# -*- coding: utf-8 -*-
# written by Ralf Biehl at the Forschungszentrum Jülich ,
# Jülich Center for Neutron Science 1 and Institute of Complex Systems 1
# Jscatter is a program to read, analyse and plot data
# Copyright (C) 2015 Ralf Biehl
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program. If not, see <http://www.gnu.org/licenses/>.
#
"""
---
Lattice objects describing a lattice of points.
Included are methods to select sublattices as parallelepiped, sphere or side of planes.
The small angle scattering is calculated by js.ff.cloudScattering.
The same method can be used to calculate the wide angle scattering with bragg peaks
using larger scattering vectors to get crystalline bragg peaks of nanoparticles.
**Examples**
A hollow sphere cut to a wedge.
::
import jscatter as js
import numpy as np
grid= js.lattice.scLattice(1/2.,2*8,b=[0])
grid.inSphere(6,b=1)
grid.inSphere(4,b=0)
grid.planeSide([1,1,1],b=0)
grid.planeSide([1,-1,-1],b=0)
grid.show()
q=js.loglist(0.01,5,600)
ffe=js.ff.cloudScattering(q,grid.points,relError=0.02,rms=0.1)
p=js.grace()
p.plot(ffe)
A cube decorated with spheres.
::
import jscatter as js
import numpy as np
grid= js.lattice.scLattice(0.2,2*15,b=[0])
v1=np.r_[4,0,0]
v2=np.r_[0,4,0]
v3=np.r_[0,0,4]
grid.inParallelepiped(v1,v2,v3,b=1)
grid.inSphere(1,center=[0,0,0],b=2)
grid.inSphere(1,center=v1,b=3)
grid.inSphere(1,center=v2,b=4)
grid.inSphere(1,center=v3,b=5)
grid.inSphere(1,center=v1+v2,b=6)
grid.inSphere(1,center=v2+v3,b=7)
grid.inSphere(1,center=v3+v1,b=8)
grid.inSphere(1,center=v3+v2+v1,b=9)
grid.show()
q=js.loglist(0.01,5,600)
ffe=js.ff.cloudScattering(q,grid.points,relError=0.02,rms=0.)
p=js.grace()
p.plot(ffe)
A comparison of sc, bcc and fcc nanoparticles (takes a while )
::
import jscatter as js
import numpy as np
q=js.loglist(0.01,35,1500)
q=np.r_[js.loglist(0.01,3,200),3:40:800j]
unitcelllength=1.5
N=8
scgrid= js.lattice.scLattice(unitcelllength,N)
sc=js.ff.cloudScattering(q,scgrid.points,relError=50,rms=0.05)
bccgrid= js.lattice.bccLattice(unitcelllength,N)
bcc=js.ff.cloudScattering(q,bccgrid.points,relError=50,rms=0.05)
fccgrid= js.lattice.fccLattice(unitcelllength,N)
fcc=js.ff.cloudScattering(q,fccgrid.points,relError=50,rms=0.05)
p=js.grace(1.5,1)
# smooth with Gaussian to include instrument resolution
p.plot(sc.X,js.formel.smooth(sc,10, window='gaussian'),legend='sc')
p.plot(bcc.X,js.formel.smooth(bcc,10, window='gaussian'),legend='bcc')
p.plot(fcc.X,js.formel.smooth(fcc,10, window='gaussian'),legend='fcc')
q=q=js.loglist(1,35,100)
p.plot(q,(1-np.exp(-q*q*0.05**2))/scgrid.shape[0],li=1,sy=0,le='sc diffusive')
p.plot(q,(1-np.exp(-q*q*0.05**2))/bccgrid.shape[0],li=2,sy=0,le='bcc diffusive')
p.plot(q,(1-np.exp(-q*q*0.05**2))/fccgrid.shape[0],li=3,sy=0,le='fcc diffusive')
p.title('Comparison sc, bcc, fcc lattice for a nano cube')
p.yaxis(scale='l',label='I(Q)')
p.xaxis(scale='l',label='Q / A\S-1')
p.legend(x=0.03,y=0.001,charsize=1.5)
p.text('cube formfactor',x=0.02,y=0.05,charsize=1.4)
p.text('Bragg peaks',x=4,y=0.05,charsize=1.4)
p.text('diffusive scattering',x=4,y=1e-6,charsize=1.4)
END
"""
from __future__ import division
from __future__ import print_function
import numpy as np
from numpy import linalg as la
from . import parallel
try:
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
except ImportError:
pass
try:
from . import fscatter
useFortran = True
except ImportError:
useFortran = False
from . import formel
# tolerance for close to zero
_atol = 1e-12
[docs]class lattice(object):
isLattice = True
def __init__(self):
"""
Create an arbitrary lattice.
Please use one of the subclasses below for creation.
pseudorandom, rhombicLattice, bravaisLattice
scLattice, bccLattice, fccLattice, diamondLattice, hexLattice,
hcpLattice, sqLattice, hexLattice, lamLattice
This base class defines methods valid for all subclasses.
"""
pass
def __getitem__(self, item):
return self.points[item]
def __setitem__(self, item, value):
if self[item].shape == np.shape(value):
self.points[item] = value
else:
raise TypeError('Wrong shape of given value')
@property
def dimension(self):
return self.points.shape[1] - 1
@property
def X(self):
"""X coordinates"""
return self.points[:, 0]
@property
def Y(self):
"""Y coordinates"""
return self.points[:, 1]
@property
def Z(self):
"""Z coordinates"""
return self.points[:, 2]
@property
def XYZ(self):
"""X,Y,Z coordinates array Nx3"""
return self.points[:, :3]
@property
def b(self):
"""Scattering length"""
return self.points[:, 3]
@property
def shape(self):
return self.array.shape
@property
def type(self):
"""Returns type of the lattice"""
return self._type
@property
def array(self):
"""Coordinates and scattering length as array"""
return np.array(self.points)
[docs] def set_b(self, b):
"""
Set all initial points to given scattering length.
