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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
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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
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7.1 Orbital Magnetic Moments and Discovery of Intrinsic Spin
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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
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Bohr Model of Ang Momentum
Note: s-states (l=0) have no Bohr model picture
Eisberg & Resnick: Fig 7-11
Classical orSemi-classical
description
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Vector Model of QM Ang. Momentum
quantum numbers
m
E&R Fig 7-12
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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
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Morrison, Estle, Lane “Understanding More QM”, Prentice-Hall, 1991
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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
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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
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Ai
Lm
eg
2
12
m
eg
mm
egzz 2
electron
neutron
proton
g
1
0
1
Orbital Magnetic MomentE&R Fig 7-11
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Lm
eg
2
22310927.02
Amm
ebohr
L
bohrg
Bohr magneton
E&R Fig 7-11
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BU z
BUnOrientatioofEnergyPotential
0 BUF z
B
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BU z
BUnOrientatioofEnergyPotential
BUF z
Different ml states experience different forces
B
dz
dBF zz
Lm
eg
2
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BU z
BUnOrientatioofEnergyPotential
Use B as z-axis.
BUF z
dz
dBF zz
Different ml states experience different forces
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Stern & Gerlach~1922
Harris Fig 7.3, 7.4
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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.
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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
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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
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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.
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Ralph Kronig ~1925
• Spinning Electron Idea
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Goudsmit & Ulhenbeck ~ 1925
• Studied high resolution spectra of alkali elements
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Ocean Optics - Helium
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Ocean Optics - Neon
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Giancoli – fig 36.21
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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.
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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
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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
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“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).
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“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).
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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
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“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).
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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
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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
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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
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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
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interesting fundamental constants
-2.002 319 304 3622 (15)
1.602 176 487 (40) x 10-19 C
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7.2 & 7.3 Complications from having Identical Particles
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Exchange Symmetry
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7.4 & 7.5 Multielectron Atoms
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,mnl YrRr
2/ eGRn
r = n2 ao / ZEn = ( 13.6 eV ) (Z2/n2)
ao = 0.529 Å
,mnl YrRr
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Prob = r2 R* R
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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
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Note: This shows how theorbitals shift as viewedfrom the perspective of an s-orbital.
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Hartree-Fock Method
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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
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r
eZkrV eff
2
)(
Using effective charge is a very crude approximation.
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r2 ~ n2 ao / Zeff
En ~ (Zeff2/n2) ( -13.6 eV )
Hartree-FockEffective Charge Effects
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7.6 Spin-Orbit Effect
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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)
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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
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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
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BU s
2
1
2
ˆ
4 r
rldIB o
vZet
lQl
t
QdlI
2
ˆ
4 r
rvZeB o
s
L
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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
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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
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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*
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LSdr
dV
rcmE
R
1
2
122
SLJ
22 SLJ
SLSLJ 2222
2222 SLJSL
222
1112
11
2
1 ssjj
dr
dV
rcmE
R
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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.
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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
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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
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Bigger atoms larger Z (central charge) ~ same size
dr
dV
r
1larger
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7.7 QM Angular Momentum
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Bohr Model of Ang Momentum
Note: s-states (l=0) have no Bohr model picture
Eisberg & Resnick: Fig 7-11
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Vector Model of Ang. Momentum
quantum numbers
m
E&R Fig 7-12
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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.”
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Morrison, Estle, Lane “Understanding More QM”, Prentice-Hall, 1991
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ADDITION OF
ANGULAR MOMENTUM
Ltot = L1 + L2
L1
L2
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Ltot = L1 + L2
11 m
,11mlY
22 m
tottot m
,22mlY ,
tottotmlY
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Ltot = L1 + L2
11 m
2121 tot
22 m
tottot m
21 mmmtot
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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
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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
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l1=1, l2=1, m’s degenerate
m1 m2 mtot
“
“
“
“
“
“
mtot Possibilities (m1,m2)
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Allowed Values of ltot mtot
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Basic A.M. Math
J = L + S
sjs
sj mmm
L
S
J
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Vector Representation of J
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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 ??
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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?
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TOTAL ANGULAR MOMEMTUM
J = L + S
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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
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Ocean Optics - HeliumBecause of the doublets, the states cannot be completely degenerate
“spin-orbit effect” + …
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Ocean Optics - NeonBecause of the doublets, the states cannot be completely degenerate
“spin-orbit effect” + …
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7.9 Multi-electron Spectra
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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
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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
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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
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7.8 Atoms in External Magnetic Fields
-- the Zeeman Effect
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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)
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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
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Weak Field Zeeman
totstotltot
tot
totstot Sm
egL
m
eg
22
)2(2 tottot SL
m
e
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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.
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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
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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
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stot=0
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Strong-Field Zeeman
• Hartree-Fock Coulomb & related procedures
• Zeeman
• Fine Structure– spin-orbit– relativistic
H’Zeeman = - tot * Bext
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Strong Field Zeeman
Bext
Ltot
Stot
H’Zeeman = - tot * Bext
)2(2 tottottot SL
m
e
exttotstotstrongZeeman Bmm
m
eE )2(
2
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