they should go and talk it over with Hendrik Lorentz, who lived close by in Haarlem.

"The Birth of Particle Physics," edited by Laurie M. Brown and Lillian Hoddeson. The essay by Paul A.M. Dirac is entitled "Origin of Quantum Field Theory."

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pauli.html

His ability to make experiments self destruct simply by being in the same room was legendary, and has been dubbed the "Pauli effect" (Frisch 1991, p. 48; Gamow 1985).

“This isn't right. This isn't even wrong.”

They did go and talk it over with Lorentz. Lorentz said, "No, it's quite impossible for the electron to have a spin. I have thought of that myself, and if the electron did have a spin, the speed of the surface of the electron would be greater than the velocity of light. So, it's quite impossible." Uhlenbeck and Goudsmit went back to Ehrenfest and said they would like to withdraw the paper that they had given to him. Ehrenfest said, "No, it's too late; I have already sent it in for publication "

"The Birth of Particle Physics," edited by Laurie M. Brown and Lillian Hoddeson. The essay by Paul A.M. Dirac is entitled "Origin of Quantum Field Theory."

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pauli.html

His ability to make experiments self destruct simply by being in the same room was legendary, and has been dubbed the "Pauli effect" (Frisch 1991, p. 48; Gamow 1985).

The calculation(using current values)

IS

2

5

21 rmss

IL

rv

r < 2.8 E-19 m

> 3 * 10 + 6

value from Bhabha scattering at CERN

“This isn't right. This isn't even wrong.”

That is how the idea of electron spin got publicized to the world. We really owe it to Ehrenfest's impetuosity and to his not allowing the younger people to be put off by the older ones. The idea of the electron having two states of spin provided a perfect answer to the duplexity.

"The Birth of Particle Physics," edited by Laurie M. Brown and Lillian Hoddeson. The essay by Paul A.M. Dirac is entitled "Origin of Quantum Field Theory."

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pauli.html

His ability to make experiments self destruct simply by being in the same room was legendary, and has been dubbed the "Pauli effect" (Frisch 1991, p. 48; Gamow 1985).

Letter fm Thomas to Goudsmit

Part of a letter by L.H. Thomas to Goudsmit (25 March 1926). Reproduced from a transparency shown by Goudsmit during his 1971 lecture. The original is presumably in the Goudsmit archive kept by the AIP Center for History of Physics.

http://www.lorentz.leidenuniv.nl/history/spin/goudsmit.html

intrinsic spin

• Fundamental objects– electron spin – ½ – neutrino spin – ½ , but LH only– photon spin – 1

• Composite objects– proton spin – ½ – neutron spin – ½ – delta spin – 3/2

How to Denote Wavefunctions(version 1)

sls smmnsmlmn YrR

the spinor has no ‘functional form’ because spin

is not a spatial feature

ss smlmnsmlmn

ss smlmnlsmlmn

Lm

eg

2

22310927.02

Amm

ebohr

L

bohrg

Two types of Magnetic Moments

Sm

egss

2

S

bohrss g

SL

electron

neutron

proton

g

1

0

1

electron

neutron

proton

g s

00.2

83.3

59.5

interesting fundamental constants

-2.002 319 304 3622 (15)

1.602 176 487 (40) x 10-19 C   

7.2 & 7.3 Complications from having Identical Particles

Exchange Symmetry

7.4 & 7.5 Multielectron Atoms

,mnl YrRr

2/ eGRn

r = n2 ao / ZEn = ( 13.6 eV ) (Z2/n2)

ao = 0.529 Å

,mnl YrRr

Prob = r2 R* R

2s

3s

2p

4s

1s

3p

3d4p

orbitals get sucked down the most

Crossings occur for the upper orbitals

0

1s sucked off bottom of page

Note: This shows how theorbitals shift as viewedfrom the perspective of an s-orbital.

