Dynamical Magnetic Xray Scattering

This notebook includes examples of the basic usage of dynamical magnetic X-ray scattering.

Setup



In [ ]:

    
import udkm1Dsim as ud
u = ud.u #  import the pint unit registry from udkm1Dsim
import scipy.constants as constants
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
u.setup_matplotlib() #  use matplotlib with pint units

Atom creation

Initialize all required atoms using the Atom class.



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Fe = ud.Atom('Fe')
Cr = ud.Atom('Cr')
Si = ud.Atom('Si')

Unit cell creation

Create all required unit cells using the UnitCell class and add Atom objects to them.



In [ ]:

    
# c-axis
c_Fe = 2.86*u.angstrom
c_Cr = 2.91*u.angstrom
c_Si = 5*u.angstrom

# we only care basic properties for now
propFe = {}
propFe['a_axis'] = c_Fe # aAxis
propFe['b_axis'] = c_Fe # bAxis

Fe_uc = ud.UnitCell('Fe', 'Fe', c_Fe, **propFe)
Fe_uc.add_atom(Fe, 0)
Fe_uc.add_atom(Fe, 0.5)

propCr = {}
propCr['a_axis'] = c_Cr # aAxis
propCr['b_axis'] = c_Cr # bAxis

Cr_uc = ud.UnitCell('Cr', 'Cr', c_Cr, **propCr)
Cr_uc.add_atom(Cr, 0)
Cr_uc.add_atom(Cr, 0.5)

propSi = {}
propSi['a_axis'] = c_Si # aAxis
propSi['b_axis'] = c_Si # bAxis

Si_uc = ud.UnitCell('Si', 'Si', c_Si, **propSi)
Si_uc.add_atom(Si, 0)

Structure creation

Create an actual sample using the Structure class and add UnitCell objects to it.



In [ ]:

    
N_Fe = 5
N_Cr = 4

S = ud.Structure('Super Lattice')

DL = ud.Structure('Double Layer Fe+Cr')
DL.add_sub_structure(Fe_uc, N_Fe)
DL.add_sub_structure(Cr_uc, N_Cr)

S.add_sub_structure(DL, 20)

# add Si substrate
substrate = ud.Structure('Si substrate')
substrate.add_sub_structure(Si_uc, 1000)
S.add_substrate(substrate)

distances, _, _ = S.get_distances_of_layers() # distance vecotor of all layers

Strain map creation

Since the Heat and Phonon simulations are not implemented, yet, the strain_map is created by a simple script, which mimics a strain wave traveling into the sample.



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delays = np.r_[-10:30:1]*u.ps #  define the delays of the simulations
strain_map = np.zeros([len(delays), S.get_number_of_layers()]) #  allocate size of the strain_map

for i, delay in enumerate(delays):
    factor = 10
    width = 100
    if delay > 0:
        end = int(delay.magnitude * factor)
        start = end - width
        if start < 0: start = 0
        if end < 0: end = 0
        strain_map[i, start:end] = 0.005

# strain_vectors are a subset of the strain_map and are required to speed up the xray simulations
strain_vectors =  [np.array(np.linspace(np.min(strain_map), np.max(strain_map), 100))]*S.get_number_of_unique_layers()

# plot the artifical strain_map
plt.figure()
plt.contourf(distances, delays, strain_map)
plt.title('Strain Map')
plt.colorbar()
plt.show()

Initialize dynamical magnetic Xray simulation

Create a dynamical Xray simulation using the XrayDynMag class and add a Structure object as sample. Also set the photon energy and $q_z$ range for the actual simulations.



In [ ]:

    
force_recalc = True #  always recalculate results and do not consider cached results
mag = ud.XrayDynMag(S, force_recalc)
mag.disp_messages = True #  enable displaying messages from within the simulations
mag.save_data = False #  do not save results for caching

Homogeneous Xray simulation

For the case of homogeneously strained samples, the dynamical Xray scattering simulations can be greatly simplyfied, which saves a lot of computational time.

