pyPRISM.core.Domain module

class pyPRISM.core.Domain.Domain(length, dr=None, dk=None)

Bases: object

Define domain and transform between Real and Fourier space

Mathematical Definition

The continuous, 1-D, radially symmetric Fourier transform is written as follows:

\[k\ \hat{f}(k) = 4 \pi \int r\ f(r) \sin(k\ r) dr\]

We define the following discretizations

\[ \begin{align}\begin{aligned}r = (i+1)\Delta r\\k = (j+1)\Delta k\\\Delta k = \frac{\pi}{\Delta r (N + 1)}\end{aligned}\end{align} \]

to yield

\[\hat{F}_j = 4 \pi \Delta r \sum_{i=0}^{N-1} F_i \sin\left(\frac{\pi}{N+1} (i+1)(j+1)\right)\]

with the following definitions:

\[\hat{F}_j = (j+1)\ \Delta k\ \hat{f}((j+1)\Delta k) = k \hat{f}(k)\]

\[F_i = (i+1)\Delta r\ f((i+1)\Delta r) = r f(r)\]

The above equations describe a Real to Real, type-I discrete sine transform (DST). To tranform to and from Fourier space we will use the type-II and type-III DST’s respectively. With Scipy’s interface to fftpack, the following functional coeffcients are

\[C^{DSTII} = 2 \pi r \Delta r\]

\[C^{DSTIII} = \frac{k \Delta k}{4 \pi^2}\]

Description

Domain describes the discretization of Real and Fourier space and also sets up the functions and coefficients for transforming data between them.

MatrixArray_to_fourier(marray)

Transform all pair-functions of a MatrixArray to Fourier space in-place

Parameters:marray (pyPRISM.core.MatrixArray.MatrixArray) – MatrixArray to be transformed Raises:*ValueError*: – If the supplied MatrixArray is already in Real-space

MatrixArray_to_real(marray)

Transform all pair-functions of a MatrixArray to Real space in-place

Parameters:marray (pyPRISM.core.MatrixArray.MatrixArray) – MatrixArray to be transformed Raises:ValueError: – If the supplied MatrixArray is already in Real-space

__init__(length, dr=None, dk=None)

Constructor

Parameters:

• length (int) – Number of gridpoints in Real and Fourier space grid
• dr,dk (float) – Grid spacing in Real space or Fourier space. Only one can be specified as it fixes the other.

build_grid()

Construct the Real and Fourier Space grids and transform coefficients

dk

Fourier grid spacing

dr

Real grid spacing

length

Number of points in grid

to_fourier(array)

Discrete Sine Transform of a numpy array

Parameters:array (float ndarray) – Real-space data to be transformed Returns:array – data transformed to fourier space Return type:float ndarray

Peforms a Real-to-Real Discrete Sine Transform of type II on a numpy array of non-complex values. For radial data that is symmetric in \(\phi\) and :math`theta`, this is a correct transform to go from Real-space to Fourier-space.

to_real(array)

Discrete Sine Transform of a numpy array

Parameters:array (float ndarray) – Fourier-space data to be transformed Returns:

• array (float ndarray) – data transformed to Real space
• Peforms a Real-to-Real Discrete Sine Transform of type III
• on a numpy array of non-complex values. For radial data that is
• symmetric in :math:`phi` and (math` heta`, this is a correct transform)
• to go from Real-space to Fourier-space.