"""
self._points[:, 3] = b
@property
def points(self):
"""Points with scattering length >0 """
return self._points[np.abs(self._points[:, 3]) > _atol]
[docs] def filter(self, funktion):
"""
Set lattice points scattering length according to a function.
All points in the lattice are changed for which funktion returns value !=0 (tolerance 1e-12).
Parameters
----------
funktion : function returning float
Function to set lattice points scattering length.
The function is applied with each i point coordinates (array) as input as .points[i,:3].
The return value is the corresponding scattering length.
Examples
--------
::
# To select points inside of a sphere with radius 5 around [1,1,1]:
from numpy import linalg as la
sc=js.sf.scLattice(0.9,10)
sc.set_b(0)
sc.filter(lambda xyz: 1 if la.norm(xyz-np.r_[1,1,1])<5 else 0)
# sphere with increase from center
from numpy import linalg as la
sc=js.sf.scLattice(0.9,10)
sc.set_b(0)
sc.filter(lambda xyz: 2*(la.norm(xyz)) if la.norm(xyz)<5 else 0)
fig=sc.show()
"""
# get float values
v = np.array([funktion(point) for point in self._points[:, :3]])
# set for v !=0, dont change others
choose = np.abs(v) > _atol
self._points[choose, 3] = v[choose]
[docs] def centerOfMass(self):
"""
Center of mass as center of geometry.
"""
return self.points[:, :3].mean(axis=0)
[docs] def numberOfAtoms(self):
"""
Number of Atoms
"""
return self.points.shape[0]
[docs] def move(self, vector):
"""
Move all points by vector.
Parameters
----------
vector : list of 3 float or array
Vector to shift the points.
"""
self._points[:, :3] = self._points[:, :3] + np.array(vector)
[docs] def inParallelepiped(self, v1, v2, v3, corner=None, b=1, invert=False):
"""
Set scattering length for points in parallelepiped.
Parameters
----------
corner : 3x float
Corner of parallelepiped
v1,v2,v3 : each 3x float
Vectors from origin to 3 corners that define the parallelepiped.
b: float
Scattering length for selected points.
invert : bool
Invert selection
Examples
--------
::
import jscatter as js
sc=js.sf.scLattice(0.2,10,b=[0])
sc.inParallelepiped([1,0,0],[0,1,0],[0,0,1],[0,0,0],1)
sc.show()
sc=js.sf.scLattice(0.1,30,b=[0])
sc.inParallelepiped([1,1,0],[0,1,1],[1,0,1],[-1,-1,-1],2)
sc.show()
"""
if corner is None:
corner = [0., 0., 0.]
a1 = np.cross(v2, v3)
b1 = np.cross(v3, v1)
c1 = np.cross(v1, v2)
# vectors perpendicular to planes
a1 = a1 /la.norm(a1)
b1 = b1 /la.norm(b1)
c1 = c1 /la.norm(c1)
da = np.dot(self._points[:, :3] - corner, a1)
da1 = np.dot(np.array(v1), a1)
db = np.dot(self._points[:, :3] - corner, b1)
db1 = np.dot(np.array(v2), b1)
dc = np.dot(self._points[:, :3] - corner, c1)
dc1 = np.dot(np.array(v3), c1)
choose = (0 <= da) & (da <= da1) & (0 <= db) & (db <= db1) & (0 <= dc) & (dc <= dc1)
if invert:
self._points[~choose, 3] = b
else:
self._points[choose, 3] = b
[docs] def planeSide(self, vector, center=None, b=1, invert=False):
"""
Set scattering length for points on one side of a plane.
Parameters
----------
center : 3x float, default [0,0,0]
Point in plane.
vector : list 3x float
Vector perpendicular to plane.
b: float
Scattering length for selected points.
invert : bool
False choose points at origin side. True other side.
Examples
--------
::
sc=js.sf.scLattice(1,10,b=[0])
sc.planeSide([1,1,1],[3,3,3],1)
sc.show()
sc.planeSide([-1,-1,0],3)
sc.show()
"""
if center is None:
center = [0, 0, 0]
v = np.array(vector)
c = np.array(center)
vv = (v ** 2).sum() ** 0.5
v = v / vv
choose = np.dot(self._points[:, :3] - c, v) > 0
if invert:
self._points[~choose, 3] = b
else:
self._points[choose, 3] = b
[docs] def inSphere(self, R, center=None, b=1, invert=False):
"""
Set scattering length for points in sphere.
Parameters
----------
center : 3 x float, default [0,0,0]
Center of the sphere.
R: float
Radius of sphere around origin.
b: float
Scattering length for selected points.
invert : bool
True to invert selection.
