KSpace Module

Modules for calculating quantities in reciprocal space, including Fourier transforms of shapes and diffraction pattern generation.

Meshgrid

freud.kspace.meshgrid2(*arrs)

Computes an n-dimensional meshgrid.

source: http://stackoverflow.com/questions/1827489/numpy-meshgrid-in-3d

Parameters:arrs – Arrays to meshgrid Returns:tuple of arrays Return type:tuple

Structure Factor

Methods for calculating the structure factor of different systems.

class freud.kspace.SFactor3DPoints(box, g)

Compute the full 3D structure factor of a given set of points.

Given a set of points \(\vec{r}_i\), SFactor3DPoints computes the static structure factor \(S \left( \vec{q} \right) = C_0 \left| {\sum_{m=1}^{N} \exp{\mathit{i}\vec{q}\cdot\vec{r_i}}} \right|^2\).

In this expression, \(C_0\) is a scaling constant chosen so that \(S\left(0\right) = 1\), and \(N\) is the number of particles.

\(S\) is evaluated on a grid of \(q\)-values \(\vec{q} = h \frac{2\pi}{L_x} \hat{i} + k \frac{2\pi}{L_y} \hat{j} + l \frac{2\pi}{L_z} \hat{k}\) for integer \(h,k,l: \left[-g,g\right]\) and \(L_x, L_y, L_z\) are the box lengths in each direction.

After calling compute(), access the \(q\) values with getQ(), the static structure factor values with getS(), and (if needed) the un-squared complex version of \(S\) with getSComplex(). All values are stored in 3D numpy.ndarray structures. They are indexed by \(a, b, c\) where \(a=h+g, b=k+g, c=l+g\).

Note

Due to the way that numpy arrays are indexed, access the returned S array as S[c,b,a] to get the value at \(q = \left(qx\left[a\right], qy\left[b\right], qz\left[c\right]\right)\).

compute(points)

Compute the static structure factor of a given set of points.

After calling compute(), you can access the results with getS(), getSComplex(), and the grid with getQ().

Parameters:points (numpy.ndarray, shape=(\(N_{particles}\), 3), dtype= numpy.float32) – points used to compute the static structure factor

getQ()

Get the \(q\) values at each point.

The structure factor S[c,b,a] is evaluated at the vector \(q = \left(qx\left[a\right], qy\left[b\right], qz\left[c\right]\right)\).

Returns:(qx, qy, qz) Return type:tuple

getS()

Get the computed static structure factor.

Returns:The computed static structure factor as a copy Return type:numpy.ndarray, shape=(X,Y), dtype= numpy.float32

getSComplex()

Get the computed complex structure factor (if you need the phase information).

Returns:The computed static structure factor, as a copy, without taking the magnitude squared Return type:numpy.ndarray, shape=(X,Y), dtype= numpy.complex64

class freud.kspace.AnalyzeSFactor3D(S)

Analyze the peaks in a 3D structure factor.

Given a structure factor \(S\left(q\right)\) computed by classes such as SFactor3DPoints, AnalyzeSFactor3D performs a variety of analysis tasks.

• Identifies peaks
• Provides a list of peaks and the vector \(\vec{q}\) positions at which they occur
• Provides a list of peaks grouped by \(q^2\)
• Provides a full list of \(S\left(\left|q\right|\right)\) values vs \(q^2\) suitable for plotting the 1D analog of the structure factor
• Scans through the full 3D peaks and reconstructs the Bravais lattice

Note

All of these operations work in an indexed integer \(q\)-space \(h,k,l\). Any peak position values returned must be multiplied by \(2\pi/L\) to get to real \(q\) values in simulation units.

getPeakDegeneracy(cut)

Get a dictionary of peaks indexed by \(q^2\).

Parameters:cut (numpy.ndarray) – All \(S\left(q\right)\) values greater than cut will be counted as peaks Returns:a dictionary with keys \(q^2\) and a list of peaks for the corresponding values Return type:dict

getPeakList(cut)

Get a list of peaks in the structure factor.

Parameters:cut – All \(S\left(q\right)\) values greater than cut will be counted as peaks Returns:peaks, \(q\) as lists Return type:list

getSvsQ()

Get a list of all \(S\left(\left|q\right|\right)\) values vs \(q^2\).

