softmatter:Structure Factor

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Some phases of structure are not easily characterized by an order parameter. Characterizing these structures often involves visual inspection of the particle positions in the simulated system. An alternative method is to calculate the Structure Factor, S(q). For a given structure, S(q) will have characteristic refraction peaks that can be used to identify the structure.

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Structure Factor

The structure factor of a crystal is a mathematical description of how the crystal scatters incident radiation due the geometric arrangement of the ions. Experimentally, structure factors are generated by x-ray diffraction. The incidence angle of the x-rays are varied while a detector measures the intensity of the reflected radiation. The ratios of the peaks in reciprocal space indicate the spacing of different planes within the crystal.

Calculating the Structure Factor in simulation

Structure factors can also be calculated for a simulation. If \mathbf{r_j} is the position of the j-th minority component of N components, then S(q) is calculated as,

S(\mathbf q) = \frac{\left( \sum_j cos(\mathbf q \cdot \mathbf r_j)\right)^2 + \left( \sum_j sin(\mathbf q \cdot \mathbf r_j)\right)^2}{N}

where q is the wave vector,

\mathbf q = 2 \pi \left(\frac{n_x}{L_x},\frac{n_y}{L_y},\frac{n_z}{L_z}\right) and nx, ny, and nz are integers greater than or equal to zero. Lx, Ly, and Lz are the appropriate dimension of the simulation box.

S(q) is often plotted against the modulus or magnitude of the wave vector. In general, the structure factor is calculated for only low values of q modulus, as higher values correspond to the small spacings between the particles in the system, rather than the spacing between crystallographic planes. Properly speaking, different species in a system will have different intensities of scattering. But by calculating the structure factor for only one species, we can ignore this effect.

This method can also be used to extract the natural length of the unit cell for a periodic structure [2].

L_{box} = \frac{2\pi}{|\mathbf q_{rp}|}m

where \mathbf q_{rp} is wave vector at which the maximum S(q) is located. and m is the first observed reflection spacing ratio of a given periodic structure.

A Peculiarity in calculating the Structure Factor of a Simulation Box

The structure factor looks for correlations in the position of particles at a length scale of integral fractions of the simulation box length. Because the simulation is periodic, particles on the simulation wall are correlated with particles on the other side of the simulation wall, i.e. the other end of the box, even though, by the structure factor calculation, these particles are a full box length away from each other. Thus, in my experience it is very common to see the first peak show up at m =1. If the structure factor calculations is not calculated to a high enough nx,ny,nz value, this may falsely be the only significant peak you see. If you look at the application of the structure factor below, you will not see this peak. That is because these particles were never given an interaction potential and permitted to dynamically evolve as a result of that interaction.

Characteristic Structure Factors

4000 spheres in an FCC crystal.
4000 spheres in an FCC crystal.
Structure Factor Calculated for crystal above
Structure Factor Calculated for crystal above
4000 spheres in an FCC crystal.
4000 spheres in an FCC crystal.
Structure Factor Calculated for crystal above
Structure Factor Calculated for crystal above

Face Centered Cubic (FCC)

Per the calculations at [1], or found in [4], the structure factor for a face centered cubic has peaks when the integer indices of the wavevector are either all even or all odd. The wave vector is defined as

\mathbf{K}=(2\pi/a)(h\hat{x}^* + k\hat{y}^* + l\hat{z}^*),

where a is the length of the side of a unit cell. This corresponds to peaks at \sqrt(h^2+k^2 +j^2) = \sqrt 3 : \sqrt 4 : \sqrt 8 : \sqrt 12 . . . ,

Lets consider an application to the perfect FCC crystal (shown on the right) of side length L = 15.87. When we apply the S(q) calculation from above to this crystal, for n_x,n_y,n_z \leq 20 , we get the structure factor S(q) vs |q| shown on the right. Lets assume that we suspected we had an fcc crystal. Then the first peak should correspond to \sqrt 3 . This peak occurs at a |q| = 6.8557. We check the ratios of the |q| values of the latter peaks to the |q| value of the first peak and verify that the ratios match \frac{\sqrt 4} {\sqrt 3} , \frac{\sqrt 8}{ \sqrt 3 }, \frac{\sqrt 12}{ \sqrt 3} , and they do! Now we use the equation from above to calculate the length of our unit box. We calculate Lbox = 1.587, or that there are 10 unit cells on a side. Now it becomes clear to us that the nx,ny,nz above were equal to 10*h, 10*k, 10*j . S(q) calculated for fractional values of h, k, and j, averages to zero.

Also on the right, we can see what happens to the structure factor as the FCC crystal is joggled randomly to introduce noise. Each sphere has been displaced a uniformly distributed random amount between +/- 0.1 in the x, y, and z, direction. As can be seen from the new structure factor, the peaks are smaller but still very apparent.

Body Centered Cubic

The Structure Factor for a body centered cubic crystal arrangement has peaks when h+k+l = even number. This corresponds to peaks at \sqrt(h^2+k^2 +j^2) = \sqrt 2 : \sqrt 4 : \sqrt 6: \sqrt 8: \sqrt 10 :\sqrt12 :\sqrt 14: \sqrt 16 . . .

Diamond Cubic

The Structure Factor for a diamond cubic crystal arrangement has peaks when [5]:

  • h,k, and l are all odd
  • h,k,l are all all even AND [h+k+l] = odd number OR [h+k+l]` = 2*[even number] (i.e. no peaks will be found if [h+k+l] = [4n+2], where n is an integer)

The latter case corresponds to the largest peaks for a perfect crystal. This corresponds to peaks at \sqrt{(h^2+k^2 +l^2)} = \sqrt 3 : \sqrt 8 : \sqrt {11} : \sqrt {16}. . .

Lamellae

Lamellaee are characterized by peaks at q modulus ratios 1:2:3:4, although only the first two may be visible. A pronounced first peak, and smaller second peak can be seen in the experimental data in Figure 1, from [1].

Gyroid

The gyroid is identified by peaks at \sqrt 3 : \sqrt 4 : \sqrt 10 : \sqrt 11 , although often only the first two peaks will be visible.

Continuous S(q) methods

see http://arxiv.org/pdf/cond-mat/0702263

References

  • [1] Martinez-Veracoechea, Escobedo, "Lattice Monte Carlo SImulations of the Gyroid Phase in Monodisperse and Bidisperse Block Copolymer Systems", Macromolecules, 2005, 38, 8522-8531 [2]
  • [2] Hajduk, et al., "The Gyroid: A New Equilibrium Morphology in Weakly Segregated Diblock Copolymers", Macromolecules, 1994, 27, 4063-4075
  • [3] Shultz, A. "Modeling and Computer Simulation of Block Copolymer/Nanoparticle Nanocomposites", Ph.D thesis, Chapter 5, North Carolina State University, Raleigh, NC, 2003
  • [4] Ashcroft, Mermin, "Solid State Physics", 1976
  • [5] De Graef M. and McHenry M.E., "Structure of Materials", Cambridge University Press, 2007 Link to Google books
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