3.23 Fall 2007 – Lecture 7 ONE BLOCH AT A TIMELast timeStudyDynamics, Lagrangian styleNewton’s second law, tooHamiltonian1-dimensional monoatomic chainPropertiesPropertiesRing geometry1-dimensional diatomic chainTranslational SymmetryBravais LatticesBravais latticesSymmetryFigure 17.1bGroup Therapy…ExamplesFigure 17.3The 4 symmetry operations of H2O form a group (called C2v)Ten crystallographic point groups in 2d32 crystallographic point groups in 3dCrystal Structure = Lattice + BasisPrimitive unit cell and conventional unit cellPeriodic boundary conditions for the ions (i.e. the ext. potential)Reciprocal lattice (I)Reciprocal lattice (II)Reciprocal lattice (III)Reciprocal lattice (IV)Examples of reciprocal latticesMIT OpenCourseWare http://ocw.mit.edu 3.23 Electrical, Optical, and Magnetic Properties of MaterialsFall 2007For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.3.23 Electronic, Optical and Magnetic Properties of Materials ‐ Nicola Marzari (MIT, Fall 2007)3.23 Fall 2007 – Lecture 7ONE BLOCH AT A TIME3.23 Electronic, Optical and Magnetic Properties of Materials ‐ Nicola Marzari (MIT, Fall 2007)Last time1. Vector space (expectation values measure the projection on different eigenvectors)2. Eigenvalues and eigenstates as a linear algebra problem3. Variational principle4. Its application to a H atom (atomic units)5. Hamiltonian for a molecular system; bonding and antibonding states6. Potential energy surface of a molecule 7. Vibrations at equilibrium; quantum harmonic oscillatorStudy• Chapter 2 of Singleton textbook – “Band theory and electronic properties of solids”3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)Dynamics, Lagrangian style• First construct L=T-V• Then, the equations of motion are given by• Why ? We can use generalized coordinates. Also, we only need to think at the two scalar functions T and V0=∂∂−⎟⎟⎠⎞⎜⎜⎝⎛∂∂jjqLqLdtd(the dot is a time derivative)3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)Newton’s second law, too• 1-d, 1 particle: T=1/2 mv2, V=V(x)0=∂∂−⎟⎟⎠⎞⎜⎜⎝⎛∂∂jjqLqLdtd2120mxdVdt x x⎛⎞⎛⎞∂⎜⎟⎜⎟∂⎝⎠⎜⎟+=∂∂⎜⎟⎜⎟⎝⎠()dVmxdt x∂=−∂3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)Hamiltonian• We could use it to derive Hamiltonian dynamics (twice the number of differential equations, but all first order). We introduce a Legendre transformationiipHq∂∂=iiqHp∂∂=−iiqLp∂∂=∑−=iiitqqLpqtpqH ),,(),,(1-dimensional monoatomicchain3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)•Unique solutions for k in the first BZ•Phase velocity and group velocityProperties3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)1ssuu+• Standing waves•Long wavelength limitProperties3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)Ring geometry3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)1-dimensional diatomic chain3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)Please replace with the credit line: Image removed due to copyright restrictions. Please see Fig. 22.10 in Ashcroft, Neil W., and N. David Mermin. Solid State Physics. Belmont, CA: Brooks/Cole, 1976. ISBN: 9780030839931.Translational Symmetryt1t1t1t1t2t2t2aTripleDoubleSingle1 Dim3 DimCl-Cu+2 DimFigure by MIT OpenCourseWare.Bravais Lattices• Infinite array of points with an arrangement and orientation that appears exactly the same regardless of the point from which the array is viewed.• 14 Bravais lattices exist in 3 dimensions (1848)• M. L. Frankenheimer in 1842 thought they were 15. So, so naïve…12312 3l,m and n in, and primitive lattice vectote rgs ers aaaaaRalmn=+ +GGGGGGG14Figure by MIT OpenCourseWare.BravaisLatticeTriclinicMonoclinicOrthorhombicTetragonalTrigonalCubicHexagonalParametersSimple(P)VolumeCentered (I)BaseCentered (C)FaceCentered (F)a1 = a2 = a3α12 = α23 = α31a1 = a2 = a3α23 = α31 = 900α12 = 900 a1 = a2 = a3α12 = α23 = α31 = 900a1 = a2 = a3α12 = α23 = α31 = 900a1 = a2 = a3α12 = 1200α23= α31= 900a1 = a2 = a3α12 = α23 = α31 = 900a1 = a2 = a3α12 = α23 = α31 < 1200a3a1a24 Lattice Types7 Crystal ClassesBravais latticesSymmetry• Symmetry operations: actions that transform an object into a new but undistinguishable configuration• Symmetry elements: geometric entities (axes, planes, points…) around which we carry out the symmetry operationsFigure 17.1bC3, S3σvσhC2C2C2Cl1Cl2Cl3Cl4Cl5PFigure by MIT OpenCourseWare.Symmetry elements and their corresponding operationsSymmetry elementsSymmetry operationsECnσiSnEIdentityn-Fold rotation axisn-Fold rotation-reflection axisMirror planeInversion centerCn, Cn2 ,....., CnnσiSnleave molecule unchangedrotate about axis by 360o /n 1, 2, .... , n times (indicated by superscript)reflect through the mirror plane(x, y, z) (-x, -y, -z)rotate about axis by 360o /n, and reflect through a plane perpendicular to axis.Figure by MIT OpenCourseWare.Group Therapy…A group G is a finite or infinite set of elements A, B, C, D…together with an operation “☼” that satisfy the four properties of:1. Closure: If A and B are two elements in G, then A☼B is also in G.2. Associativity: For all elements in G, (A☼B) ☼C==A☼ (B☼C).3. Identity: There is an identity element I such that I☼A=A☼I=A for every element A in G.4. Inverse: There is an inverse or reciprocal of each element. Therefore, the set must contain an element B=inv(A) such that A☼inv(A)=inv(A) ☼A=I for each element of G.Examples• Integer numbers, and addition• Integer numbers, and multiplication• Real numbers, and multiplication• Rotations around an axis by 360/nFigure by MIT OpenCourseWare.C2vFigures by MIT OpenCourseWare.Symmetriesof H O 2Figure by MIT OpenCourseWare.Symmetriesof H