Time-frequency Analysis in Physics

This page demonstrates the use of our time-frequency analysis add-on package for Mathematica. For an introduction to time-frequency analysis, read our tutorial.

Physics example

For the last century, X-ray crystallography has been used to determine the 3D atomic structure of crystals. It is well known that no technique can exist which is capable of extracting the same information from amorphous materials. However, CWT-based time-frequency analysis can be used to extract far more information from the diffraction data of non-crystalline materials than was previously thought possible.

Load some experimentally measured X-ray diffraction data for amorphous silicon, as the static structure factor S(k):

Sdata = ReadList["Physics/sqgr.dat", {Number, Number}] ;

{kmin, kmax} = {Sdata〚1, 1〛, Sdata〚 -1, 1〛} ;

Calculate the reduced static structure factor F(k)=k(S(k)-1):

Fdata = {#〚1〛, #〚1〛 (#〚2〛 - 1)} &/@Sdata ;

Create an odd-symmetric interpolation of this function for -k_max<k<k_max:

Finterp = Interpolation[Join[-Reverse @ Fdata, Fdata]] ;

Plot[Finterp[k], {k, -kmax, kmax}, AxesLabel {"k", "F(k)"}] ;

[Graphics:HTMLFiles/tutorial_166.gif]

Calculate the average separation δk of the samples of F(k):

δk = (kmax - kmin)/(Length[Sdata] - 1)

0.0248372

Calculate the expected number of samples n in the range -k_max+δk<k<k_max:

n = Round[2kmax/δk]

1852

Compute the real-reciprocal space representation as the TFR of F(k):

rrsr = FunctionTFR[Finterp[k], {k, -kmax + δk, kmax, n}, Parameter7π] ;//Timing

{76.3854 Second, Null}

{rmin, rmax} = GetTFRωRange[rrsr] ;

Plot3D[Log @ Abs @ rrsr[k, r], {r, rmin, 25}, {k, 0, 12}, PlotRange {-4, .5}, AxesLa ...                                                                                                F

[Graphics:HTMLFiles/tutorial_177.gif]

ContourPlot[Log @ Abs @ rrsr[k, r], {r, rmin, 25}, {k, 0, 12}, PlotRange {-4, .5}, FrameLabel {"r", "k"}] ;

[Graphics:HTMLFiles/tutorial_179.gif]

The ridge at k=2Å^(-1) is of particular interest as this corresponds to the first sharp diffraction peak in the static structure factor. Further CWT-based analysis has shown that this ridge corresponds to a sinusoidal component with exponentially decaying amplitude in the real-space reduced radial distribution function. Naturally, the frequency and instantaneous amplitude of this (and other) components can be quantified via the TFR.