$q_z$-scan



In [ ]:

    
mag.energy = np.r_[600, 710]*u.eV #  set two photon energies
mag.qz = np.r_[0.01:5:0.01]/u.nm #  qz range

R_hom = mag.homogeneous_reflectivity() #  this is the actual calculation

plt.figure()
plt.semilogy(mag.qz[0, :], R_hom[0, :], label='{}'.format(mag.energy[0]), alpha=0.5)
plt.semilogy(mag.qz[1, :], R_hom[1, :], label='{}'.format(mag.energy[1]), alpha=0.5)
plt.ylabel('Reflectivity')
plt.legend()
plt.show()

Post-Processing

Simple convolution of the results with an arbitrary function handle.



In [ ]:

    
FWHM = 0.01/1e-10 # Angstrom
sigma = FWHM /2.3548
        
handle = lambda x: np.exp(-((x)/sigma)**2/2)
y_conv = mag.conv_with_function(R_hom[0,:], mag._qz[0,:], handle)

plt.figure()
plt.semilogy(mag.qz[0,:], R_hom[0,:], label='{}'.format(mag.energy[0]))
plt.semilogy(mag.qz[0,:], y_conv, label='{} convoluted'.format(mag.energy[0]))
plt.ylabel('Reflectivity')
plt.legend()
plt.show()

Energy-scan



In [ ]:

    
mag.energy = np.r_[680:750:0.01]*u.eV #  set the energy range
mag.qz = np.r_[2.5]/u.nm # qz range

R_hom= mag.homogeneous_reflectivity() #  this is the actual calculation

plt.figure()
plt.plot(mag.energy, R_hom[:, 0])
plt.ylabel('Reflectivity')
plt.show()

Sequential dynamical Xray simulation

Do a time-resolved magnetic xray scatting simulation for the above defined strain_map without parallelization.



In [ ]:

    
%%time
mag.energy = np.r_[600, 710]*u.eV #  set two photon energies
mag.qz = np.r_[0.1:5:0.1]/u.nm #  qz range

R_seq = mag.inhomogeneous_reflectivity(strain_map, strain_vectors)

Sequential Results



In [ ]:

    
for i, energy in enumerate(mag.energy):
    plt.figure()
    plt.contourf(mag.qz[i,:], delays, np.log10(R_seq[:, i, :]), levels=100)
    plt.title('{:0.1f} eV'.format(energy.magnitude))
    plt.show()

Parallel dynamical Xray scattering

Parallelization needs still to be implemented, but will work similarly as with XrayDyn.

You need to install the udkm1Dsim with the parallel option which essentially add the Dask package to the requirements:

> pip install udkm1Dsim[parallel]

You can also install/add Dask manually, e.g. via pip:

> pip install dask

Please refer to the Dask documentation for more details on parallel computing in Python.

Polarization Dependence

Non-Magnetic sample

The current sample is currently not magnetic, therefore, there should no scattering from sigma to pi and vice versa.



In [ ]:

    
mag.energy = np.r_[710]*u.eV #  set two photon energies
mag.qz = np.r_[0.01:5:0.01]/u.nm #  qz range


plt.figure()
mag.set_polarization(3, 3)
R_hom = mag.homogeneous_reflectivity() #  this is the actual calculation
plt.semilogy(mag.qz[0, :], R_hom[0, :], label='{}: sigma -> sigma'.format(mag.energy[0]), alpha=0.5)

mag.set_polarization(3, 4)
R_hom = mag.homogeneous_reflectivity() #  this is the actual calculation
plt.semilogy(mag.qz[0, :], R_hom[0, :], label='{}: sigma -> pi'.format(mag.energy[0]), alpha=0.5)

plt.ylabel('Reflectivity')
plt.title('non-magnetic sample: sigma')
plt.legend()
plt.show()



In [ ]:

    
mag.energy = np.r_[710]*u.eV #  set two photon energies
mag.qz = np.r_[0.01:5:0.01]/u.nm #  qz range


plt.figure()
mag.set_polarization(4, 4)
R_hom = mag.homogeneous_reflectivity() #  this is the actual calculation
plt.semilogy(mag.qz[0, :], R_hom[0, :], label='{}: pi -> pi'.format(mag.energy[0]), alpha=0.5)

mag.set_polarization(4, 3)
R_hom = mag.homogeneous_reflectivity() #  this is the actual calculation
plt.semilogy(mag.qz[0, :], R_hom[0, :], label='{}: pi -> sigma'.format(mag.energy[0]), alpha=0.5)

plt.ylabel('Reflectivity')
plt.title('non-magnetic sample: pi')
plt.legend()
plt.show()

Magnetic Sample

Now compare the scattering with a ferromagnetic and antiferromagnetic sample structure.



