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

HydrogenFigure 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