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Computational Physics (Lecture 3)

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  1. Computational Physics(Lecture 3) PHY4370

  2. Interpolation • Computer is a system with finite number of discrete states. • In numerical analysis, the results obtained from computations are always approximationsof the desired quantities and in most cases are within some uncertainties. • Interpolation is needed • When we need to infer some information from discrete data.

  3. Presentation topic 1: interpolation • The simplest way to obtain the approximation of f (x) for x ∈ [xi , xi+1] is to construct a straight line between xi and xi+1. • Lagrange interpolation and Aitken method. • How to obtain the generalized interpolation formula passing through n data points? • Give the description of linear interpolation, Describe what is a Lagrange interpolation and the algorithm of Aitken method. • 10 min

  4. Least-square approximation • The global behavior of a set of data in order to understand the trend. • The most common approximation: based on the least squares of the differences between the approximation pm(x) and the data f (x).

  5. Presentation topic 2 • Least square method • Describe the concept, the algorithm and one example. • 10 min talk

  6. Spline approximation • Aset of data that varies rapidly over the range of interest • A typical spectral measurement that contains many peaks and dips. • fit the function locally and to connect each piece of the function smoothly. • A spline • interpolates the data locally through a polynomial • fits the data overall by connecting each segment of the interpolation polynomial by matching the function and its derivatives at the data points.

  7. Numerical Calculus • the heart of describing physical phenomena. • The velocity and the acceleration of a particle are the first-order and second-order time derivatives of the corresponding position vector…

  8. Numerical differentiation • Taylor exapnsion: • f (x) = f (x0) + (x − x0) f ‘(x0) + (x − x0)2/2! f’’ (x0)+ · · • The first-order derivative of a single-variable function f (x) around a point xi is defined from the limit • f ‘(xi ) = lim (x→0) [f (xi + Δx) − f (xi )] /Δ x • divide the space into discrete points xiwith evenly spaced intervals, h. • f i’= fi+1 − fi+ O(h). • Can be improved if we expand around i+1 and i-1: • f i’= (fi+1− fi-1)/2h+ O(h). • A three point formula: • For a second-order derivative. A three point formula is given by the combination:

  9. Presentation Topic 3 • First order and second order of derivative with a three-point formula • Introduce the concept and algorithm • Explain the problems of this method and solutions. • 10 min.

  10. Numerical Integrations • For a integral: • We just divide the region [a,b] into n slices with an interval of h.

  11. Random method • Just take N points randomly in the region, evaluate the function on those points and take average, times the integration area. Simple sampling method.

  12. Two Problems: Calculate: accurate value:

  13. Relative error as a function of N F(x)

  14. Example 2: Calculate: Accurate result: Using the above method:

  15. In this example The function is significant in the range of [2,4] So it’s no good to eventually divide [0,10]

  16. Random Variable, density function and distribution function Random Variable: Value is not predetermined, but the value distribution is known. (Probability of each value is known. P(a ≤ x ≤ b)= , f(x) is the distribution density function. Distribution function F(x)= f(x)=dF(x)/dx If f(x) is normalized, the integration is 1.

  17. Calculate it intelligently: Importance sampling: Take some distribution function P(x). (normalized). The integration is now: If we take,  is zero!! Of course, we don’t know I…. However, we know the shape of f(x). Therefore, we still can construct a P(x), to reduce the error.

  18. Consider this algorithm: construct a series of random numbers: 1, Randomly pick: 2, Suppose, for each ; a random number in : adjustable parameter. evaluate If let If generate another random number: If let Otherwise,

  19. With { }, Recalculate the problem.

  20. Introduction to Crystal structure -continued

  21. reciprocal lattice Important to study reciprocal lattice Primitive translation vectors t1, t2 and t3 In the reciprocal space, we have g1, g2 and g3 ti∙gj =2 πδij 2 π factor is to simplify some expressions. If a crystal rotation of t1, t2, t3 is performed in the direct space, the same rotation of g1, g2, g3 occurs in the reciprocal space. The propagation of wave-vector k of a general plane wave exp(ik∙r) has the reciprocal length dimension!

  22. reciprocal space • All the points defined bye the vectors of the type: • gm =m1 gi +m2 g2 +m3 gj • Reciprocal lattice • Note: Only related to the translation properties of the crystal and not to the basis. • Solve that general equation, we have: • g1=2 (t2x t3) / Ω Ω = t1 ·(t2 х t3) volume of the primitive cell • g2=2 (t3 x t1) / Ω • g3=2 (t1x t2) / Ω • Examples:sc<==> scfcc<==> bcc bcc<==> fcc

  23. Useful Properties The direct and reciprocal lattices obey some simple useful properties 1, the volume Ωkof the unit cell in the reciprocal space is (2π)3times the reciprocal of the volume of the unit cell in the direct lattice. Will be assigned as a homework to prove this 2, gm∙tn =integer∙2π 3, If a vector q satisfies the relation , q∙t n =integer∙2πfor any tn , q has to be a reciprocacl lattice vector. 4, A plan wave exp(ik∙r) has the lattice periodicity if and only if the wavevector k equals a reciprocal lattice vector. W(r) = exp(i g m∙r)

  24. Fourier expansion W(r) = exp(i g m ∙r) remain unchanged if we replace r==> r+tn. A function f(r) periodic in the direct lattice can be expanded in the form F(r)= (i g m ∙r) Where, the sum is over reciprocal lattice vectors.

  25. Distance between lattice planes gm∙t n =integer∙2π Consider a family of planes in the direct space defined by the equations: gm∙r=integer∙2π All translation vectors belong to the family of planes. The distance between two consecutive planes is d= 2π/ gm Every reciprocal lattice vector is normal to a family of parallel and equidistant planes containing all the direct lattice points.

  26. Laue Condition and Braggrule Introduce Fourier Components of Charge density Laue Condition

  27. MAX VON LAUE • 1914 Nobel Laureate in Physics • for his discovery of the diffraction of X-rays by crystals.

  28. 1915 Nobel Laureate in Physics for their services in the analysis of crystal structure by means of X-rays • SIR WILLIAM HENRY BRAGG(1862-1942) • SIR WILLIAM LAWRENCE BRAGG(1890-1971)

  29. k-k0=G elastic diffraction: |k0|= |k|= |k - G| Squared 2 k •G = G2 Bragg plane Bragglaw n2dhkl sin 