In [ ]:

    
# Fe atom with magnetization in-plane to the left
Fe_left = ud.Atom('Fe', mag_amplitude=1, mag_phi=90*u.deg, mag_gamma=90*u.deg, id='Fe left')
# Fe atom with magnetization in-plane to the right
Fe_right = ud.Atom('Fe', mag_amplitude=1, mag_phi=90*u.deg, mag_gamma=270*u.deg, id='Fe right')

# Fe unit cell with left in-plane magnetization
Fe_uc_left = ud.UnitCell('Fe_left', 'Fe', c_Fe, **propFe)
Fe_uc_left.add_atom(Fe_left, 0)
Fe_uc_left.add_atom(Fe_left, 0.5)

# Fe unit cell with right in-plane magnetization
Fe_uc_right = ud.UnitCell('Fe_right', 'Fe', c_Fe, **propFe)
Fe_uc_right.add_atom(Fe_right, 0)
Fe_uc_right.add_atom(Fe_right, 0.5)

FM Sample



In [ ]:

    
N_Fe = 5
N_Cr = 4

SFM = ud.Structure('FM Super Lattice')

DLFM = ud.Structure('FM Double Layer Fe+Cr')
DLFM.add_sub_structure(Fe_uc_left, N_Fe)
DLFM.add_sub_structure(Cr_uc, N_Cr)

SFM.add_sub_structure(DLFM, 20)

# add Si substrate
SFM.add_substrate(substrate)

distances, _, _ = SFM.get_distances_of_layers() # distance vecotor of all unit cells

mag.S = SFM # replace the sample in the current simulation



In [ ]:

    
mag.energy = np.r_[710]*u.eV #  set two photon energies
mag.qz = np.r_[0.01:5:0.01]/u.nm #  qz range


plt.figure()
mag.set_polarization(3, 3)
R_hom = mag.homogeneous_reflectivity() #  this is the actual calculation
plt.semilogy(mag.qz[0, :], R_hom[0, :], label='{}: sigma -> sigma'.format(mag.energy[0]), alpha=0.5)

mag.set_polarization(3, 4)
R_hom = mag.homogeneous_reflectivity() #  this is the actual calculation
plt.semilogy(mag.qz[0, :], R_hom[0, :], label='{}: sigma -> pi'.format(mag.energy[0]), alpha=0.5)

plt.ylabel('Reflectivity')
plt.title('FM sample: sigma')
plt.legend()
plt.show()



In [ ]:

    
mag.energy = np.r_[710]*u.eV #  set two photon energies
mag.qz = np.r_[0.01:5:0.01]/u.nm #  qz range


plt.figure()
mag.set_polarization(4, 4)
R_hom = mag.homogeneous_reflectivity() #  this is the actual calculation
plt.semilogy(mag.qz[0, :], R_hom[0, :], label='{}: pi -> pi'.format(mag.energy[0]), alpha=0.5)

mag.set_polarization(4, 3)
R_hom = mag.homogeneous_reflectivity() #  this is the actual calculation
plt.semilogy(mag.qz[0, :], R_hom[0, :], label='{}: pi -> sigma'.format(mag.energy[0]), alpha=0.5)

plt.ylabel('Reflectivity')
plt.title('FM sample: pi')
plt.legend()
plt.show()



In [ ]:

    
mag.energy = np.r_[710]*u.eV #  set two photon energies
mag.qz = np.r_[0.01:5:0.01]/u.nm #  qz range


plt.figure()
mag.set_polarization(1, 0)
R_hom = mag.homogeneous_reflectivity() #  this is the actual calculation
plt.semilogy(mag.qz[0, :], R_hom[0, :], label='{}: circ +'.format(mag.energy[0]), alpha=0.5)

mag.set_polarization(2, 0)
R_hom = mag.homogeneous_reflectivity() #  this is the actual calculation
plt.semilogy(mag.qz[0, :], R_hom[0, :], label='{}: circ -'.format(mag.energy[0]), alpha=0.5)

plt.ylabel('Reflectivity')
plt.title('FM sample: circular')
plt.legend()
plt.show()