Hartree-Fock Method

Hartree-Fock MethodsChoose initial shape

For Coulomb Potl V(r) Solve Schro Eqnfor En n

Build atom according toThis set of orbital energies En

Use the collection of n*n to

Get new electron charge distrib

Use Gauss’ Law toget new V(r) shape

Lo

op

un

til V

(r)

do

esn

’t ch

an

ge

mu

ch

Insert fine structurecorrections

r

eZkrV eff

2

)(

Using effective charge is a very crude approximation.

r2 ~ n2 ao / Zeff

En ~ (Zeff2/n2) ( -13.6 eV )

Hartree-FockEffective Charge Effects

7.6 Spin-Orbit Effect

Corrections to the Coulomb Potlfor H-atom

• Central Potential• Spin-Orbit (electron viewpoint)• Relativistic Spin (Thomas precession)• Relativistic Kinetic Energy• Spin-Orbit (nucleus viewpoint)• Spin-Spin• Impact of External Fields

– Zeeman Effect (applied B-field)– Stark Effect (applied E-field)

Spin-Orbit Interaction

Note: L.H. Thomas showed that in the x-form between non-inertial reference frames a factor of ½ appears.

s

L

s

L

Goal: find expression for the orientational potential energy of electron intrinsic mag moment (s) in terms of orbital motion (L) and forces (~ dV/dr).

s

L

BU s

Note: L.H. Thomas showed that in the x-form between non-inertial reference frames a factor of ½ appears.

BU s

2

1

BU s

2

1

2

ˆ

4 r

rldIB o

vZet

lQl

t

QdlI

2

ˆ

4 r

rvZeB o

s

L

2

ˆ

4 r

rvZeB o

2

ˆ

4 r

rZeE

o

EvB oo

FEe

rdr

dVVF ˆEv

cB

2

1

r

rv

dr

dV

ecrv

dr

dV

ecB

22

1ˆ

1

rvmvmrL

L

dr

dV

remcB

112

E

BU s

2

1

Ldr

dV

remcB

112

Sm

egss

2

LSdr

dV

rcmU

1

2

122

SL

NRG shift depends on relative orientation

of L and S

How to evaluate E and S·L

LSdr

dV

rcmU

1

2

122 S

L

involved in radial

integrations

depends on A.M. qu. no.s

dddrrUUEspaceall

njmlsnjmlstot sin2*

LSdr

dV

rcmE

R

1

2

122

SLJ

22 SLJ

SLSLJ 2222

2222 SLJSL

222

1112

11

2

1 ssjj

dr

dV

rcmE

R

electronSpin-Orbit “locks” the angle between L & S J is now a well-defined direction.

LSdr

dV

rcmE

R

1

2

122

S

SL

L

JNOTE

Lz

is no longer

well-defined

ml not a good q. no.

Revised H-atom Level Scheme

add in spin-orbitcorrection

2/11s

2/12s 2/12 p

2/32 p

s1

s2 p2

not required to specify NRG j mj l ml s ms

not required to specify NRG mj ml s ms

2/13s 2/13p

2/33p

2/33d

2/53d

s3 p3 d3

nlj

absolutely worthless

electron Spin-Orbit is more important in higher-Z atoms

222

1112

11

2

1 ssjj

dr

dV

rcmE

R

fn’l expression only for H-atom, for all others, must come fromHartree procedure

Li Na K Rb Cs

Splitting

(eV)

0.42E-4 21.E-4 72.E-4 295.E-4 687.E-4

Bigger atoms larger Z (central charge) ~ same size

dr

dV

r

1larger

7.7 QM Angular Momentum

Bohr Model of Ang Momentum

Note: s-states (l=0) have no Bohr model picture

Eisberg & Resnick: Fig 7-11

Vector Model of Ang. Momentum

quantum numbers

m

E&R Fig 7-12

pg 19: “We might imagine the vector moving in an unobservable way about the z-axis...”

Edmonds“A.M. in QM”

pg 29: “The QM probability density, not being time dependent, gives us no information about the motion of the particle in it’s orbit.”

Morrison, Estle, Lane “Understanding More QM”, Prentice-Hall, 1991

ADDITION OF

ANGULAR MOMENTUM

Ltot = L1 + L2

L1

L2

Ltot = L1 + L2

11 m

,11mlY

22 m

tottot m

,22mlY ,

tottotmlY

Ltot = L1 + L2

11 m

2121 tot

22 m

tottot m

21 mmmtot

Addition of Angular Momentum

aligned configuration

jack-knife configuration

www.bokerusa.com

www.cartowning.co.za/DBNRECGC.htm

“aligned” does not mean straight

“jack-knife” does not mean antiparallel

Detailed Example

Problem: Two objects each travel in a p-orbit ( l=1 ). The total energy of each object is degenerate wrt ml, so

we have no detailed knowledge of ml.

What are the allowed values of ltot, mtot ?