Returns:S, qsquared Return type:numpy.ndarray

class freud.kspace.SingleCell3D(k, ndiv, dK, boxMatrix)

SingleCell3D objects manage data structures necessary to call the Fourier Transform functions that evaluate FTs for given form factors at a list of \(K\) points. SingleCell3D provides an interface to helper functions to calculate \(K\) points for a desired grid from the reciprocal lattice vectors calculated from an input boxMatrix. State is maintained as set_ and update_ functions invalidate internal data structures and as fresh data is restored with update_ function calls. This should facilitate management with a higher-level UI such as a GUI with an event queue.

I’m not sure what sort of error checking would be most useful, so I’m mostly allowing ValueErrors and such exceptions to just occur and then propagate up through the calling functions to be dealt with by the user.

add_ptype(name)

Create internal data structures for new particle type by name.

Particle type is inactive when added because parameters must be set before FT can be performed.

Parameters:name (str) – particle name

calculate(*args, **kwargs)

Calculate FT. The details and arguments will vary depending on the form factor chosen for the particles.

For any particle type-dependent parameters passed as keyword arguments, the parameter must be passed as a list of length max(p_type)+1 with indices corresponding to the particle types defined. In other words, type-dependent parameters are optional (depending on the set of form factors being calculated), but if included must be defined for all particle types.

Parameters:

• position (numpy.ndarray, shape=(\(N_{particles}\), 3), dtype= numpy.float32) – array of particle positions in nm
• orientation (numpy.ndarray, shape=(\(N_{particles}\), 4), dtype= numpy.float32) – array of orientation quaternions
• kwargs – additional keyword arguments passed on to form-factor-specific FT calculator

get_form_factors()

Get form factor names and indices.

Returns:list of factor names and indices Return type:list

get_ptypes()

Get ordered list of particle names.

Returns:list of particle names Return type:list

remove_ptype(name)

Remove internal data structures associated with ptype name.

Parameters:name (str) – particle name

Note

This shouldn’t usually be necessary, since particle types may be set inactive or have any of their properties updated through set_ methods.

set_active(name)

Set particle type active.

Parameters:name (str) – particle name

set_box(boxMatrix)

Set box matrix.

Parameters:boxMatrix (numpy.ndarray, shape=(3, 3), dtype= numpy.float32) – unit cell box matrix

set_dK(dK)

Set grid spacing in diffraction image.

Parameters:dK (float) – difference in \(K\) vector between two adjacent diffraction image grid points

set_form_factor(name, ff)

Set scattering form factor.

Parameters:

• name (str) – particle type name
• ff (list) – scattering form factor named in get_form_factors()

set_inactive(name)

Set particle type inactive.

Parameters:name (str) – particle name

set_k(k)

Set angular wave number of plane wave probe.

Parameters:k (float) – \(\left|k_0\right|\)

set_ndiv(ndiv)

Set number of grid divisions in diffraction image.

Parameters:ndiv (int) – define diffraction image as ndiv x ndiv grid

set_param(particle, param, value)

Set named parameter for named particle.

Parameters:

• particle (str) – particle name
• param (str) – parameter name
• value (float) – parameter value

set_rq(name, position, orientation)

Set positions and orientations for a particle type.

To best maintain valid state in the event of changing numbers of particles, position and orientation are updated in a single method.

Parameters:

• name (str) – particle type name
• position (numpy.ndarray, shape=(\(N_{particles}\), 3), dtype= numpy.float32) – (N,3) array of particle positions
• orientation (numpy.ndarray, shape=(\(N_{particles}\), 4), dtype= numpy.float32) – (N,4) array of particle quaternions

set_scale(scale)

Set scale factor. Store global value and set for each particle type.

Parameters:scale (float) – nm per unit for input file coordinates

update_K_constraint()

Recalculate constraint used to select \(K\) values.

The constraint used is a slab of epsilon thickness in a plane perpendicular to the \(k_0\) propagation, intended to provide easy emulation of TEM or relatively high-energy scattering.

update_Kpoints()

Update \(K\) points at which to evaluate FT.

Note

If the diffraction image dimensions change relative to the reciprocal lattice, the \(K\) points need to be recalculated.

update_bases()

Update the direct and reciprocal space lattice vectors.

Note

If scale or boxMatrix is updated, the lattice vectors in direct and reciprocal space need to be recalculated.

class freud.kspace.FTfactory

Factory to return an FT object of the requested type.

addFT(name, constructor, args=None)

Add an FT class to the factory.

Parameters:

• name (str) – identifying string to be returned by getFTlist()
• constructor (object) – class / function name to be used to create new FT objects
• args (list) – set default argument object to be used to construct FT objects

getFTlist()

Get an ordered list of named FT types.