Same again with the non-magnetic sample



In [ ]:

    
mag.S = S # change again to the inital sample

mag.energy = np.r_[710]*u.eV #  set two photon energies
mag.qz = np.r_[0.01:5:0.01]/u.nm #  qz range


plt.figure()
mag.set_polarization(1, 0)
R_hom = mag.homogeneous_reflectivity() #  this is the actual calculation
plt.semilogy(mag.qz[0, :], R_hom[0, :], label='{}: circ +'.format(mag.energy[0]), alpha=0.5)

mag.set_polarization(2, 0)
R_hom = mag.homogeneous_reflectivity() #  this is the actual calculation
plt.semilogy(mag.qz[0, :], R_hom[0, :], label='{}: circ -'.format(mag.energy[0]), alpha=0.5)

plt.ylabel('Reflectivity')
plt.title('FM sample: circular')
plt.legend()
plt.show()

AFM Sample



In [ ]:

    
N_Fe = 5
N_Cr = 4

SAFM = ud.Structure('AFM Super Lattice')

DLAFM = ud.Structure('AFM Double Layer Fe+Cr')
DLAFM.add_sub_structure(Fe_uc_left, N_Fe)
DLAFM.add_sub_structure(Cr_uc, N_Cr)
DLAFM.add_sub_structure(Fe_uc_right, N_Fe)
DLAFM.add_sub_structure(Cr_uc, N_Cr)

SAFM.add_sub_structure(DLAFM, 10)

# add Si substrate
SAFM.add_substrate(substrate)

distances, _, _ = SAFM.get_distances_of_layers() # distance vecotor of all unit cells

mag.S = SAFM # replace the sample in the current simulation



In [ ]:

    
mag.energy = np.r_[710]*u.eV #  set two photon energies
mag.qz = np.r_[0.01:5:0.01]/u.nm #  qz range


plt.figure()

mag.set_polarization(3, 0)
R_hom = mag.homogeneous_reflectivity() #  this is the actual calculation
plt.semilogy(mag.qz[0, :], R_hom[0, :], label='{}: sigma -> unpolarized'.format(mag.energy[0]), alpha=0.5)

mag.set_polarization(3, 3)
R_hom = mag.homogeneous_reflectivity() #  this is the actual calculation
plt.semilogy(mag.qz[0, :], R_hom[0, :], label='{}: sigma -> sigma'.format(mag.energy[0]), alpha=0.5)

mag.set_polarization(3, 4)
R_hom = mag.homogeneous_reflectivity() #  this is the actual calculation
plt.semilogy(mag.qz[0, :], R_hom[0, :], label='{}: sigma -> pi'.format(mag.energy[0]), alpha=0.5)

plt.ylabel('Reflectivity')
plt.title('AFM sample: sigma')
plt.legend()
plt.show()



In [ ]:

    
mag.energy = np.r_[710]*u.eV #  set two photon energies
mag.qz = np.r_[0.01:5:0.01]/u.nm #  qz range


plt.figure()

mag.set_polarization(4, 0)
R_hom = mag.homogeneous_reflectivity() #  this is the actual calculation
plt.semilogy(mag.qz[0, :], R_hom[0, :], label='{}: pi -> unpolarized'.format(mag.energy[0]), alpha=0.5)

mag.set_polarization(4, 4)
R_hom = mag.homogeneous_reflectivity() #  this is the actual calculation
plt.semilogy(mag.qz[0, :], R_hom[0, :], label='{}: pi -> pi'.format(mag.energy[0]), alpha=0.5)

mag.set_polarization(4, 3)
R_hom = mag.homogeneous_reflectivity() #  this is the actual calculation
plt.semilogy(mag.qz[0, :], R_hom[0, :], label='{}: pi -> sigma'.format(mag.energy[0]), alpha=0.5)

plt.ylabel('Reflectivity')
plt.title('AFM sample: pi')
plt.legend()
plt.show()



