Dynamical Magnetic X-ray Scattering¶

In this example static and transient magnetic X-ray simulations are carried out employing a dynamical magnetic X-ray scattering formalism which was adapted from Project Dyna.

Setup¶

Do all necessary imports and settings.

[1]:

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

Structure¶

Refer to the structure-example for more details.

In this example the sample Structure consists of AmorphousLayers.

In order to build an antiferromagnetic sample two different types of Fe Atoms and AmorphousLayers are created with opposite in-plane magnetization.

Here a local file for the atomic scattering factor is read only for Fe atoms.

[2]:

Fe_right = ud.Atom('Fe', mag_amplitude=1, mag_phi=90*u.deg, mag_gamma=90*u.deg,
                  atomic_form_factor_path='./Fe.cf')
Fe_left = ud.Atom('Fe', mag_amplitude=1, mag_phi=90*u.deg, mag_gamma=270*u.deg,
                  atomic_form_factor_path='./Fe.cf')
Cr = ud.Atom('Cr')
Si = ud.Atom('Si')

[3]:

density_Fe = 7874*u.kg/u.m**3

prop_Fe = {}
prop_Fe['heat_capacity'] = 449*u.J/u.kg/u.K
prop_Fe['therm_cond'] = 80*u.W/(u.m *u.K)
prop_Fe['lin_therm_exp'] = 11.8e-6
prop_Fe['sound_vel'] = 4.910*u.nm/u.ps
prop_Fe['opt_ref_index'] = 2.9174+3.3545j

layer_Fe_left = ud.AmorphousLayer('Fe_left', 'Fe left amorphous', 1*u.nm, density_Fe, atom=Fe_left, **prop_Fe)
layer_Fe_right = ud.AmorphousLayer('Fe_right', 'Fe right amorphous', 1*u.nm, density_Fe, atom=Fe_right, **prop_Fe)

[4]:

density_Cr = 7140*u.kg/u.m**3

prop_Cr = {}
prop_Cr['heat_capacity'] = 449*u.J/u.kg/u.K
prop_Cr['therm_cond'] = 94*u.W/(u.m *u.K)
prop_Cr['lin_therm_exp'] = 6.2e-6
prop_Cr['sound_vel'] = 5.940*u.nm/u.ps
prop_Cr['opt_ref_index'] = 3.1612+3.4606j

layer_Cr = ud.AmorphousLayer('Cr', "Cr amorphous", 1*u.nm, density_Cr, atom=Cr, **prop_Cr)

[5]:

density_Si = 2336*u.kg/u.m**3

prop_Si = {}
prop_Si['heat_capacity'] = 703*u.J/u.kg/u.K
prop_Si['therm_cond'] = 150*u.W/(u.m *u.K)
prop_Si['lin_therm_exp'] = 2.6e-6
prop_Si['sound_vel'] = 8.433*u.nm/u.ps
prop_Si['opt_ref_index'] = 3.6941+0.0065435j

layer_Si = ud.AmorphousLayer('Si', "Si amorphous", 1*u.nm, density_Si, atom=Si, **prop_Si)

[6]:

S = ud.Structure('Fe/Cr AFM Super Lattice')

# create a sub-structure
DL = ud.Structure('Two Fe/Cr Double Layers')
DL.add_sub_structure(layer_Fe_left, 1)
DL.add_sub_structure(layer_Cr, 1)
DL.add_sub_structure(layer_Fe_right, 1)
DL.add_sub_structure(layer_Cr, 1)

S.add_sub_structure(DL, 10)
S.add_sub_structure(layer_Fe_left, 10)
S.add_sub_structure(layer_Si, 200)

[7]:

S.visualize()

../_images/examples_dynamical_magnetic_xray_9_0.png

Heat¶

Refer to the heat-example for more details.