Returns:list of FT names Return type:list

getFTobject(i, args=None)

Get a new instance of an FT type from list returned by getFTlist().

Parameters:

• i (int) – index into list returned by getFTlist()
• args (list) – argument object used to initialize FT, overriding default set at addFT()

class freud.kspace.FTbase

Base class for FT calculation classes.

getFT()

Return Fourier Transform.

Returns:Fourier Transform Return type:numpy.ndarray

get_density(density)

Get density.

Returns:density Return type:numpy.complex64

get_parambyname(name)

Get named parameter for object.

Parameters:name (str) – parameter name. Must exist in list returned by get_params() Returns:parameter value Return type:float

get_params()

Get the parameter names accessible with set_parambyname().

Returns:parameter names Return type:list

get_scale()

Get scale.

Returns:scale Return type:float

set_K(K)

Set \(K\) points to be evaluated.

Parameters:K (numpy.ndarray) – list of \(K\) vectors at which to evaluate FT

set_density(density)

Set density.

Parameters:density (numpy.complex64) – density

set_parambyname(name, value)

Set named parameter for object.

Parameters:

• name (str) – parameter name. Must exist in list returned by get_params()
• value (float) – parameter value to set

set_rq(r, q)

Set \(r\), \(q\) values.

Parameters:

• r (numpy.ndarray) – \(r\)
• q (numpy.ndarray) – \(q\)

set_scale(scale)

Set scale.

Parameters:scale (float) – scale

class freud.kspace.FTdelta

Fourier transform a list of delta functions.

compute(*args, **kwargs)

Compute FT.

Calculate \(S = \sum_{\alpha} \exp^{-i \mathbf{K} \cdot \mathbf{r}_{\alpha}}\)

set_K(K)

Set \(K\) points to be evaluated.

Parameters:K (numpy.ndarray) – list of \(K\) vectors at which to evaluate FT

set_density(density)

Set density.

Parameters:density (numpy.complex64) – density

set_rq(r, q)

Set \(r\), \(q\) values.

Parameters:

• r (numpy.ndarray) – \(r\)
• q (numpy.ndarray) – \(q\)

set_scale(scale)

Set scale.

Parameters:scale (float) – scale

Note

For a scale factor, \(\lambda\), affecting the scattering density \(\rho\left(r\right)\), \(S_{\lambda}\left (k\right) == \lambda^3 * S\left(\lambda * k\right)\)

class freud.kspace.FTsphere

Fourier transform for sphere.

Calculate \(S = \sum_{\alpha} \exp^{-i \mathbf{K} \cdot \mathbf{r}_{\alpha}}\)

get_radius()

Get radius parameter.

If appropriate, return value should be scaled by get_parambyname('scale') for interpretation.

Returns:unscaled radius Return type:float

set_radius(radius)

Set radius parameter.

Parameters:radius (float) – sphere radius will be stored as given, but scaled by scale parameter when used by methods

class freud.kspace.FTpolyhedron

Fourier Transform for polyhedra.

compute(*args, **kwargs)

Compute FT.

Calculate \(S = \sum_{\alpha} \exp^{-i \mathbf{K} \cdot \mathbf{r}_{\alpha}}\)

get_radius()

Get radius parameter.

If appropriate, return value should be scaled by get_parambyname('scale') for interpretation.

Returns:unscaled radius Return type:float

set_K(K)

Set \(K\) points to be evaluated.

Parameters:K (numpy.ndarray) – list of \(K\) vectors at which to evaluate FT

set_density(density)

Set density.

Parameters:density (numpy.complex64) – density

set_params(verts, facets, norms, d, areas, volume)

Construct list of facet offsets.

Parameters:

• verts (numpy.ndarray, shape=(\(N_{verts}\), 3), dtype= numpy.float32) – list of vertices
• facets (numpy.ndarray, shape=(\(N_{facets}\), \(N_{verts}\)), dtype= numpy.float32) – list of facets
• norms (numpy.ndarray, shape=(\(N_{facets}\), 3), dtype= numpy.float32) – list of norms
• d (numpy.ndarray, shape=(\(N_{facets}\)), dtype= numpy.float32) – list of d values
• areas (numpy.ndarray, shape=(\(N_{facets}\)), dtype= numpy.float32) – list of areas
• volumes (numpy.ndarray) – list of volumes

set_radius(radius)

Set radius of in-sphere.

Parameters:radius (float) – radius inscribed sphere radius without scale applied

set_rq(r, q)

Set \(r\), \(q\) values.