In [ ]:

    
mag.energy = np.r_[710]*u.eV #  set two photon energies
mag.qz = np.r_[0.01:5:0.01]/u.nm #  qz range


plt.figure()
mag.set_polarization(1, 0)
R_hom = mag.homogeneous_reflectivity() #  this is the actual calculation
plt.semilogy(mag.qz[0, :], R_hom[0, :], label='{}: circ +'.format(mag.energy[0]), alpha=0.5)

mag.set_polarization(2, 0)
R_hom = mag.homogeneous_reflectivity() #  this is the actual calculation
plt.semilogy(mag.qz[0, :], R_hom[0, :], label='{}: circ -'.format(mag.energy[0]), alpha=0.5)

plt.ylabel('Reflectivity')
plt.title('AFM sample: circular')
plt.legend()
plt.show()

Engery vs qz



In [ ]:

    
mag.energy = np.r_[680:750]*u.eV #  set two photon energies
mag.qz = np.r_[0.01:5:0.01]/u.nm #  qz range
mag.set_polarization(3, 0)

R_hom = mag.homogeneous_reflectivity() #  this is the actual calculation

plt.figure()

plt.figure(figsize=[12,12])
plt.contourf(mag.qz[0,:], mag.energy, np.log10(R_hom[:, :]), levels=100)
plt.colorbar()

plt.ylabel('Reflectivity')
plt.title('AFM sample: circular')
plt.show()

Amorphous Magnetic Sample

Check that we get the same results in magnetic scattering as from a crystalline sample.



In [ ]:

    
N_Fe = 5
N_Cr = 4

amAFM = ud.Structure('amorphous AFM Super Lattice')
amDLAFM = ud.Structure('amourphous AFM Double Layer Fe+Cr')

amFeLeftLay = ud.AmorphousLayer('Fe_left_amorphous', "Fe left amorphous", Fe_uc.c_axis*N_Fe, Fe_uc.density, atom=Fe_left)
amFeRightLay = ud.AmorphousLayer('Fe_right_amorphous', "Fe right amorphous", Fe_uc.c_axis*N_Fe, Fe_uc.density, atom=Fe_right)

amCrLay = ud.AmorphousLayer('Cr_amorphous', "Cr amorphous", Cr_uc.c_axis * N_Cr, Cr_uc.density, atom=Cr)


amDLAFM.add_sub_structure(amFeLeftLay)
amDLAFM.add_sub_structure(amCrLay)
amDLAFM.add_sub_structure(amFeRightLay)
amDLAFM.add_sub_structure(amCrLay)

amAFM.add_sub_structure(amDLAFM, 10)



In [ ]:

    
# add Si substrate
amAFM.add_substrate(substrate)

distances, _, _ = amAFM.get_distances_of_layers() # distance vecotor of all unit cells

mag.S = amAFM # replace the sample in the current simulation



In [ ]:

    
mag.energy = np.r_[710]*u.eV #  set two photon energies
mag.qz = np.r_[0.01:5:0.01]/u.nm #  qz range


plt.figure()

mag.set_polarization(3, 0)
R_hom = mag.homogeneous_reflectivity() #  this is the actual calculation
plt.semilogy(mag.qz[0, :], R_hom[0, :], label='{}: sigma -> unpolarized'.format(mag.energy[0]), alpha=0.5)

mag.set_polarization(3, 3)
R_hom = mag.homogeneous_reflectivity() #  this is the actual calculation
plt.semilogy(mag.qz[0, :], R_hom[0, :], label='{}: sigma -> sigma'.format(mag.energy[0]), alpha=0.5)

mag.set_polarization(3, 4)
R_hom = mag.homogeneous_reflectivity() #  this is the actual calculation
plt.semilogy(mag.qz[0, :], R_hom[0, :], label='{}: sigma -> pi'.format(mag.energy[0]), alpha=0.5)

plt.ylabel('Reflectivity')
plt.title('AFM sample: sigma')
plt.legend()
plt.show()