Examples
--------
::
import jscatter as js
sc=js.sf.scLattice(1,15,b=[0])
sc.inSphere(6,[2,2,2],b=1)
sc.show()
sc.inSphere(6,[-2,-2,-2],b=2)
sc.show()
sc=js.sf.scLattice(0.8,20,b=[0])
sc.inSphere(3,[2,2,2],b=1)
sc.inSphere(3,[-2,-2,-2],b=1)
sc.show()
sc=js.sf.scLattice(0.8,20,b=[0])
sc.inSphere(3,[2,2,2],b=1)
sc.inSphere(4,[0,0,0],b=2)
sc.show()
"""
if center is None:
center = [0, 0, 0]
choose = la.norm(self._points[:, :3] - np.array(center), axis=1) < abs(R)
if invert:
self._points[~choose, 3] = b
else:
self._points[choose, 3] = b
def _inEllipsoid(self, center, v, Rmajor, Rminor, b=1, invert=False):
# only for prolate ellipsoid
v = np.array(v)
center = np.array(center)
v = v /la.norm(v)
c = abs(Rmajor ** 2 - Rminor ** 2) ** 0.5
# 2 Foci
P1 = center - v * c
P2 = center + v * c
a2 = abs(Rmajor * 2)
print(P1, P2, c, a2)
choose = (la.norm(self._points[:, :3] - P1, axis=1) + la.norm(self._points[:, :3] - P2, axis=1)) < a2
if invert:
self._points[~choose, 3] = b
else:
self._points[choose, 3] = b
[docs] def show(self, R=None, cmap='rainbow', fig=None, ax=None):
"""
Show the lattice in matplotlib with scattering length color coded.
Parameters
----------
R : float,None
Radius around origin to show.
cmap : colormap
Colormap. E.g. 'rainbow', 'winter','autumn','gray'
Use js.mpl.showColors() for all possibilities.
fig : matplotlib Figure
Figure to plot in. If None a new figure is created.
ax : Axes
If given this axes is used for plotting.
Returns
-------
fig handle
"""
if R is None:
points = self.points
else:
choose = la.norm(self.points[:, :3], axis=1) < R
points = self.points[choose]
if len(points) == 0:
raise AttributeError('No points with b>0 to show')
bmax = points[:, 3].max()
bmin = points[:, 3].min()
if fig is None:
fig = plt.figure()
if ax is None:
ax = fig.add_subplot(1, 1, 1, projection='3d')
else:
if isinstance(ax, int):
if len(fig.axes) == 0:
ax = fig.add_subplot(1, ax, ax, projection='3d')
elif len(fig.axes) > ax:
ax = fig.axes[ax - 1]
ax.clear()
else:
r, c, i = fig.axes[-1].get_geometry()
if r * c < ax:
r += 1
c += 1
ax = fig.add_subplot(r, c, ax, projection='3d')
else:
# r,c,i = ax.get_geometry()
ax.clear()
ax.scatter(points[:, 0], points[:, 1], points[:, 2], c=points[:, 3], s=10, cmap=cmap, vmin=bmin, vmax=bmax,
depthshade=False)
try:
for v in self.latticeVectors:
ax.plot([0, v[0]], [0, v[1]], [0, v[2]], color='g')
except (AttributeError, IndexError):
pass
try:
for v in self.unitCellAtomPositions:
if (v ** 2).sum() == 0:
pass
ax.plot([0, v[0]], [0, v[1]], [0, v[2]], color='b')
except (AttributeError, IndexError):
pass
ax.set_xlabel('x axis')
ax.set_ylabel('y axis')
ax.set_zlabel('z axis')
# ax.set_aspect("equal")
xyzmin = self.XYZ.min()
xyzmax = self.XYZ.max()
ax.set_xlim(xyzmin, xyzmax)
ax.set_ylim(xyzmin, xyzmax)
ax.set_zlim(xyzmin, xyzmax)
plt.tight_layout()
plt.show(block=False)
return fig
def getReciprocalLattice(self, size=2):
print('Only for rhombic lattices')
return None
# noinspection PyMethodMayBeStatic,PyUnusedLocal,PyUnusedLocal
def rotateGrid2hkl(self, grid, hkl):
print('Only for rhombic lattices')
return None
[docs] def rotate(self, axis, angle):
"""
Rotate points in lattice around axis by angle.
Parameters
----------
axis : list 3xfloat
Axis of rotation
angle :
Rotation angle in rad
"""
R = formel.rotationMatrix(axis, angle)
self._points[:, 3] = np.einsum('ij,kj->ki', R, self._points[:, 3])
[docs]class pseudoRandomLattice(lattice):
def __init__(self, size, numberOfPoints, b=None, seed=None):
"""
Create a lattice with a pseudo random distribution of points.
Allows to create 1D, 2D or 3D pseudo random latices.
The Halton sequence is used with skipping the first seed elements of the Halton sequence.
Parameters
----------
size :3x float
Size of the lattice for each dimension relative to origin.
numberOfPoints : int
Number of points.
b : float,array
Scattering length of atoms. If array the sequence is repeated to fill N atoms.
seed : None, int
Seed for the Halton sequence by skipping the first seed elements of the sequence.
If None an random integer between 10 and 1e6 is chosen.
Returns
-------
lattice object
.array grid points as numpy array
Examples
--------
::
import jscatter as js
grid=js.sf.pseudoRandomLattice([5,5,5],3000)
fig=grid.show()
"""
super(pseudoRandomLattice, self).__init__()
self._makeLattice(size, numberOfPoints, b, seed)
self._type = 'pseudorandom'
def _makeLattice(self, size, N, pb=None, skip=None):
if pb is None:
pb = 1
dim = np.shape(size)[0]
if skip is None:
skip = np.random.randint(10, 1000000)
seq=formel.randomPointsInCube(N, skip=skip, dim=dim)
seq *= np.array(size)
if dim in [1, 2]:
seq = np.c_[seq, np.zeros((N, 3 - dim))]
self._points = np.c_[seq, np.tile(pb, N)[:N]]
# noinspection PyMissingConstructor
[docs]class rhombicLattice(lattice):
isRhombic = True
def __init__(self, latticeVectors, size, unitCellAtoms=None, b=None):
"""
Create a rhombic lattice with specified unit cell atoms.
