Principles of Feedback Control in Engineering Applications
The principles of feedback control seen in various engineering applications, from cruise control in cars to smart grid control. Learn about process characteristics, properties, and controller functionalities in dynamic systems. Dive into closed-loop feedback control systems for effective financial management and societal impact.
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PHY 752 Solid State Physics 11-11:50 AM MWF Olin 103 Plan for Lecture 20: Review: Chapters 1-6 in GGGPP Bloch theorem Kronig-Penny model Group theory 1. 2. 3. 10/09/2015 PHY 752 Fall 2015 -- Lecture 20 1
10/09/2015 PHY 752 Fall 2015 -- Lecture 20 2
( ) to the Schroedinger equation for r Bloch theorem for solu t ion k T an electron in a periodic solid in terms of translation ( ) =e ( ) + k k r T r Consider an electron moving in a one-dimensional model potential (Kronig and Penney, Proc. Roy. Soc. (London) 130, 499 (1931) vector : k T i a V0 Schroedinger equation for electron: + 2 2 = ( ) ( ) x ( ) x V x E 2 m PHY 752 Fall 2015 -- Lecture 20 10/09/2015 3
a V0 b Matching conditions redu cos( ) ( )cos ka F E = ce to: ( ) ( ) E b 1/2 2 V ( ) + 2 ( ) E 1 sinh ( 0 F a b ( ) 4 E E V 0 2 V E V E ( ) = tan ( ) E tanh ( ) 0 a b ( ) 2 E 0 10/09/2015 PHY 752 Fall 2015 -- Lecture 20 4
a V0 b ( ) b 2 2 49 2 16 17 = = Details for ; V b a 0 2 2 16 m E V Let f 0 ( ) 1/2 2 sinh 0.687 1 f ( ) = + cos( ) 1 cos 11.7 ( ) f k a f ( ) 4 1 f f ( ) 1 2 1 f = t an ( ) tanh 0.687 1 f f ( ) 2 f f 10/09/2015 PHY 752 Fall 2015 -- Lecture 20 5
( ) 1/2 2 sinh 0.687 1 f ( ) = + cos( ) 1 co s 11.7 ( ) f k a f ( ) 4 1 f f ( ) 1 2 1 f = t an ( ) f tanh 0.687 1 f ( ) 2 f f Forbidden states v f Forbidden states v ( ) 1/2 2 sinh 0.687 1 f ( ) = + ( ) 1 cos 11.7 ( ) f X f f ( ) 4 1 f f 10/09/2015 PHY 752 Fall 2015 -- Lecture 20 6
f Band gap Band gap ka/ 10/09/2015 PHY 752 Fall 2015 -- Lecture 20 7
Classification of lattice types 14 Bravais lattices 230 space groups 10/09/2015 PHY 752 Fall 2015 -- Lecture 20 8
Short digression on abstract group theory What is group theory ? 10/09/2015 PHY 752 Fall 2015 -- Lecture 20 9
Group theory some comments The elements of the group may be abstract; in general, we will use them to describe symmetry properties of our system Representations of a group A representation of a group is a for each group element) -- set of matrices (on e ( ), ( )... that satisfies A B the multip of the ma representation. lication table of the group. The trices is called the dimension dimension of the 10/09/2015 PHY 752 Fall 2015 -- Lecture 20 10
The great orthogonality theorem Notation: R order of the group element of the group ( ) th representation of R i h i R denote matrix i ndices ension of the representation h = ( R Analysis shows: dim ) l i * i j ( ) R ( ) R ij l i = 2 i l h i Note: only irreducible representations are used. 10/09/2015 PHY 752 Fall 2015 -- Lecture 20 11
Simplified analysis in terms of the characters of the representations jl j j R = ( ) ( ) R 1 Character orthogonality theorem ( R ) * = i j ( ) R ( ) R h ij Note that all members of a class have the same character for any given representation i. 10/09/2015 PHY 752 Fall 2015 -- Lecture 20 12
Reciprocal lattice real lattice reciprocal lattice Unit vectors of the reciprocal lattice General reciprocal lattice vector 10/09/2015 PHY 752 Fall 2015 -- Lecture 20 13
Reciprocal lattice real lattice reciprocal lattice Unit vectors of the reciprocal lattice General reciprocal lattice vector 10/09/2015 PHY 752 Fall 2015 -- Lecture 20 14
Quantum Theory of materials Electronic coordinates Atomic coordinates Exact Schrodinger equation: ({ },{ }) i r R H = a a a ({ },{ i r R ({ },{ i r R }) }) E where ({ },{ r H = Nuclei Electrons a a a H +H R R ({ },{ i r R }) ({ }) }) i Born-Oppenheimer approximation Born & Huang, Dynamical Theory of Crystal Lattices, Oxford (1954) Approximate factorization: ({ },{ i r R = Electrons Nuclei a a a R ({ },{ i r R }) ({ }) }) X 10/09/2015 PHY 752 Fall 2015 -- Lecture 20 15
Quantum Theory of materials -- continued Electronic Schrodinger equation: ({ },{ i r R H Electrons = Electrons Electrons a a a a ({ },{ i r R R r R }) }) ({ }) ({ },{ i e r }) U 2 2 2 a Z e r = + Electro n s 2 i a H ({ },{ i r R }) a R r 2 | | | | m , i a i i j i i j Nuclear Hamiltonian: (Often treated classically) ( ) = Nuclei Nuclei Nuclei a a a H R R R { } ({ }) ({ }) X W X 2 a P ( ) = + Nuclei a a H R R { } ({ }) U a 2 M a Effective nuclear interaction provided by electrons 10/09/2015 PHY 752 Fall 2015 -- Lecture 20 16
First consider electronic Hamiltonian Electronic Schrodinger equation: ({ },{ i r R H Electrons = Electrons Electrons a a a a ({ },{ i r R R r R }) }) ({ }) ({ },{ i e r }) U 2 2 2 a Z e r = + Electro n s 2 i a H ({ },{ i r R }) a R r 2 | | | | m , i a i i j i i j Replace by jellium Independent electron contributions 10/09/2015 PHY 752 Fall 2015 -- Lecture 20 17
Jellium model of metals Nuclear potential represented by a uniform background of positive charge with charge density n0=Z/V (V representing volume per atom) Electrons represented as independent free electrons occupying free-electron states 2 2 k = ( ) for 0 E k k k F 2 m Figure from GGGPP 10/09/2015 PHY 752 Fall 2015 -- Lecture 20 18
Calculation of the Fermi level for jellium 10/09/2015 PHY 752 Fall 2015 -- Lecture 20 19
Density of states analysis of free electron gas 10/09/2015 PHY 752 Fall 2015 -- Lecture 20 20
Practical modeling schemes based on density functional theory Kohn-Sham formulation of density functional theory Results of self-consistent calculations Variationally determined -- Ground state energ Electron density y [ ] ( ) n r E vn Some remaining issues Theory for Eexc[n] still underdevelopment This formalism does not access excited states Strongly correlated electron systems are not well approximated 10/09/2015 PHY 752 Fall 2015 -- Lecture 20 21
Practical solution of Kohn-Sham equations in solids 10/09/2015 PHY 752 Fall 2015 -- Lecture 20 22