Parameters:

• r (numpy.ndarray) – \(r\)
• q (numpy.ndarray) – \(q\)

class freud.kspace.FTconvexPolyhedron

Fourier Transform for convex polyhedra.

Spoly2D(i, k)

Calculate Fourier transform of polygon.

Parameters:

• i (int) – face index into self.hull simplex list
• k (int) – angular wave vector at which to calculate \(S\left(i\right)\)

Spoly3D(k)

Calculate Fourier transform of polyhedron.

Parameters:k (int) – angular wave vector at which to calculate \(S\left(i\right)\)

compute_py(*args, **kwargs)

Compute FT.

Calculate \(P = F * S\):

• \(S = \sum_{\alpha} \exp^{-i \mathbf{K} \cdot \mathbf{r}_{\alpha}}\)
• F is the analytical form factor for a polyhedron, computed with Spoly3D()

get_radius()

Get radius parameter.

If appropriate, return value should be scaled by get_parambyname(‘scale’) for interpretation.

Returns:unscaled radius Return type:float

set_radius(radius)

Set radius of in-sphere.

Parameters:radius (float) – radius inscribed sphere radius without scale applied

Diffraction Patterns

Methods for calculating diffraction patterns of various systems.

class freud.kspace.DeltaSpot

Base class for drawing diffraction spots on a 2D grid.

Based on the dimensions of a grid, determines which grid points need to be modified to represent a diffraction spot and generates the values in that subgrid. Spot is a single pixel at the closest grid point.

get_gridPoints()

Get indices of sub-grid.

Based on the type of spot and its center, return the grid mask of points containing the spot

makeSpot(cval)

Generate intensity value(s) at sub-grid points.

Parameters:cval (numpy.complex64) – complex valued amplitude used to generate spot intensity

set_xy(x, y)

Set \(x\), \(y\) values of spot center.

Parameters:

• x (float) – x value of spot center
• y (float) – y value of spot center

class freud.kspace.GaussianSpot

Draw diffraction spot as a Gaussian blur.

Grid points filled according to Gaussian at spot center.

makeSpot(cval)

Generate intensity value(s) at sub-grid points.

Parameters:cval (numpy.complex64) – complex valued amplitude used to generate spot intensity

set_sigma(sigma)

Define Gaussian.

Parameters:sigma (float) – width of the Gaussian spot

set_xy(x, y)

Set \(x\), \(y\) values of spot center.

Parameters:

• x (float) – x value of spot center
• y (float) – y value of spot center

Utilities

Classes and methods used by other kspace modules.

class freud.kspace.Constraint

Constraint base class.

Base class for constraints on vectors to define the API. All constraints should have a ‘radius’ defining a bounding sphere and a ‘satisfies’ method to determine whether an input vector satisfies the constraint.

satisfies(v)

Constraint test.

Parameters:v (numpy.ndarray, shape=(3), dtype= numpy.float32) – vector to test against constraint

class freud.kspace.AlignedBoxConstraint

Axis-aligned Box constraint.

Tetragonal box aligned with the coordinate system. Consider using a small z dimension to serve as a plane plus or minus some epsilon. Set R < L for a cylinder

satisfies(v)

Constraint test.

Parameters:v (numpy.ndarray, shape=(3), dtype= numpy.float32) – vector to test against constraint

freud.kspace.constrainedLatticePoints()

Generate a list of points satisfying a constraint.

Parameters:

• v1 (numpy.ndarray, shape=(3), dtype= numpy.float32) – lattice vector 1 along which to test points
• v2 (numpy.ndarray, shape=(3), dtype= numpy.float32) – lattice vector 2 along which to test points
• v3 (numpy.ndarray, shape=(3), dtype= numpy.float32) – lattice vector 3 along which to test points
• constraint (Constraint) – constraint object to test lattice points against

freud.kspace.reciprocalLattice3D()

Calculate reciprocal lattice vectors.

3D reciprocal lattice vectors with magnitude equal to angular wave number.

Parameters:

• a1 (numpy.ndarray, shape=(3), dtype= numpy.float32) – real space lattice vector 1
• a2 (numpy.ndarray, shape=(3), dtype= numpy.float32) – real space lattice vector 2
• a3 (numpy.ndarray, shape=(3), dtype= numpy.float32) – real space lattice vector 3

Returns:

list of reciprocal lattice vectors

Return type:

list

Note

For unit test, dot(g[i], a[j]) = 2 * pi * diracDelta(i, j)