Allows to create 1D, 2D or 3D latices by using 1, 2 or 3 latticeVectors.
Parameters
----------
latticeVectors : list of array 3x1
Lattice vectors defining the translation of the unit cell along its principal axes.
size :3x integer or integer
A list of integers describing the size in direction of the respective latticeVectors.
Size is symmetric around zero in interval [-i,..,i] with length 2i+1.
If one integer is given it is used for all 3 dimensions.
unitCellAtoms : list of 3x1 array, None=[0,0,0]
Position vectors vi of atoms in the unit cell in relative units of the lattice vectors [0<x<1].
For 2D and 1D the unit cell atoms vectors are len(vi)=2 and len(vi)=1.
b : list of float
Corresponding scattering length of atoms in the unit cell.
Returns
-------
lattice object
.array : grid points as numpy array
.unitCellVolume : V = a1*a2 x a3 with latticeVectors a1, a2, a3; if existing.
.dim : dimensionality
.unitCellAtoms : Unit cell atoms in relative coordinates
.unitCellAtoms_b : Scattering length of specific unit cell atoms
Examples
--------
::
import jscatter as js
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
# cubic lattice with diatomic base
grid=js.sf.rhombicLattice([[1,0,0],[0,1,0],[0,0,1]],[3,3,3],[[-0.1,-0.1,-0.1],[0.1,0.1,0.1]],[1,2])
grid.show(1.5)
"""
if len(latticeVectors) != len(size):
raise TypeError('size and latticeVectors not compatible. Check dimension!')
if unitCellAtoms is None:
unitCellAtoms = [np.r_[0, 0, 0]]
size = np.trunc(size)
if b is None:
b = [1] * len(unitCellAtoms)
self.unitCellAtoms_b = b
self.unitCellAtoms = unitCellAtoms
self.latticeVectors = latticeVectors
self.size = size
self.dim = len(latticeVectors)
self._makeLattice()
self._makeReciprocalVectors()
self._type = 'rhombic'
def _makeLattice(self):
latticeVectors = self.latticeVectors
size = self.size
pb = self.unitCellAtoms_b
abc = latticeVectors[0] * np.r_[-size[0]:size[0] + 1][:, None]
if self.dim > 1:
abc = abc + (latticeVectors[1] * np.r_[-size[1]:size[1] + 1][:, None])[:, None]
if self.dim > 2:
abc = abc + (latticeVectors[2] * np.r_[-size[2]:size[2] + 1][:, None, None])[:, None, None]
# abc are basis atoms positions of all unit cells
abc = abc.reshape(-1, 3)
abc = np.c_[(abc, np.zeros(abc.shape[0]))] # add b
# unit cell atoms in real coordinates
# uCA=np.einsum('il,ji',latticeVectors,self.unitCellAtoms)
uCA = self.unitCellAtomPositions
# build lattice with all atoms in full grid with b
self._points = np.vstack([abc + np.r_[ev, b] for ev, b in zip(uCA, pb)])
def _makeReciprocalVectors(self):
"""
Creates the reciprocal vectors
"""
latticeVectors = self.latticeVectors
# calc reciprocal vectors
if len(latticeVectors) == 3:
V = np.dot(latticeVectors[0], np.cross(latticeVectors[1], latticeVectors[2]))
self.unitCellVolume = V
self.reciprocalVectors = []
self.reciprocalVectors.append(2 * np.pi / V * np.cross(latticeVectors[1], latticeVectors[2]))
self.reciprocalVectors.append(2 * np.pi / V * np.cross(latticeVectors[2], latticeVectors[0]))
self.reciprocalVectors.append(2 * np.pi / V * np.cross(latticeVectors[0], latticeVectors[1]))
elif len(latticeVectors) == 2:
R = formel.rotationMatrix(np.r_[0, 0, 1], np.pi / 2)
self.reciprocalVectors = []
v0 = np.dot(R, latticeVectors[0])
v1 = np.dot(R, latticeVectors[1])
self.unitCellVolume = np.dot(v0, v1)
self.reciprocalVectors.append(2 * np.pi * v1 / np.dot(latticeVectors[0], v1))
self.reciprocalVectors.append(2 * np.pi * v0 / np.dot(latticeVectors[1], v0))
elif len(latticeVectors) == 1:
R = formel.rotationMatrix(np.r_[0, 0, 1], np.pi / 2)
v0 = np.dot(R, latticeVectors[0])
self.unitCellVolume = la.norm(v0)
self.reciprocalVectors = []
self.reciprocalVectors.append(2 * np.pi * v0 / la.norm(v0) ** 2)
return
@property
def unitCellAtomPositions(self):
"""
Absolute positions of unit cell atoms.
"""
return np.einsum('il,ji', self.latticeVectors, np.array(self.unitCellAtoms)[:, :self.dim])
[docs] def getReciprocalLattice(self, size=2):
"""
Reciprocal lattice of given size with peak scattering intensity.
Parameters
----------
size : 3x int or int, default 2
Number of reciprocal lattice points in each direction (+- direction).
