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3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
3.012 Fund of Mat Sci: Bonding – Lecture 7
ALPHABET SOUP
Photograph of a bowl of alphabet soup removed for copyright reasons.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Homework for Wed Oct 5
• Study: 21.5, 21.6• Read: 17.2 (Stern-Gerlach)• Non-graded PS3 will be posted today
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Last time: 1. Hamiltonian in a spherical potential:
2. all commute with each other3. We can find common eigenfunctions
labeled with the 3 quantum numbers n, l, m
2 22
2 2
1ˆ ( )2 2e e
d d LH r V rm r dr dr m r
⎛ ⎞= − + +⎜ ⎟⎝ ⎠
h
2, , andˆ ˆ ˆzH L L
( ) ( ) ( , )nlm nl lmr R r Yψ ϑ ϕ=r
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Three Quantum Numbers• Principal quantum number n (energy,
accidental degeneracy)
• Angular momentum quantum number ll = 0,1,…,n-1 (a.k.a. s, p, d… orbitals)
• Magnetic quantum number mm = -l,-l+1,…,l-1,l
( ) ( )2 2 2 2
2 2 20 0
13.6058 eV 1 Ry8ne Z Z ZE
a n n nπε= − = − = −
H ↔
ˆzL ↔
2L ↔
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
2
2
1
10
0 3 4
R10
00
0.2
0.4
1 2 3 4
r2R220
r2R210
r2R221
r2R230
r2R231
r2R232
rr
r
r
r
r
r
r
r
-0.20
0
0
0.20.4
0.040.080.12
0.6
2
r
r
r
6 10
2 6 10
00
0.10.05
0.05
0.15
2 6 10
0
00
0
0
00
02 6 10
R20
R21
R30
R31
R32
0.1
4
4
8
8
12
12
16
16
4
4
8
8
12
12
16
16
00
4 8 12 16
r0 4 8 12 16
-0.1
0.2
0.4
0.04
0.08
-0.04
0.04
0.08
0.02
0.04
0.4
0.8
0
0.4
0.8
Radial functions Rnl(r) and radial distribution functions r2R2nl(r) for
atomic hydrogen. The unit of length is aµ = (m/µ) a0, where a0 is thefirst Bohr radius.
0.15
Figure by MIT OCW.Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
The Full Alphabet Soup
Courtesy of David Manthey. Used with permission. Source: http://www.orbitals.com/orb/orbtable.htm
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Orbital levels in hydrogenoid atomsO
rbita
l Ene
rgy
(kJ /
mol
)
0-82
-146
-328
-13131s
4s
2s
3s
4p
3p
2p
4d
3d
4f
Hydrogen
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Balmer lines in a hot star
Courtesy of the Department of Physics and Astronomy at the University of Tennessee. Used with permission.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Emission and absorption lines
Courtesy of the Department of Physics and Astronomy at the University of Tennessee. Used with permission.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
XPS in Condensed Matter
Diagram of Moon composition as seen in X-rays removed for copyright reasons.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Composition Analysis
Images of X-Ray element maps of mine waste soil particles removed for copyright reasons.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Orbital levels in multi-electron atomsO
rbita
l Ene
rgy
(kJ /
mol
)0
-82
-146
-328
-1313
0-82
-146
-328
-13131s 1s
4s4s
2s
2s
3s3s
4p 4p
3p 3p
2p 2p
4d 4d
3d3d
4f4f
Hydrogen Multielectron Atoms
Figure by MIT OCW.
(I) “Centripetal” repulsion
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
2
1
-1
-2
1 2 3 4 5 6
Vcentripetal(r)
1010r(m)
Veff(r) VCoulomb(r)
1018
v(r)
(J)
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
(II) Screening
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
(III) Screening
ˆˆ ˆ ˆ ˆˆ ˆ( 2) )( 2 Bz z z
BH H L S B H L S Bµµ+ +→ + + ⋅ =
r
h hGoudsmit and Uhlenbeck
Stern-Gerlach Experiment
Image courtesy of Theresa Knott.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Spin
• Dirac derived the relativistic extension of Schrödinger’s equation; for a free particle he found two independent solutions for a given energy
• There is an operator (spin S) that commutes with the Hamiltonian and that can only have two eigenvalues
• In a magnetic field, the spin combines with the angular momentum, and they couple via
ˆˆ ˆ ˆ( 2 )BH H L S Bµ→ + + ⋅
r
h
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Spin Eigenvalues/Eigenfunctions
• Norm (s integer → bosons, half-integer → fermions)
• Z-axis projection (electron is a fermion with s=1/2)
• Spin-orbital: product of the “space” wavefunction and the “spin” wavefunction
( )2 2ˆ 1
ˆ2
spin spin
z spin spin
S s s
S
Ψ = + Ψ
Ψ = ± Ψ
h
h
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