Summary

Simulation d'un diffractogramme par rayons X du quartz fortement bruité, et sa transformée de Fourier.

La partie gauche du spectre contient le signal ; pour des fréquences supérieures à l'inverse de la largeur des pics, le spectre ne contient que du bruit.

Simulation of the X-ray diffratogram of quartz with a strong noise, and its Fourier transform.

The left part of the spectrum contains the signal; for the frequencies higher than the inverse of the width of the peaks, the spectrum contains only noise.

Auteur/author : Christophe Dang Ngoc Chan (cdang) Réalisé avec/made with Scilab

clear;clf;

// domaine : 20 à 50°, pas de 0,02°

pas=0.02; // step size
freq=1/pas; // sampling frequency
x=[20:pas:50]; // 2-theta range

// hauteur maximale, largeur et fond continu

h=1000; // maximum height
s=0.1; // peak width (standard variation)
fond=50; // continuous background

// Fonction de Gauss

function [y]=gauss(x)
   y=2.5*h/(sqrt(2*%pi))*exp(-x^2/(2*s))
endfunction

// pics, désignés par les indices de Miller hkl

p100 = 0.16*gauss(x-20.86);
p101 = gauss(x-26.64);
p110 = 0.09*gauss(x-36.544);
p102 = 0.08*gauss(x-39.465);
p111 = 0.04*gauss(x-40.3);
p200 = 0.06*gauss(x-42.45);
p201 = 0.04*gauss(x-45.793);
p112 = 0.13*gauss(x-50.139);

// diffractogramme

d = p100 + p101 + p110 + p102 + p111 + p200 + p201 + p112 + fond;

// bruit

N = size(d); n=N(1,2);
b = rand(1,n,'normal');
b1 = b.*sqrt(d);

// Calcul

f=d+b1; // diffractogramme
ptf = abs(fft(f)); // puissance de la transformée de Fourier
for i=1:n
  if ptf(i)>2e+9 then ptf(i)=2e+9; // saturation
  end
end

k=freq*(1:n)/n; // échelle x TF

// Affichage

subplot(2,1,1)
plot2d(x,f,style=2,rect=[20,0,50,h+fond])
xtitle('Simulation du diffractogramme du quartz','2 thêta  (°)','intensité (coups)')

subplot(2,1,2)
plot2d(k,ptf,style=2,rect=[0,100,20,1e5],logflag='nl')
xtitle('Transformée de Fourier du diffractogramme','fréq.','Puissance (éch. log.)')

Licensing

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