Returns
-------
Array [N x 4] with
reciprocal lattice vectors [:,:3]
corresponding structure factor fhkl**2>0 [:, 3]
"""
if isinstance(size, (int, float)):
size = [size] * 3
size = np.trunc(size)
# create lattice
bbb = self.reciprocalVectors[0] * np.r_[-size[0]:size[0] + 1][:, None]
hkl = np.r_[1, 0, 0] * np.r_[-size[0]:size[0] + 1][:, None]
if len(self.reciprocalVectors) > 1:
bbb = bbb + (self.reciprocalVectors[1] * np.r_[-size[1]:size[1] + 1][:, None])[:, None]
hkl = hkl + (np.r_[0, 1, 0] * np.r_[-size[1]:size[1] + 1][:, None])[:, None]
if len(self.reciprocalVectors) > 2:
bbb = bbb + (self.reciprocalVectors[2] * np.r_[-size[2]:size[2] + 1][:, None, None])[:, None, None]
hkl = hkl + (np.r_[0, 0, 1] * np.r_[-size[2]:size[2] + 1][:, None, None])[:, None, None]
bbb = bbb.reshape(-1, 3)
hkl = hkl.reshape(-1, 3)
# calc structure factor
f2hkl = self._f2hkl(hkl)
# selection rule
choose = (f2hkl > 1e-10 * f2hkl.max())
return np.c_[bbb[choose], f2hkl[choose]]
[docs] def getRadialReciprocalLattice(self, size, includeZero=False):
"""
Get radial distribution of Bragg peaks with structure factor and multiplicity.
Parameters
----------
size : int
Size of the lattice as maximum included Miller indices.
includeZero : bool
Include q=0 peak
Returns
-------
3x list of [unique q values, structure factor fhkl(q)**2, multiplicity mhkl(q)]
"""
qxyzb = self.getReciprocalLattice(size)
qr = la.norm(qxyzb[:, :3], axis=1)
f2hkl = qxyzb[:, 3]
tol = 1e7
qrunique, qrindex, qrcount = np.unique(np.floor(qr * tol) / tol, return_index=True, return_inverse=False,
return_counts=True)
# q values of unique peaks, scattering strength f2hkl, multiplicity as number of unique peaks from 3D count
if includeZero:
return qrunique, f2hkl[qrindex], qrcount
else:
return qrunique[1:], f2hkl[qrindex][1:], qrcount[1:]
def _f2hkl(self, hkl):
"""
Structure factor f**2_hkl which includes the extinction rules.
"""
pb = np.array(self.unitCellAtoms_b)[:, None]
hxkylz = np.einsum('ij,lj', np.array(self.unitCellAtoms)[:, :self.dim], hkl[:, :self.dim])
fhkl = (pb * np.exp(2j * np.pi * hxkylz)).sum(axis=0)
return (fhkl * fhkl.conj()).real
[docs] def vectorhkl(self, hkl):
"""
Get vector corresponding to hkl direction.
Parameters
----------
hkl : 3x float
Miller indices
Returns
-------
array 3x float
"""
h, k, l = hkl
vhkl = h * self.latticeVectors[0] + k * self.latticeVectors[1] + l * self.latticeVectors[2]
return vhkl
[docs] def rotatePlane2hkl(self, plane, hkl):
"""
Rotate plane points that plane is perpendicular to hkl direction.
Parameters
----------
plane : array Nx3
3D points of plane
hkl : list of int, float
Miller indices as [1,1,1] indicating the lattice direction where to rotate to.
Returns
-------
plane points array 3xN
Examples
--------
::
import jscatter as js
import numpy as np
R=8
N=10
qxy=np.mgrid[-R:R:N*1j, -R:R:N*1j].reshape(2,-1).T
qxyz=np.c_[qxy,np.zeros(N**2)]
fccgrid = js.lattice.fccLattice(2.1, 3)
xyz=fccgrid.rotatePlane2hkl(qxyz,[1,1,1])
p=js.mpl.scatter3d(xyz[:,0],xyz[:,1],xyz[:,2])
p.axes[0].scatter(fccgrid.X,fccgrid.Y,fccgrid.Z)
"""
# hkl direction
vhkl = self.vectorhkl(hkl)
vhkl = vhkl /la.norm(vhkl)
# search for vector v3 perpendicular to plane
# first vector in grid close to point, then next with cross >0
v1 = plane[0] - plane[1]
i = 2
while True:
v2 = plane[0] - plane[i]
v3 = np.cross(v1, v2)
# test if >0 then it is perpendicular to plane as v1not parallel to v2
if la.norm(v3) > 0:
break
else:
i += 1
v3 = v3 /la.norm(v3)
rotvector = np.cross(vhkl, v3)
if la.norm(rotvector) < 1e-8:
# is parallel
return plane
angle = np.arccos(np.clip(np.dot(vhkl / la.norm(vhkl), v3 / la.norm(v3)), -1.0, 1.0))
R = formel.rotationMatrix(rotvector, -angle)
Rplane = np.einsum('ij,kj->ki', R, plane)
return Rplane
[docs] def rotatePlaneAroundhkl(self, plane, hkl, angle):
"""
Rotate plane points around hkl direction.