[8]:

h = ud.Heat(S, True)

h.save_data = False
h.disp_messages = True

h.excitation = {'fluence': [40]*u.mJ/u.cm**2,
                'delay_pump':  [0]*u.ps,
                'pulse_width':  [1]*u.ps,
                'multilayer_absorption': True,
                'wavelength': 800*u.nm,
                'theta': 45*u.deg}

# enable heat diffusion
h.heat_diffusion = True

# temporal and spatial grid
delays = np.r_[-2:40:0.1]*u.ps
_, _, distances = S.get_distances_of_layers()

[9]:

temp_map, delta_temp_map = h.get_temp_map(delays, 0*u.K)

Surface incidence fluence scaled by factor 0.7071 due to incidence angle theta=45.00 deg
Calculating _heat_diffusion_ for excitation 1:1 ...
Absorption profile is calculated by multilayer formalism.
Total reflectivity of 47.7 % and transmission of 6.4 %.


Elapsed time for _heat_diffusion_ with 1 excitation(s): 71.088708 s
Calculating _heat_diffusion_ ...
Absorption profile is calculated by multilayer formalism.
Total reflectivity of 47.7 % and transmission of 6.4 %.


Elapsed time for _heat_diffusion_: 29.224858 s
Elapsed time for _temp_map_: 101.010451 s

[10]:

plt.figure(figsize=[6, 8])
plt.subplot(2, 1, 1)
plt.plot(distances.to('nm').magnitude, temp_map[20, :], label='{:0.2f} ps'.format(delays[20].to('ps').magnitude))
plt.plot(distances.to('nm').magnitude, temp_map[-1, :], label='{:0.2f} ps'.format(delays[-1].to('ps').magnitude))
plt.xlim([0, distances.to('nm').magnitude[-1]])
plt.xlabel('Distance [nm]')
plt.ylabel('Temperature [K]')
plt.legend()
plt.title('Temperature Profile')

plt.subplot(2, 1, 2)
plt.pcolormesh(distances.to('nm').magnitude, delays.to('ps').magnitude, temp_map)
plt.colorbar()
plt.xlabel('Distance [nm]')
plt.ylabel('Delay [ps]')
plt.title('Temperature Map')

plt.tight_layout()
plt.show()

../_images/examples_dynamical_magnetic_xray_13_0.png

Numerical Phonons¶

Refer to the phonons-example for more details.

[11]:

p = ud.PhononNum(S, True)
p.save_data = False
p.disp_messages = True

[12]:

strain_map = p.get_strain_map(delays, temp_map, delta_temp_map)

Calculating linear thermal expansion ...
Calculating coherent dynamics with ODE solver ...


Elapsed time for _strain_map_: 3.246968 s

[13]:

plt.figure(figsize=[6, 8])
plt.subplot(2, 1, 1)
plt.plot(distances.to('nm').magnitude, strain_map[130, :], label=np.round(delays[130]))
plt.plot(distances.to('nm').magnitude, strain_map[-1, :], label=np.round(delays[-1]))
plt.xlim([0, distances.to('nm').magnitude[-1]])
plt.xlabel('Distance [nm]')
plt.ylabel('Strain')
plt.legend()
plt.title('Strain Profile')

plt.subplot(2, 1, 2)
plt.pcolormesh(distances.to('nm').magnitude, delays.to('ps').magnitude,
               strain_map, cmap='RdBu',
               vmin=-np.max(strain_map), vmax=np.max(strain_map))
plt.colorbar()
plt.xlabel('Distance [nm]')
plt.ylabel('Delay [ps]')
plt.title('Strain Map')

plt.tight_layout()
plt.show()

../_images/examples_dynamical_magnetic_xray_17_0.png

Magnetization¶

The Magnetization class is currently only an interface to allow the user for defining specific magnetization dynamics depending on the strain_map and temp_map.

Here the magnetization as function of temperature is used as a simplified model to alter the transient magnetization amplitude.

