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Re: 2D FFT of HCP lattice -- Inconsistency with line plot?

Posted by Mason Guffey on Aug 15, 2008; 4:43pm
URL: http://imagej.273.s1.nabble.com/2D-FFT-of-HCP-lattice-Inconsistency-with-line-plot-tp3695379p3695381.html

Hi Michael,

Thanks so much for your help! I have to admit, I'm still a little
confused on this, can you provide me with a reference for this? Or would
any solid state physics text have an explanation?

Is this a general property of a Fourier transform (i.e in a mathematical
sense) or something particular to the 2D FFT algorithm? What I'm using
this for is to determine the average interparticle separation for a TEM
image of a nanoparticle array. The array is a 2d close-packed (HCP)
lattice of spheres. Does what you say hold true for arrays of spheres
rather than the triangle example?

Also, what if I wanted to use the FFT to determine the average
interparticle separation of a disordered complex of spherical particles
(in this case the FFT yields rings rather than peaks). Would the same
constant factor apply?

I realize that this is an ImageJ forum rather than an FFT forum, but I'm
having a hard time finding any discussion of this and I think this may
be a problem that has tripped up other people working with nanoparticle
superlattices. So your comments are very helpful.

thanks again,
Mason

Michael Schmid wrote:

> Hi Mason,
>
> a 2D FFT of a non-rectangular lattice gives the distance between
> parallel *lines*, which is different from the lattice constant
> measured along a lattice direction.
> In physics, this is known as reciprocal lattice.
> Also note that the reciprocal lattice vectors are perpendicular to
> the lines of the original lattice, and their length corresponds to
> the distance measured in that direction.
>
> So you get the side length of the triangles multiplied by sqrt(3)/2.
>
> Michael
> ____________________________________________________________________
>
> not that reciprocal lattic;
>
> On Thu, 14 Aug 2008 15:46:05 -0500 Mason Guffey <[hidden email]>
> wrote:
>  
>> Hi all,
>>
>> I've been having a problem with ImageJ related to 2-d fourier transform
>> of ordered images. Essentially, if I take an FFT of any hexagonally
>> close-packed lattice (e.g.
>> http://content.answers.com/main/content/wp/en-commons/thumb/c/c0/180px-Tile_3,6.svg.png 
>> ) and compare that with a line plot of the objects I get inconsistent
>> results.
>>
>> Specifically, take a line plot down any lattice direction of that image.
>> Divide the length of the line plot by the number of triangles it passes
>> through (i.e. the total number of lattice spacings). Compare that with
>> what you get from the FFT. The line plot yields a lattice constant of
>> about 14.5 pixels per cycle while the fourier transform peaks are all
>> right around 12 pixels / cycle.
>>
>> Does anyone know what's going on here? This same phenomena has repeated
>> for just about any image of hexagonal close-packed circles that I load
>> into it.... The line plot method reveals X for a lattice constant, while
>> the FFT peak is at Y. There's a factor of about 1.2 between them.
>>
>> However, if you do the same thing with an array of straight lines, the
>> FFT and the line profile match eachother perfectly. (As is shown on the
>> imageJ website for the FFT demo).
>>
>> Thanks in advance
>>
>> --
>> Mason Guffey | Gordon Center for Integrative Science ESB09B
>> Scherer Group | [hidden email]
>> Department of Chemistry | (773) 834-1877
>> The University of Chicago | http://schererlab.uchicago.edu
>>
>>