Part A: Plane waves

Associated videos: None. Wikipedia

File: Plane Waves

A.1 Initialization: Open the Plane Wave file in Mathematica and select "Evaluation-Evaluate Notebook". This reads in all the code in sequence and initializes the interactive parts. Wave your hands if Mathematica is giving you trouble.

A.2 1D Plane Waves: Consider the first block of code. The most important part is the function under consideration, which is the real part of a complex exponential.

i - What happens when the magnitude of k changes?

ii - When the sign of k changes, what happens to the progression of the wave with time?

iii - What is the impact of changing w?

iv - Rather than considering the real amplitude, what is the intensity of the wave as a function of position?

v - How would you manipulate the function so that at time = 0, x = 0, the amplitude was zero?

A.3 2D Plane Waves: How does the k vector direction relate to the direction of propagation for the plane wave?

 

Part B: Fourier Series & Transforms

Associated videos: 3.0

File: Fourier

B.1 Initialization: Open the file in Mathematica and execute the notebook through "Evaluate-Evaluate Notebook". For the first example, expand the x-range to 3 pi to see the periodicity. Wave your hands if Mathematica is giving you trouble.

B.2 Number of terms: How many terms in the Fourier series are needed for a 'good' approximation? Does the value change with the shape you're trying to approximate?

B.3 Fourier coefficients: How do you expect the magnitude of the Fourier coefficients to vary as you go out to farther terms?

B.4 Orthogonality: Are the terms within a Fourier series orthogonal and/or orthonormal? Justify.

B.5 - Fourier transforms -

i - When you take the Fourier transform of the series PeriodicFunction, why did the plot look like it did?

ii - If you change all the Fourier coefficients to be unity, what do you expect the plot to look like?

iii - If you change the phase offsets, what do you expect the plot to look like?

B.6 - Gaussians: We know that Guassians can be guild from an infinite series of plane waves. When we take the transform of a Gaussian, we obtain another Gaussian in this space of Fourier coefficients vs wavenumber. Why does a broader Gaussian lead to a narrower peak following the transform?

Part C: Scattering density

Associated videos: 3.1, 3.2, 3.3

In this exercise, we consider the limits of scattering phenomena by investigating two different scattering densities n(r).

C.1 Concept setup: Let's make sure we're all on the same page.

i - What are delta k and G and how can they be modified?

ii - What space is G in?

iii - Most source and detector combinations do not lead to a peak in intensity. In your own words, describe what happens when the source and detector lead to a delta k which does not equal a G of the lattice.

C.2 Jellium solid - Imagine a material with uniform scattering density, n(r) = n'.

i - Do any solids behave like this? Other phases of matter?

ii - Develop an expression for the scattering amplitude as a function of delta k. You can leave this in integral form

iii - Solve the integral for the scattering density. In words, what result do you obtain?

C.3 Dirac comb - Imagine a material with a scattering density built out of a periodic array of delta functions.

i - Do any solids behave like this? Hint: The answer depends on scattering source.

ii - Develop an expression for the scattering density.

iii - Develop an expression for the scattering amplitude as a function of delta k. You can leave this in integral form

iv - Solve the integral for the scattering density. In words, what result do you obtain?

C.4 Liquids - What sort of scattering do you expect from a liquid?

 

Part D: Scattering examples

Associated videos: 3.1, 3.2, 3.3

The goal of this activity is to apply the delta k = G condition for diffraction to real and model materials.

D.1 Chain of atoms - Imagine a single strand of atoms in a near-infinite linear chain. Both your incident beam and detector are off the chain axis by some angle theta. We'll treat the scattering density of each atom as a Gaussian sphere.

i - Develop an expression for the scattering density. Let's describe the atomic scattering density as n_alpha.

ii - Sketch the Fourier transform of this chain. Hint: You can treat the axial and radial components separately. Think about the individual waves you would need to build up this scattering density.

iii - As you sweep across theta with your source and detector, what will you observe at your detector?

iv - If you rotate the dashed line, above, by 90 degrees, what will you observe at your detector as you sweep theta?

v - How can you use the data generated above to tell you about the structure?

 

D.2 3D structure determination - You're working with a single crystal chunk of polonium to determine its structure.. Before it kills you from radiation, you do a series of scans with your X-ray diffractometer.

i - What do you know about reciprocal space at the start of the experiment?

ii - The first scan picks an arbitrary source angle and then sweeps the detector. Sketch what happens to delta k in reciprocal space.

iii - Once you've found a peak, you use your magical goniometer to rotate the sample across all orientations and pick up 7 additional peaks, 70.5 degrees apart. Form a hypothesis on what the structure is and suggest a second position for the source and detector to pick up another set of peaks to test this hypothesis.

iv - To measure the coefficient of thermal expansion of polonium in this structure, what would you do?

 

Part E: Structure factor

Associated videos: 3.4

E.1 Return of the 1D chain - In Part D, we considered scattering from a 1D chain. Let's repeat the question for an alternating chain of low and high electron density atoms.

i - Develop an expression for the scattering density. Let's describe the atomic scattering density as n_alpha.

ii - Sketch the Fourier transform of this chain. Hint: You can treat the axial and radial components separately. Think about the individual waves you would need to build up this scattering density. What new terms are required to describe the chain?

iii - As you sweep across theta with your source and detector, what will you observe at your detector? You'll want to consider the structure factor of the chain.

iv - If you rotate the dashed line, above, by 90 degrees, what will you observe at your detector as you sweep theta?

v - How can you use the data generated above to tell you about the structure?

E.2 Trends across a series - You've got single crystals of Ge, GaAs, and ZnSe, all in the zinc blende structure.

For the diamond structure, the structure factor rules are:

for mixed values (odds and even values combined) of h, k, and l, F^2 will be 0

if the values are unmixed and... h+k+l is odd then F=4f(1+i) or 4f(1-i), F^2=32 f^2 h+k+l is even and exactly divisible by 4 (satisfies h+k+l=4n) then F^2 = 64 f^2 h+k+l is even but not exactly divisible by 4(doesn't satisfy h+k+l=4n) then F^2 = 0

 

i - What is the average electron count per atom in these three compounds?

ii - What will happen w/ respect to constructive and destructive interference of the (h00) reflections as we move from Ge to ZnSe?