L1

L2

l1=1, l2=1, m’s degenerate

m1 m2 mtot

“

“

“

“

“

“

mtot Possibilities (m1,m2)

Allowed Values of ltot mtot

Basic A.M. Math

J = L + S

sjs

sj mmm

L

S

J

Vector Representation of J

Annoying Pictures #1

Jeff’s Qs: i) what am I supposed to think about the S & L cones as drawn? ii) I thought I was told earlier that L & S were about z ??

Annoying Pictures #2

Jeff: Pictures such as this confuse the vector symbols L and S with the quantum numbers ℓ and s .For instance, how could L and S ever point in the same direction?

TOTAL ANGULAR MOMEMTUM

J = L + S

More Detailed H-atom Level Scheme

2/11s

2/12s 2/12 p

2/32 p

s1

s2 p2

Energies & Spectra not sensitive to l ml

2/13s 2/13p

2/33p

2/33d

2/53d

s3 p3 d3

Energies & Spectra not sensitive to

j mj l ml s ms

till next page

Ocean Optics - HeliumBecause of the doublets, the states cannot be completely degenerate

“spin-orbit effect” + …

Ocean Optics - NeonBecause of the doublets, the states cannot be completely degenerate

“spin-orbit effect” + …

7.9 Multi-electron Spectra

QUANTUM NUMBERS principal: n ltot , stot

jtot .

Stot = S1 + S2 + …

Ltot = L1 + L2 + …

Jtot = Ltot + Stot

Multi-e Spectroscopic Notation

tottot j

tots 12

stot = 1, ltot=0, jtot=1

2S1

Two Kinds of Notation

• Where an individ electron is at

• n l j

– 1s1/2

– 2s1/2

– 2p1/2

– 2p3/2

• A.M. for whole atom

• 2Stot+1 ltot jtot

– 1S0

– 3S1

– 3P0 , 3P1, 3P2

Curious Things That Happen:Ground State of Helium

0tot

1

0tots

1s

1s

Ltot = L1 + L2

Stot = S1 + S2

system = (spatial wfn) (spin wfn)

2

1

2

1 )1()2()2()1( 1111 sssssys

2

1

2

1 )1()2()2()1( 1111 sssssys

sym

asym

1S0

3S1

7.8 Atoms in External Magnetic Fields

-- the Zeeman Effect

Corrections to the Coulomb Potlfor H-atom

• Central Potential• Spin-Orbit (electron viewpoint)• Relativistic Spin (Thomas precession)• Relativistic Kinetic Energy• Spin-Orbit (nucleus viewpoint)• Spin-Spin• Impact of External Fields

– Zeeman Effect (applied B-field)– Stark Effect (applied E-field)

Weak-Field Zeeman

• Hartree-Fock Coulomb & related Procedures

• Fine Structure– spin-orbit ( jtot becomes important )

– relativistic

• Zeeman

H’Zeeman = - tot * Bext

Bext < few 0.1’s Tesla

Weak Field Zeeman

totstotltot

tot

totstot Sm

egL

m

eg

22

)2(2 tottot SL

m

e

electronSpin-Orbit “locks” the angle between L & S J is now a well-defined direction.

LSdr

dV

rcmE

R

1

2

122

S

SL

L

JNOTE

Lz

is no longer

well-defined

ml not a good q. no.

Weak-Field Zeeman

totJtot

project average tot onto Jtot

J

Jμtottot

Jontotot

cos

)1(

)()2(2

jj

SLSLme

tot

tot

eSO makes jtot good quantum number,

mltot & mstot become ‘confused’ (near worthless).

Jtot is ‘well-defined’ direction; jtot mjtot

Weak Field Zeeman

cosJontoBonto

Jonto

projection of tot onto J onto B

extj B

jj

m

jj

SLSL

m

e

)1()1(

)()2(

2))((

EZeeman = - tot * Bext

Bext

Jtot

onto J

J

J zJonto

BontoJonto

stot=0

Strong-Field Zeeman

• Hartree-Fock Coulomb & related procedures

• Zeeman

• Fine Structure– spin-orbit– relativistic

H’Zeeman = - tot * Bext

Strong Field Zeeman

Bext

Ltot

Stot

H’Zeeman = - tot * Bext

)2(2 tottottot SL

m

e

exttotstotstrongZeeman Bmm

m

eE )2(

2