[14]:

def magnetization_bloch(x,Tc, M0, beta):
    m = M0*((1-(x/Tc)**(3/2))**beta)*np.heaviside(Tc-x, 0.5)
    m[np.isnan(m)] = 0
    return m

plt.figure()
temperatures = np.r_[10:1000:0.1]
plt.plot(temperatures, magnetization_bloch(temperatures, 770, 1, 0.75))
plt.xlabel('Temperature [K]')
plt.ylabel('Magneitzation')
plt.show()

c:\users\schick\general\python\wpy64-3770\python-3.7.7.amd64\lib\site-packages\ipykernel_launcher.py:2: RuntimeWarning: invalid value encountered in power

../_images/examples_dynamical_magnetic_xray_19_1.png

The calc_magnetization_map method must be overwritten to allow for calculating magnetization dynamics.

[15]:

class SimpleMagnetization(ud.Magnetization):

    def calc_magnetization_map(self, delays, **kwargs):
        strain_map = kwargs['strain_map']
        temp_map = kwargs['temp_map']
        T_c = kwargs['T_c']
        magnetization_map = np.zeros([len(delays), self.S.get_number_of_layers(), 3])

        handles = self.S.get_layer_vectors()[2]

        for i, handle in enumerate(handles):
            if handle.id in ['Fe_left', 'Fe_right']:
                magnetization_map[:, i, 0] = magnetization_bloch(temp_map[:, i], T_c, 1, .75)
            handle.magnetization['phi']
            magnetization_map[:, i, 1] = handle.magnetization['phi'].to_base_units().magnitude
            magnetization_map[:, i, 2] = handle.magnetization['gamma'].to_base_units().magnitude

        return magnetization_map

[16]:

mag = SimpleMagnetization(S, True)
mag.save_data = False

magnetization_map = mag.get_magnetization_map(delays,
                                              strain_map=strain_map,
                                              temp_map=temp_map,
                                              T_c = 770)

Calculating _magnetization_map_ ...
Elapsed time for _magnetization_map_: 0.097714 s

c:\users\schick\general\python\wpy64-3770\python-3.7.7.amd64\lib\site-packages\ipykernel_launcher.py:2: RuntimeWarning: invalid value encountered in power

[17]:

plt.figure()
plt.pcolormesh(distances.to('nm').magnitude, delays.to('ps').magnitude, magnetization_map[:, :, 0])
plt.title('Magnetization Amplitude Map')
plt.ylabel('Delay [ps]')
plt.xlabel('Distance [nm]')
plt.colorbar()
plt.show()

../_images/examples_dynamical_magnetic_xray_23_0.png

Initialize dynamical magnetic X-ray simulations¶

The XrayDynMag class requires a Structure object and a boolean force_recalc in order overwrite previous simulation results.

These results are saved in the cache_dir when save_data is enabled. Printing simulation messages can be en-/disabled using disp_messages and progress bars can using the boolean switch progress_bar.

[18]:

dyn_mag = ud.XrayDynMag(S, True)
dyn_mag.disp_messages = True
dyn_mag.save_data = False

incoming polarizations set to: sigma
analyzer polarizations set to: unpolarized

Homogeneous magnetic X-ray scattering¶

For the case of homogeneously strained/magnetized samples, the dynamical magnetic X-ray scattering simulations can be greatly simplified, which saves a lot of computational time.

\(q_z\)-scan¶

The XrayDynMag object requires an energy and scattering vector qz to run the simulations.

Both parameters can be arrays and the resulting reflectivity has a first dimension for the photon energy and the a second for the scattering vector.

The resulting reflectivity is always calculated for the actual magnetization (R_hom) of the sample, as well as for opposite magnetization (R_hom_phi).

[19]:

dyn_mag.energy = np.r_[600, 708]*u.eV #  set two photon energies
dyn_mag.qz = np.r_[0.01:5:0.01]/u.nm #  qz range

R_hom, R_hom_phi, _, _ = dyn_mag.homogeneous_reflectivity() #  this is the actual calculation

Calculating _homogeneous_reflectivity_ ...
Elapsed time for _homogeneous_reflectivity_: 0.953312 s

In this example half-order antiferromagnetic Bragg peaks appear only at the resonance.