Parameters
----------
plane : array Nx3
3D points of plane
hkl : list of int, float
Miller indices as [1,1,1] indicating the lattice direction to rotate around.
angle : float
Angle in rad
Returns
-------
plane points array 3xN
Examples
--------
::
import jscatter as js
import numpy as np
R=8
N=10
qxy=np.mgrid[-R:R:N*1j, -R:R:N*1j].reshape(2,-1).T
qxyz=np.c_[qxy,np.zeros(N**2)]
fccgrid = js.lattice.fccLattice(2.1, 1)
xyz=fccgrid.rotatePlane2hkl(qxyz,[1,1,1])
xyz2=fccgrid.rotatePlaneAroundhkl(xyz,[1,1,1],np.deg2rad(30))
p=js.mpl.scatter3d(xyz[:,0],xyz[:,1],xyz[:,2])
p.axes[0].scatter(xyz2[:,0],xyz2[:,1],xyz2[:,2])
p.axes[0].scatter(fccgrid.X,fccgrid.Y,fccgrid.Z)
"""
vhkl = self.vectorhkl(hkl)
R = formel.rotationMatrix(vhkl, angle)
Rplane = np.einsum('ij,kj->ki', R, plane)
return Rplane
[docs] def rotatehkl2Vector(self, hkl, vector):
"""
Rotate lattice that hkl direction is parallel to vector.
Includes rotation of latticeVectors.
Parameters
----------
hkl : 3x float
Direction given as Miller indices.
vector : 3x float
Direction to align to.
Examples
--------
::
import jscatter as js
import numpy as np
R=8
N=10
qxy=np.mgrid[-R:R:N*1j, -R:R:N*1j].reshape(2,-1).T
qxyz=np.c_[qxy,np.zeros(N**2)]
fccgrid = js.lattice.fccLattice(2.1, 1)
p=js.mpl.scatter3d(fccgrid.X,fccgrid.Y,fccgrid.Z)
fccgrid.rotatehkl2Vector([1,1,1],[1,0,0])
p.axes[0].scatter(fccgrid.X,fccgrid.Y,fccgrid.Z)
fccgrid.rotateAroundhkl([1,1,1],np.deg2rad(30))
p.axes[0].scatter(fccgrid.X,fccgrid.Y,fccgrid.Z)
"""
vv = np.asarray(vector, dtype=np.float)
vv = vv /la.norm(vv)
vhkl = self.vectorhkl(hkl)
vhkl = vhkl /la.norm(vhkl)
rotvector = np.cross(vhkl, vv)
angle = np.arccos(np.clip(np.dot(vhkl, vv), -1.0, 1.0))
R = formel.rotationMatrix(rotvector, -angle)
# rotate lattice vectors
self.latticeVectors = list(np.einsum('ij,kj->ki', R, self.latticeVectors))
# rotate points
# recreating of points by _makeLattice() is a bit faster for small lattices (N<20)
# for larger it is about the same (N=100)
self._points[:, :3] = np.einsum('ij,kj->ki', R, self._points[:, :3])
# update reciprocal vectors
self._makeReciprocalVectors()
[docs] def rotateAroundhkl(self, hkl, angle=None, vector=None, hkl2=None):
"""
Rotate lattice around hkl direction by angle or to align to vector.
Uses angle or aligns hkl2 to vector.
Includes rotation of latticeVectors.
Parameters
----------
hkl : 3x float
Direction given as Miller indices.
angle : float
Rotation angle in rad.
vector : 3x float
Vector to align hkl2 to. Overrides angle.
Should not be parallel to hkl direction.
hkl2 : 3x float
Direction to align along vector. Overrides angle.
Should not be parallel to hkl direction.
Examples
--------
::
import jscatter as js
import numpy as np
import matplotlib.pyplot as plt
R=8
N=10
qxy=np.mgrid[-R:R:N*1j, -R:R:N*1j].reshape(2,-1).T
qxyz=np.c_[qxy,np.zeros(N**2)]
fccgrid = js.lattice.fccLattice(2.1, 1)
fig=plt.figure( )
# create subplot to define geometry
fig.add_subplot(2,2,1,projection='3d')
fccgrid.show(fig=fig,ax=1)
fccgrid.rotatehkl2Vector([1,1,1],[1,0,0])
fccgrid.show(fig=fig,ax=2)
fccgrid.rotateAroundhkl([1,1,1],np.deg2rad(30))
fccgrid.show(fig=fig,ax=3)
fccgrid.rotateAroundhkl([1,1,1],[1,0,0],[1,0,0])
fccgrid.show(fig=fig,ax=4)
"""
vhkl = self.vectorhkl(hkl)
vhkl = vhkl /la.norm(vhkl)
if vector is not None and hkl2 is not None:
vv = np.asarray(vector)
vv = vv /la.norm(vv)
vhkl2 = self.vectorhkl(hkl2)
vhkl2 = vhkl2 /la.norm(vhkl2)
if np.cross(vhkl, vhkl2) < 1e-8 or np.cross(vhkl, vv) < 1e-8:
# parallel to hkl
raise Exception('vector or hkl2 parallel to hkl')
else:
angle = np.arccos(np.clip(np.dot(vhkl, vv), -1.0, 1.0))
R = formel.rotationMatrix(vhkl, angle)
# rotate points
self._points[:, :3] = np.einsum('ij,kj->ki', R, self._points[:, :3])
# rotate lattice vectors
self.latticeVectors = list(np.einsum('ij,kj->ki', R, self.latticeVectors))
# update reciprocal vectors to reflect rotation
self._makeReciprocalVectors()
[docs]class bravaisLattice(rhombicLattice):
def __init__(self, latticeVectors, size, b=None):
"""
Create a Bravais lattice. Lattice with one atom in the unit cell.
See rhombicLattice for methods and attributes.
