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
2
1
d
d
L
h
⎛
⎞
2
Hˆ = −
r
+ V (r )
⎜
⎟+
2
2
2me r dr ⎝ dr ⎠ 2me r
2
ˆ
ˆ
2. H , L , and Lˆ z all commute with each other
3. We can find common eigenfunctions
r
ψ nlm (r ) = Rnl (r )Ylm (ϑ , ϕ )
labeled with the 3 quantum numbers n, l, m
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Three Quantum Numbers
• Ĥ ↔ Principal quantum number n (energy,
accidental degeneracy)
e2
Z2
Z2
Z2
En = −
= − (13.6058 eV ) 2 = − (1 Ry ) 2
2
n
n
8πε 0 a0 n
• L̂ ↔ Angular momentum quantum number l
l = 0,1,…,n-1 (a.k.a. s, p, d… orbitals)
2
• Lˆ z ↔ Magnetic quantum number m
m = -l,-l+1,…,l-1,l
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
R10
r2R210
2
0.4
1
0
0.2
0
1
2
3
4
R20
rR
2
0.6
0.4
0.2
0
-0.2
2
6
10
r
R30
0
2
6
10
r
0
1
2
3
4
r
2
20
0.15
0.1
0.05
0
r2R221
R21
0.12
0.08
0.04
0
0.15
0.1
0.05
0
0
2
6
10
r
0
2
6
10
r
r2R230
0.4
0.08
0.2
4
0
-0.1
8
12
16
0.04
r
R31
0
0
4
8
12
16
r
0
4
8
12
16
r
0
4
8
12
16
r
r2R231
0.08
0.8
0.04
4
0
-0.04
8
12
16
r
0.4
0
r2R232
R32
0.04
0.8
0.02
0
Figure by MIT OCW.
0
r
0.4
0
4
8
12
16
r
0
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 the
first Bohr radius.
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 atoms
0
-82
Orbital Energy (kJ / mol)
-146
-328
-1313
Figure by MIT OCW.
4s
4p
4d
3s
3p
3d
2s
2p
1s
4f
Hydrogen
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 atoms
0
-82
-146
4s
4p
4d
3s
3p
3d
4f
0
-82
-146
4p
4s
3p
4d
3d
Orbital Energy (kJ / mol)
3s
-328
2s
2p
-328
2p
2s
-1313
Figure by MIT OCW.
1s
-1313
Hydrogen
1s
Multielectron Atoms
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
4f
(I) “Centripetal” repulsion
2
Vcentripetal(r)
1
1018v(r) (J)
1010r(m)
1
-1
2
Veff(r)
3
4
5
6
VCoulomb(r)
-2
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
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Stern-Gerlach Experiment
r
µ
µB ˆ
B
ˆ
ˆ
ˆ
ˆ
ˆ
H →H+
( L + 2S ) ⋅ B = H +
( Lz + 2Sˆz ) Bz
h
Image courtesy of Theresa Knott.
h
Goudsmit and Uhlenbeck
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
r
µ
Hˆ → Hˆ + B ( Lˆ + 2Sˆ ) ⋅ B
h
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Spin Eigenvalues/Eigenfunctions
• Norm (s integer → bosons, half-integer → fermions)
Sˆ 2 Ψ spin = h 2 s ( s + 1) Ψ spin
• Z-axis projection (electron is a fermion with s=1/2)
h
ˆ
S z Ψ spin = ± Ψ spin
2
• Spin-orbital: product of the “space” wavefunction
and the “spin” wavefunction
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)