[20]:

plt.figure()
plt.semilogy(dyn_mag.qz[0, :], R_hom[0, :], label='{}'.format(dyn_mag.energy[0]))
plt.semilogy(dyn_mag.qz[1, :], R_hom[1, :], label='{}'.format(dyn_mag.energy[1]))
plt.ylabel('Reflectivity')
plt.xlabel(r'$q_z$ [1/nm]')
plt.legend()
plt.show()

../_images/examples_dynamical_magnetic_xray_30_0.png

Post-Processing¶

Simple convolution of the results with an arbitrary function handle.

[21]:

FWHM = 0.01/1e-10 # Angstrom
sigma = FWHM /2.3548

handle = lambda x: np.exp(-((x)/sigma)**2/2)
y_conv = dyn_mag.conv_with_function(R_hom[1,:], dyn_mag._qz[1,:], handle)

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

../_images/examples_dynamical_magnetic_xray_32_0.png

Energy- and \(q_z\)-scan¶

[22]:

dyn_mag.energy = np.r_[690:730:0.1]*u.eV #  set the energy range
dyn_mag.qz = np.r_[0.01:5:0.01]/u.nm # qz range

R_hom, R_hom_phi, _, _ = dyn_mag.homogeneous_reflectivity() #  this is the actual calculation

Calculating _homogeneous_reflectivity_ ...
Elapsed time for _homogeneous_reflectivity_: 193.556878 s

[23]:

plt.figure(figsize=[6, 8])
plt.subplot(2, 1, 1)
plt.semilogy(dyn_mag.qz[0, :].to('1/nm'), R_hom[0, :], label=np.round(dyn_mag.energy[0]))
plt.semilogy(dyn_mag.qz[0, :].to('1/nm'), R_hom[180, :], label=np.round(dyn_mag.energy[180]))
plt.semilogy(dyn_mag.qz[0, :].to('1/nm'), R_hom[-1, :], label=np.round(dyn_mag.energy[-1]))

plt.xlabel('$q_z$ [nm$^{-1}$]')
plt.ylabel('Reflectivity')
plt.legend()
plt.title('Dynamical Magnetic X-ray')

plt.subplot(2, 1, 2)
plt.pcolormesh(dyn_mag.qz[0, :].to('1/nm').magnitude, dyn_mag.energy.magnitude, np.log10(R_hom[:, :]))
plt.title('Dynamical Magnetic X-ray')
plt.ylabel('Energy [eV]')
plt.xlabel('$q_z$ [nm$^{-1}$]')

plt.tight_layout()
plt.show()

../_images/examples_dynamical_magnetic_xray_35_0.png

Polarization dependence¶

The XrayDynMag allows to set the incoming and outgoing polarization of the X-rays.

[24]:

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

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

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

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

plt.xlabel('$q_z$ [nm$^{-1}$]')
plt.ylabel('Reflectivity')
plt.title('Dynamical X-ray Polarization Dependence')
plt.legend()
plt.show()

incoming polarizations set to: sigma
analyzer polarizations set to: unpolarized
Calculating _homogeneous_reflectivity_ ...
Elapsed time for _homogeneous_reflectivity_: 0.620994 s
incoming polarizations set to: sigma
analyzer polarizations set to: sigma
Calculating _homogeneous_reflectivity_ ...
Elapsed time for _homogeneous_reflectivity_: 0.402254 s
incoming polarizations set to: sigma
analyzer polarizations set to: pi
Calculating _homogeneous_reflectivity_ ...
Elapsed time for _homogeneous_reflectivity_: 0.388865 s

../_images/examples_dynamical_magnetic_xray_37_1.png

Inhomogeneous dynamical magnetic X-ray scattering¶

The inhomogeneous_reflectivity() method allows to calculate the transient magnetic X-ray reflectivity according to a strain_map and/or magnetization_map.