Parameters
----------
latticeVectors : list of array 1x3
Lattice vectors defining the translation of the unit cell along its principal axes.
size :3x integer, integer
A list of integers describing the size in direction of the respective latticeVectors.
Size is symmetric around zero in interval [-i,..,i] with length 2i+1.
If one integer is given it is used for all 3 dimensions.
b : list of float
Corresponding scattering length of atoms in the unit cell.
Returns
-------
lattice object
.array grid points as numpy array
"""
rhombicLattice.__init__(self, latticeVectors, size, unitCellAtoms=[np.r_[0, 0, 0]], b=b)
self._type = 'bravais'
[docs]class scLattice(bravaisLattice):
def __init__(self, abc, size, b=None):
"""
Simple Cubic lattice.
See rhombicLattice for methods.
Parameters
----------
abc : float
Point distance.
size : 3x integer, integer
A list of integers describing the size in direction of the respective latticeVectors.
Size is symmetric around zero in interval [-i,..,i] with length 2i+1.
If one integer is given it is used for all 3 dimensions.
b : list of float
Corresponding scattering length of atoms in the unit cell.
Returns
-------
lattice object
.array grid points as numpy array
Examples
--------
::
import jscatter as js
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
grid=js.sf.bccLattice(1.2,1)
grid.show(2)
"""
latticeVectors = [abc * np.r_[1., 0., 0.],
abc * np.r_[0., 1., 0.],
abc * np.r_[0., 0., 1.]]
if isinstance(size, (int, float)):
size = [size] * 3
bravaisLattice.__init__(self, latticeVectors, size, b=b)
self._type = 'sc'
[docs]class bccLattice(rhombicLattice):
def __init__(self, abc, size, b=None):
"""
Body centered cubic lattice.
See rhombicLattice for methods.
Parameters
----------
abc : float
Point distance.
size : 3x integer, integer
A list of integers describing the size in direction of the respective latticeVectors.
Size is symmetric around zero in interval [-i,..,i] with length 2i+1.
If one integer is given it is used for all 3 dimensions.
b : list of float
Corresponding scattering length of atoms in the unit cell.
Returns
-------
lattice object
.array grid points as numpy array
Examples
--------
::
import jscatter as js
grid=js.sf.bccLattice(1.2,1)
grid.show(2)
"""
unitCellAtoms = [np.r_[0, 0, 0], np.r_[0.5, 0.5, 0.5]]
latticeVectors = [abc * np.r_[1., 0., 0.],
abc * np.r_[0., 1., 0.],
abc * np.r_[0., 0., 1.]]
if isinstance(size, (int, float)):
size = [size] * 3
rhombicLattice.__init__(self, latticeVectors, size, unitCellAtoms, b=b)
self._type = 'bcc'
[docs]class fccLattice(rhombicLattice):
def __init__(self, abc, size, b=None):
"""
Face centered cubic lattice.
See rhombicLattice for methods.
Parameters
----------
abc : float
Point distance.
size : 3x integer, integer
A list of integers describing the size in direction of the respective latticeVectors.
Size is symmetric around zero in interval [-i,..,i] with length 2i+1.
If one integer is given it is used for all 3 dimensions.
b : list of float
Corresponding scattering length of atoms in the unit cell.
Returns
-------
lattice object
.array grid points as numpy array
Examples
--------
::
import jscatter as js
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
grid=js.sf.fccLattice(1.2,1)
grid.show(2)
"""
unitCellAtoms = [np.r_[0, 0, 0],
np.r_[0, 0.5, 0.5],
np.r_[0.5, 0, 0.5],
np.r_[0.5, 0.5, 0]]
latticeVectors = [abc * np.r_[1., 0., 0.],
abc * np.r_[0., 1., 0.],
abc * np.r_[0., 0., 1.]]
if isinstance(size, (int, float)):
size = [size] * 3
rhombicLattice.__init__(self, latticeVectors, size, unitCellAtoms, b=b)
self._type = 'fcc'
[docs]class diamondLattice(rhombicLattice):
def __init__(self, abc, size, b=None):
"""
Diamond cubic lattice with 8 atoms in unit cell.
See rhombicLattice for methods.
Parameters
----------
abc : float
Point distance.
size : 3x integer, integer
A list of integers describing the size in direction of the respective latticeVectors.
Size is symmetric around zero in interval [-i,..,i] with length 2i+1.
If one integer is given it is used for all 3 dimensions.
b : list of float
Corresponding scattering length of atoms in the unit cell.
Returns
-------
lattice object
.array grid points as numpy array
Examples
--------
::
import jscatter as js
grid=js.sf.diamondLattice(1.2,1)
grid.show(2)
"""
unitCellAtoms = [np.r_[0, 0, 0],
np.r_[0.5, 0.5, 0],
np.r_[0, 0.5, 0.5],
np.r_[0.5, 0, 0.5],
np.r_[1 / 4., 1 / 4., 1 / 4.],
np.r_[3 / 4., 3 / 4., 1 / 4.],
np.r_[1 / 4., 3 / 4., 3 / 4.],
np.r_[3 / 4., 1 / 4., 3 / 4.]]
latticeVectors = [abc * np.r_[1., 0., 0.],
abc * np.r_[0., 1., 0.],
abc * np.r_[0., 0., 1.]]
if isinstance(size, (int, float)):
size = [size] * 3
rhombicLattice.__init__(self, latticeVectors, size, unitCellAtoms, b=b)
self._type = 'diamond'
[docs]class hexLattice(rhombicLattice):
def __init__(self, ab, c, size, b=None):
"""
Hexagonal lattice.
