Correlation Functions and Fourier Transforms

Transcription

1 Correlation Functions and Fourier Transforms Introduction The importance of these functions in condensed matter physics Correlation functions (aside convolution) Fourier transforms The diffraction pattern of Ag 3 SI as a Fourier transform of the simulated total atom pair correlation function This week s coursework atom pair correlation functions and neutron scattering from your computer simulations of Ag 3 SI

2 MC Generated Configurations T =0 T =5.5 T =100

3 Correlation Functions Correlation functions are used widely in signal processing, astronomy and condensed matter physics They are formulated in many different ways Here we are using the atomic pair distribution function as an example of a correlation function, i.e. the probability of finding an atom of type j a distance r from an atom of type i in a material For a multicomponent material, the partial pair distribution function, g ij (r) is given by: where n ij (r) is the number of atoms j between nij( r) r and r+dr from an atom i, averaged over all gij( r) = 2 atoms i as centres and r 4p r drr j is the average j atomic density r multiplied by the proportion of atoms j

4 Pair Distribution Functions by Rosalind Franklin R E Franklin, Acta Cryst (1950)

5 Partial Pair Distribution Function g n ij ij( r) = 2 ( r) 4p r drr j ~n(r) r r For a monatomic ideal 2D square lattice

6 Total Pair Distribution Function This is simply a weighted sum of all the partial pdf s from a material, i.e. G( r) = n å i, j= 1 c c i ( r) - 1) for a neutron weighted _ pdf. c i is the proportion of atom i in the material and b i is the neutron scattering length for atom i. For Ag 3 SI, a 3-component material, there are 9 different terms in the summation, but only 6 distinct g ij (r), since g ij (r) = g ji (r), i.e. j b b i j ( g ij G( r) = c 2 Ag + c + c b 2 S 2 I 2 Ag b b 2 S 2 I ( g ( g ( g AgAg SS II ( r) - 1) + 2c ( r) - 1) + 2c ( r) - 1) S Ag c I c b S S b b I Ag b ( g S SI ( g AgS ( r) - 1) ( r) - 1) + 2c Ag c I b Ag b I ( g AgI ( r) - 1) (Keen J. Appl. Cryst. 34 (2001) 172)

7 Broadening Correlation Functions Convoluting G(r) with a Gaussian broadening function s = 0.0Å s = 0.1Å s = 0.5Å

8 Partial Pair Distribution Functions in b - Ag3SI a/2 3a/2 a 2a 6. CORRELATION FUNCTIONS AND FOURIER TRANSFORMS COMPUTATIONAL METHODS IN CONDENSED MATTER PHYSICS RESEARCH, DAVID KEEN, MT 2017

9 Total Pair Distribution Function in b -Ag 3 SI r*g(r) Ag S I Ag S I

10 Introducing Fourier Transforms The Fourier transform, F(s), of a function f(x) is defined as: F( s) with the inverse relationship: f ( x) ò - 2p = f ( x)e ixs dx - 2p = F( s)e ixs ds ò - They are widely used in condensed matter physics since they relate the time and frequency domains (or, in our example here, real and reciprocal space) Hence the link between measurement and physical interpretation is often forged via a Fourier transform Here we are only looking at the practicalities of these functions, not the theory

11 The Fourier Transform in Diffraction A reciprocal space total scattering function, F(Q), is related to a real space correlation function, G(r), via a sine Fourier transform: with the inverse relationship: = ò 2 sin( Qr) F( Q) r 0 4p r G( r) dr 0 Qr 1 sin( Qr) G( r) = Q (2p ) ò 2 4p Q F( Q) d 3 r 0 0 Qr In practical terms the Fourier transform is limited: G( r j 1 ) = 3 (2p ) r 0 N å i= 1 4p Q where Q min <Q i <Q max and F(Q) suffers from experimental errors 2 i sin( Qirj F( Qi ) Q r i j ) d Q i (Keen J. Appl. Cryst. 34 (2001) 172)

12 Fourier Relationship in a *-Ag 3 SI at 10K Ideal situation Reciprocal space Real space

13 Fourier Relationship in a *- Ag3SI at 10K Actual situation Real space Reciprocal space FT Q x F(Q) r x G(r) 6. CORRELATION FUNCTIONS AND FOURIER TRANSFORMS COMPUTATIONAL METHODS IN CONDENSED MATTER PHYSICS RESEARCH, DAVID KEEN, MT 2017

14 Effect of Termination in a FT Truncation broadens sharp features by 3.791/r max and introduces ripples from the step function at r max

15 Using Modification Functions Modification function = ò sin( 2 Qr F( Q) r ) 0 4p r G( r) M ( r) dr 0 Qr Where (for example): M ( r) = sin( p r / r ( p r / r max max ) ) Before modification (offset by +1) After modification

16 FT of G(r) Using a Modification Function With modification Without modification

17 Total Scattering S(Q) from Ag 3 SI

18 Total Pair Distribution Functions of Ag 3 SI

19 This Week s Coursework Calculate total pair distribution functions (pdfs) from representative Monte Carlo generated configurations of S-I ordering Fourier transform these pdfs to obtain total scattering structure factors, F(Q). Investigate: _ 1. The effect of different sized configurations/r-ranges 2. The use of a modification function on the pdf prior to the Fourier transform _ Use b S =-b I to highlight the local deviation in your models from the average scattering and, by comparison with the data from a *-Ag 3 SI, estimate the degree of ordering found experimentally You will need to use a configuration which is at least 20x20x20 unit cells.

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21 Code to Convolute with a Gaussian Function dx=x(2)-x(1) nsig=3.0*sig/dx fgau0=(1/((2.0*pi)**0.5*sig)) do i=1,nsig fgau(i)=(1/((2.0*pi)**0.5*sig))*exp(-(real(i)*dx)**2/(2.0*sig**2)) enddo Defining Gaussian using x-spacing from data and out to 3s do i=1,npts yin(i)=yin(i)*dx yout(i)=yout(i)+fgau0*yin(i) do j=1,nsig if ((i+j).le.npts) then yout(i+j)=yout(i+j)+yin(i)*fgau(j) endif if((i-j).ge.1) then yout(i-j)=yout(i-j)+yin(i)*fgau(j) endif enddo enddo Looping through all data points and spreading their intensity into neighbouring data points using the pre-determined (symmetric) Gaussian values. (N.B. does not treat ends of the data correctly.) Of course, de-convoluting resolution functions from data is often more important