alphabet soup - mit opencourseware · 2020-07-09 · alphabet soup photograph of a bowl of alphabet...
TRANSCRIPT
![Page 1: ALPHABET SOUP - MIT OpenCourseWare · 2020-07-09 · ALPHABET SOUP Photograph of a bowl of alphabet soup removed for copyright reasons. 3.012 Fundamentals of Materials Science: Bonding](https://reader034.vdocuments.net/reader034/viewer/2022042711/5f82f3ef3957dd48795486e4/html5/thumbnails/1.jpg)
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.
![Page 2: ALPHABET SOUP - MIT OpenCourseWare · 2020-07-09 · ALPHABET SOUP Photograph of a bowl of alphabet soup removed for copyright reasons. 3.012 Fundamentals of Materials Science: Bonding](https://reader034.vdocuments.net/reader034/viewer/2022042711/5f82f3ef3957dd48795486e4/html5/thumbnails/2.jpg)
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
![Page 3: ALPHABET SOUP - MIT OpenCourseWare · 2020-07-09 · ALPHABET SOUP Photograph of a bowl of alphabet soup removed for copyright reasons. 3.012 Fundamentals of Materials Science: Bonding](https://reader034.vdocuments.net/reader034/viewer/2022042711/5f82f3ef3957dd48795486e4/html5/thumbnails/3.jpg)
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
![Page 4: ALPHABET SOUP - MIT OpenCourseWare · 2020-07-09 · ALPHABET SOUP Photograph of a bowl of alphabet soup removed for copyright reasons. 3.012 Fundamentals of Materials Science: Bonding](https://reader034.vdocuments.net/reader034/viewer/2022042711/5f82f3ef3957dd48795486e4/html5/thumbnails/4.jpg)
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 ↔
![Page 5: ALPHABET SOUP - MIT OpenCourseWare · 2020-07-09 · ALPHABET SOUP Photograph of a bowl of alphabet soup removed for copyright reasons. 3.012 Fundamentals of Materials Science: Bonding](https://reader034.vdocuments.net/reader034/viewer/2022042711/5f82f3ef3957dd48795486e4/html5/thumbnails/5.jpg)
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.
![Page 6: ALPHABET SOUP - MIT OpenCourseWare · 2020-07-09 · ALPHABET SOUP Photograph of a bowl of alphabet soup removed for copyright reasons. 3.012 Fundamentals of Materials Science: Bonding](https://reader034.vdocuments.net/reader034/viewer/2022042711/5f82f3ef3957dd48795486e4/html5/thumbnails/6.jpg)
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
![Page 7: ALPHABET SOUP - MIT OpenCourseWare · 2020-07-09 · ALPHABET SOUP Photograph of a bowl of alphabet soup removed for copyright reasons. 3.012 Fundamentals of Materials Science: Bonding](https://reader034.vdocuments.net/reader034/viewer/2022042711/5f82f3ef3957dd48795486e4/html5/thumbnails/7.jpg)
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
HydrogenFigure by MIT OCW.
![Page 8: ALPHABET SOUP - MIT OpenCourseWare · 2020-07-09 · ALPHABET SOUP Photograph of a bowl of alphabet soup removed for copyright reasons. 3.012 Fundamentals of Materials Science: Bonding](https://reader034.vdocuments.net/reader034/viewer/2022042711/5f82f3ef3957dd48795486e4/html5/thumbnails/8.jpg)
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.
![Page 9: ALPHABET SOUP - MIT OpenCourseWare · 2020-07-09 · ALPHABET SOUP Photograph of a bowl of alphabet soup removed for copyright reasons. 3.012 Fundamentals of Materials Science: Bonding](https://reader034.vdocuments.net/reader034/viewer/2022042711/5f82f3ef3957dd48795486e4/html5/thumbnails/9.jpg)
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.
![Page 10: ALPHABET SOUP - MIT OpenCourseWare · 2020-07-09 · ALPHABET SOUP Photograph of a bowl of alphabet soup removed for copyright reasons. 3.012 Fundamentals of Materials Science: Bonding](https://reader034.vdocuments.net/reader034/viewer/2022042711/5f82f3ef3957dd48795486e4/html5/thumbnails/10.jpg)
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.
![Page 11: ALPHABET SOUP - MIT OpenCourseWare · 2020-07-09 · ALPHABET SOUP Photograph of a bowl of alphabet soup removed for copyright reasons. 3.012 Fundamentals of Materials Science: Bonding](https://reader034.vdocuments.net/reader034/viewer/2022042711/5f82f3ef3957dd48795486e4/html5/thumbnails/11.jpg)
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.
![Page 12: ALPHABET SOUP - MIT OpenCourseWare · 2020-07-09 · ALPHABET SOUP Photograph of a bowl of alphabet soup removed for copyright reasons. 3.012 Fundamentals of Materials Science: Bonding](https://reader034.vdocuments.net/reader034/viewer/2022042711/5f82f3ef3957dd48795486e4/html5/thumbnails/12.jpg)
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.
![Page 13: ALPHABET SOUP - MIT OpenCourseWare · 2020-07-09 · ALPHABET SOUP Photograph of a bowl of alphabet soup removed for copyright reasons. 3.012 Fundamentals of Materials Science: Bonding](https://reader034.vdocuments.net/reader034/viewer/2022042711/5f82f3ef3957dd48795486e4/html5/thumbnails/13.jpg)
(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.
![Page 14: ALPHABET SOUP - MIT OpenCourseWare · 2020-07-09 · ALPHABET SOUP Photograph of a bowl of alphabet soup removed for copyright reasons. 3.012 Fundamentals of Materials Science: Bonding](https://reader034.vdocuments.net/reader034/viewer/2022042711/5f82f3ef3957dd48795486e4/html5/thumbnails/14.jpg)
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
(II) Screening
![Page 15: ALPHABET SOUP - MIT OpenCourseWare · 2020-07-09 · ALPHABET SOUP Photograph of a bowl of alphabet soup removed for copyright reasons. 3.012 Fundamentals of Materials Science: Bonding](https://reader034.vdocuments.net/reader034/viewer/2022042711/5f82f3ef3957dd48795486e4/html5/thumbnails/15.jpg)
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
(III) Screening
![Page 16: ALPHABET SOUP - MIT OpenCourseWare · 2020-07-09 · ALPHABET SOUP Photograph of a bowl of alphabet soup removed for copyright reasons. 3.012 Fundamentals of Materials Science: Bonding](https://reader034.vdocuments.net/reader034/viewer/2022042711/5f82f3ef3957dd48795486e4/html5/thumbnails/16.jpg)
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)
![Page 17: ALPHABET SOUP - MIT OpenCourseWare · 2020-07-09 · ALPHABET SOUP Photograph of a bowl of alphabet soup removed for copyright reasons. 3.012 Fundamentals of Materials Science: Bonding](https://reader034.vdocuments.net/reader034/viewer/2022042711/5f82f3ef3957dd48795486e4/html5/thumbnails/17.jpg)
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
![Page 18: ALPHABET SOUP - MIT OpenCourseWare · 2020-07-09 · ALPHABET SOUP Photograph of a bowl of alphabet soup removed for copyright reasons. 3.012 Fundamentals of Materials Science: Bonding](https://reader034.vdocuments.net/reader034/viewer/2022042711/5f82f3ef3957dd48795486e4/html5/thumbnails/18.jpg)
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