See rhombicLattice for methods.
Parameters
----------
ab,c : float
Point distance.
ab is distance in hexagonal plane, c perpendicular.
For c/a = (8/3)**0.5 the hcp structure
size : 3x integer, integer
A list of integers describing the size in direction of the respective latticeVectors.
Size is symmetric around zero in interval [-i,..,i] with length 2i+1.
If one integer is given it is used for all 3 dimensions.
b : list of float
Corresponding scattering length of atoms in the unit cell.
Returns
-------
lattice object
.array grid points as numpy array
Examples
--------
::
import jscatter as js
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
grid=js.sf.hexLattice(1.,2,[2,2,2])
grid.show(2)
"""
latticeVectors = [np.r_[ab, 0., 0.],
np.r_[0.5 * ab, 3 ** 0.5 / 2 * ab, 0.],
np.r_[0., 0., c]]
unitCellAtoms = [np.r_[0, 0, 0]]
if isinstance(size, (int, float)):
size = [size] * 3
rhombicLattice.__init__(self, latticeVectors, size, unitCellAtoms, b=b)
self._type = 'hex'
[docs]class hcpLattice(rhombicLattice):
def __init__(self, ab, size, b=None):
"""
Hexagonal closed packed lattice.
See rhombicLattice for methods.
Parameters
----------
ab : float
Point distance.
ab is distance in hexagonal plane, c = ab* (8/3)**0.5
size : 3x integer, integer
A list of integers describing the size in direction of the respective latticeVectors.
Size is symmetric around zero in interval [-i,..,i] with length 2i+1.
If one integer is given it is used for all 3 dimensions.
b : list of float
Corresponding scattering length of atoms in the unit cell.
Returns
-------
lattice object
.array grid points as numpy array
Examples
--------
::
import jscatter as js
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
grid=js.sf.hcpLattice(1.2,[3,3,1])
grid.show(2)
"""
c = ab * (8 / 3.) ** 0.5
latticeVectors = [np.r_[ab, 0., 0.],
np.r_[0.5 * ab, 3 ** 0.5 / 2 * ab, 0.],
np.r_[0., 0., c]]
unitCellAtoms = [np.r_[0, 0, 0],
np.r_[1 / 3., 1 / 3., 0.5]]
if isinstance(size, (int, float)):
size = [size] * 3
rhombicLattice.__init__(self, latticeVectors, size, unitCellAtoms, b=b)
self._type = 'hcp'
[docs]class sqLattice(bravaisLattice):
def __init__(self, ab, size, b=None):
"""
Simple 2D square lattice.
See rhombicLattice for methods.
Parameters
----------
ab : float
Point distance.
size : 2x integer, integer
A list of integers describing the size in direction of the respective latticeVectors.
Size is symmetric around zero in interval [-i,..,i] with length 2i+1.
If one integer is given it is used for all 3 dimensions.
b : list of float
Corresponding scattering length of atoms in the unit cell.
Returns
-------
lattice object
.array grid points as numpy array
Examples
--------
::
import jscatter as js
grid=js.sf.sqLattice(1.2,1)
grid.show(2)
"""
latticeVectors = [ab * np.r_[1., 0., 0.],
ab * np.r_[0., 1., 0.]]
unitCellAtoms = [np.r_[0, 0]] # only 2D
if isinstance(size, (int, float)):
size = [size] * 2
rhombicLattice.__init__(self, latticeVectors, size, unitCellAtoms, b=b)
self._type = 'sq'
[docs]class hex2DLattice(bravaisLattice):
def __init__(self, ab, size, b=None):
"""
Simple 2D hexagonal lattice.
See rhombicLattice for methods.
Parameters
----------
ab : float
Point distance.
size : 2x integer, integer
A list of integers describing the size in direction of the respective latticeVectors.
Size is symmetric around zero in interval [-i,..,i] with length 2i+1.
If one integer is given it is used for all 3 dimensions.
b : list of float
Corresponding scattering length of atoms in the unit cell.
Returns
-------
lattice object
.array grid points as numpy array
Examples
--------
::
import jscatter as js
grid=js.sf.hexLattice(1.2,1)
grid.show(2)
"""
latticeVectors = [np.r_[ab, 0., 0.],
np.r_[0.5 * ab, 3 ** 0.5 / 2 * ab, 0.]]
unitCellAtoms = [np.r_[0, 0]] # only 2D
if isinstance(size, (int, float)):
size = [size] * 2
rhombicLattice.__init__(self, latticeVectors, size[:2], unitCellAtoms, b=b)
self._type = 'hex'
[docs]class lamLattice(bravaisLattice):
def __init__(self, a, size, b=None):
"""
1D lamellar lattice.
See rhombicLattice for methods.
Parameters
----------
a : float
Point distance.
size : 1x integer, integer
A list of integers describing the size in direction of the respective latticeVectors.
Size is symmetric around zero in interval [-i,..,i] with length 2i+1.
If one integer is given it is used for all 3 dimensions.
b : list of float
Corresponding scattering length of atoms in the unit cell.
Returns
-------
lattice object
.array grid points as numpy array
Examples
--------
::
import jscatter as js
grid=js.sf.lamLattice(1.2,1)
grid.show(2)
"""
latticeVectors = [np.r_[a, 0., 0.]]
unitCellAtoms = [np.r_[0]] # only 1D
if isinstance(size, (int, float)):
size = [size] * 1
rhombicLattice.__init__(self, latticeVectors, size[:1], unitCellAtoms, b=b)
self._type = 'lam'