Wikiadmin: Wikiadmin moved page Problem: Fourier transform to Problem:Fourier transform

2020-09-20T15:29:35Z

Wikiadmin moved page Problem: Fourier transform to Problem:Fourier transform

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ucph>Tommy: Created page with " One-dimensional crystal. Mathematically the scattering amplitude is the Fourier transform of the dist..."

2019-07-14T21:28:28Z

Created page with " thumb | 400px | <caption>One-dimensional crystal.</caption> Mathematically the scattering amplitude is the Fourier transform of the dist..."

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[[File:problemfouriertransform.png | thumb | 400px | <caption>One-dimensional crystal.</caption>]]
Mathematically the scattering amplitude is the Fourier transform of the distribution of scattering centers (nuclei, electrons, spins) within the material. The scattered intensity (the scattering function) is the square of the scattering amplitude.

The Fourier transform of a function \(\rho(r)\) is written as

:\( F(q) = \displaystyle\int \rho(r) \exp{(iqr)} dr , \)

where \(\rho(r)\) is the function in real space given by positions \(r\), and \(q\) is a coordinate in Fourier space (which in scattering terms usually is called "reciprocal space"). \(\rho(r)\) is in case of scattering theory the position sensitive scattering length density within the sample.

We will consider a one-dimensional space, ''i.e.'' all particles (scattering centers) are positioned on a line, and correspondingly only calculate the one-dimensional Fourier transform. We assume further that all particles are points (size = 0).

=====Question 1=====
Calculate the Fourier transform and the scattering intensity of a sample with only one particle, and plot the normalized scattered intensity \(I(q)=|F(q)|^2/N^2\) versus \(qR\).

{{hidden begin|toggle=right|title=Hint|titlestyle=background:#ccccff}}
A point-particle may mathematically be described as a Dirac \(\delta\)-function with the property

:\( \displaystyle\int \delta(r_0) f(r) d r = f(r_0) .\)
{{hidden end}}

{{hidden begin|toggle=right|title=Hint|titlestyle=background:#ccccff}}
Place the particle in origo \((0,0)\).
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{{hidden begin|toggle=right|title=Solution|titlestyle=background:#ccccff}}
Fourier transform:

:\( F(q) = \displaystyle\int \delta(0) \exp{(iqr)} dr = \exp(0) = 1.\)

Scattering intensity:

:\( I(q) = F^2(q) = 1^2 = 1. \,\)

[[File:Fouriersolution1.png | border | <caption>Sample and normalized intensity \(I(q)/N^2\).</caption>]]
{{hidden end}}

=====Question 2=====
Calculate the Fourier transform and the scattering intensity of a one-dimensional crystal with two particles separated with a distance \(R\), and plot the normalized scattered intensity \(I(q)\) versus \(qR\).

{{hidden begin|toggle=right|title=Hint|titlestyle=background:#ccccff}}
Place the particles symmetric around origo, ''i.e.'' in \(-R/2\) and \(+R/2\).
{{hidden end}}

{{hidden begin|toggle=right|title=Solution|titlestyle=background:#ccccff}}
Fourier transform:

:\( F(q) = \displaystyle\int \big( \delta(-R/2) + \delta(R/2) \big) \exp{(iqr)} dr
= \exp{(iqR/2)} + \exp{(-iqR/2)}
= 2 \cos{(qR/2)} . \)

Scattering intensity:

:\( I(q) = F^2(q) = 4\cos^2{(qR/2)} . \,\)

[[File:Fouriersolution2.png | border | <caption>Sample and normalized intensity \(I(q)/N^2\).</caption>]]
{{hidden end}}

=====Question 3=====
Calculate the Fourier transform and the scattering intensity of a one-dimensional crystal with three particles separated with a distance \(R\), and plot the normalized scattered intensity \(I(q)\) versus \(qR\).

{{hidden begin|toggle=right|title=Hint|titlestyle=background:#ccccff}}
Place the particles in origo and in \(-R\) and \(+R\).
{{hidden end}}

{{hidden begin|toggle=right|title=Solution|titlestyle=background:#ccccff}}
[[File:Fouriersolution3.png | border | <caption>Sample and normalized intensity \(I(q)/N^2\).</caption>]]
{{hidden end}}

=====Question 4=====
Calculate the Fourier transform and the scattering intensity of a one-dimensional crystal with more particles (4, 5, 6, or ...) separated with a distance \(R\), and plot the normalized scattered intensity \(I(q)\) versus \(qR\).

{{hidden begin|toggle=right|title=Hint|titlestyle=background:#ccccff}}
Place the particles symmetric with respect to origo.
{{hidden end}}

{{hidden begin|toggle=right|title=Solution|titlestyle=background:#ccccff}}
[[File:Fouriersolution4-4particle.png | border | <caption>Sample with 4 particles, and normalized intensity \(I(q)/N^2\).</caption>]] [[File:Fouriersolution4-8particle.png | border | <caption>Sample with 8 particles, and normalized intensity \(I(q)/N^2\).</caption>]]
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=====Question 5=====
Sketch the normalized scattering intensity of a one-dimensional sample of a very large number of particles (\(\sim\)infinite) separated with a distance \(R\).

{{hidden begin|toggle=right|title=Solution|titlestyle=background:#ccccff}}
[[File:Fouriersolution5.png | border | <caption>Sample and normalized intensity \(I(q)/N^2\).</caption>]]
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Problem:Fourier transform - Revision history

Wikiadmin: Wikiadmin moved page Problem: Fourier transform to Problem:Fourier transform

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ucph>Tommy: Created page with " One-dimensional crystal. Mathematically the scattering amplitude is the Fourier transform of the dist..."