PH427 are periodic oscillations ubiquitous or merely just a paradigm?

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PH427are periodic oscillations ubiquitous or
merely just a paradigm?
Paradigm Periodic Systems Instructor Matt
Graham Winter 2016
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smaller goal full mathematical description of
physically electronically coupled systemsBIG
GOAL show how symmetry, oscillations and quantum
mechanics really describe everyday stuffi.e.
quantum mechanics you can get paid to do!!
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35 ? problem sets (3)15 ? pick a solid state
physics journal article, give a 10 min.
talk50 ? final exam (M of exam week)
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Feb 2016 Feb 2016 Feb 2016 Feb 2016 Feb 2016
Mon Tue Wed Thu Fri
22Coupled Oscillators 23Coupled Oscillators 24-PS1 (0,1,2) dueDispersion for 1D mass/spring system 25Dispersion for 1D mass/spring system 26Lattices at finite temperature-PS1(3,4)due
29 Phonon Dispersion heat capacity-Journal proposals due 1Lattices (diatomic materials, final topics) 2-PS2 (1,2) dueDouble potential well 3Linear combination of atomic orbitals (LCAO) 4-PS2 due Electronic band structure
7 Journal Presentations 8 Journal Presentations- LCAO model 9 -PS3 (1,2) dueTight-Binding Model 10Electronic Band Structure 11Review, session I -PS3 due
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Roadmap

• DAYS 1- 7
• Coupled pendulum, railroad cars, atoms, etc.
• From atoms to crystals ? extend coupling to
  infinity and define a dispersion relation for an
  atomic system

• DAYS 7- 15
• Quantum wavestates in periodic systems
• 5 minute journal presentations

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Translational Symmetry and Noethers Theorem
Any system with translational symmetry has an
associated momentum conservation law. ? An
electron moving through a perfectly periodic
crystal maintains its momentum like the electron
was travelling though a vaccuum
Graphene
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Draw the Unit Cell
Graphene
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Draw the Unit Cell
Graphene
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Draw the Unit Cell
Celtic knot
Protein/DNA
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Draw the Unit Cell
Celtic knot
Protein/DNA
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?
?
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PERIODIC SYSTEM IN BIOLOGYLight Harvesting
Complex II
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Bacterial Light Harvesting
Bahatyrova, et al. Nature (2004) 430 1058
Hu, et al. J. Phys. Chem. B (1997) 101 3854
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Beats in Motion of Coupled Oscillator
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?1
?2
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Chain of Many (N) Spring-Coupled Masses
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Two Masses Connected to Hookes Law Springs
Force on m1
Force on m2
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Low frequency mode
Symmetric mode A1 A2
High frequency mode
Antisymmetric mode A1 ? A2
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symmetric mode (low frequency)
anti-symmetric mode (high frequency)
From Fig. 8.3, I. G. Main, Vibrations and Waves
in Physics
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5 Mass Chain Mode n1
l 12a ? k 2p / l p/ 6a
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5 Mass Chain Mode n2

• 6a ? k2 2p / l p/ 3a
• 2k

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5 Mass Chain Mode n3

• 4a ? k3 2p / l p/ 2a
• 3k

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5 Mass Chain Mode n4

• 3a ? k4 2p / l 2p/ 3a
• 4k

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5 Mass Chain Mode 5

• 12a/5 ? k5 2p / l 5p/6a
• 5k

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5 Mass Chain Mode 6

• 2a ? k6 2p / l p/a
• 6k

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5 Mass Chain Mode 7

• 12a/7 ? k7 2p / l 7p/6a
• 7k

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5 Mass Chain Dispersion Relation
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Diatomic chain, 16 masses, 8 unit cells
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Dispersion Relation for m 1, M 2
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Dispersion Relation for m 1, M 1.1
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Amplitude (m)/Amplitude (M) ? ?(m 1, M 2)
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Amplitude (m)/Amplitude (M) ? ?(m 1, M 1.1)
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Damped Wave in Forbidden Frequency
Range(Evanescent Wave)