[25]:

dyn_mag.energy = np.r_[708]*u.eV #  set the energy range
dyn_mag.qz = np.r_[1:3.5:0.01]/u.nm # qz range
dyn_mag.set_polarization(3, 0)

incoming polarizations set to: sigma
analyzer polarizations set to: unpolarized

[26]:

R_seq, R_seq_phi, _, _ = dyn_mag.inhomogeneous_reflectivity(strain_map=strain_map,
                                                            magnetization_map=magnetization_map)

Calculating _inhomogeneous_reflectivity_ ...


Elapsed time for _inhomogeneous_reflectivity_: 769.194134 s

[27]:

plt.figure(figsize=[6, 8])
plt.subplot(2, 1, 1)
plt.semilogy(dyn_mag.qz[0, :].to('1/nm'), R_seq[0, 0, :], label=np.round(delays[0]))
plt.semilogy(dyn_mag.qz[0, :].to('1/nm'), R_seq[40, 0, :], label=np.round(delays[40]))
plt.semilogy(dyn_mag.qz[0, :].to('1/nm'), R_seq[-1, 0, :], label=np.round(delays[-1]))

plt.xlabel('$q_z$ [nm$^{-1}$]')
plt.ylabel('Reflectivity')
plt.legend()
plt.title('Dynamical Magnetic X-ray')

plt.subplot(2, 1, 2)
plt.pcolormesh(dyn_mag.qz[0, :].to('1/nm').magnitude,
               delays.to('ps').magnitude, np.log10(R_seq[:, 0, :]))

plt.ylabel('Delay [ps]')
plt.xlabel('$q_z$ [nm$^{-1}$]')
plt.colorbar()
plt.show()

../_images/examples_dynamical_magnetic_xray_41_0.png

Parallel dynamical X-ray 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.

[28]:

try:
    from dask.distributed import Client
    client = Client()
    R_seq_par, R_seq_phi_par, _, _ = dyn_mag.inhomogeneous_reflectivity(strain_map=strain_map,
                                                                        magnetization_map=magnetization_map,
                                                                        calc_type='parallel', dask_client=client)
    client.close()
except:
    pass

Calculating _inhomogeneous_reflectivity_ ...
Elapsed time for _inhomogeneous_reflectivity_: 1293.669474 s

distributed.nanny - WARNING - Worker process still alive after 4 seconds, killing
distributed.nanny - WARNING - Worker process still alive after 4 seconds, killing
distributed.nanny - WARNING - Worker process still alive after 4 seconds, killing
distributed.nanny - WARNING - Worker process still alive after 4 seconds, killing

[29]:

plt.figure(figsize=[6, 8])
plt.subplot(2, 1, 1)
plt.semilogy(dyn_mag.qz[0, :].to('1/nm'), R_seq_par[0, 0, :], label=np.round(delays[0]))
plt.semilogy(dyn_mag.qz[0, :].to('1/nm'), R_seq_par[40, 0, :], label=np.round(delays[40]))
plt.semilogy(dyn_mag.qz[0, :].to('1/nm'), R_seq_par[-1, 0, :], label=np.round(delays[-1]))

plt.xlabel('$q_z$ [nm$^{-1}$]')
plt.ylabel('Reflectivity')
plt.legend()
plt.title('Parallel Dynamical Magnetic X-ray')

plt.subplot(2, 1, 2)
plt.pcolormesh(dyn_mag.qz[0, :].to('1/nm').magnitude,
               delays.to('ps').magnitude, np.log10(R_seq_par[:, 0, :]))

plt.ylabel('Delay [ps]')
plt.xlabel('$q_z$ [nm$^{-1}$]')
plt.colorbar()
plt.show()

../_images/examples_dynamical_magnetic_xray_44_0.png