Chapter 4: Spectroscopy
4.4. Introduction to Core-Loss Spectroscopy#
Working with X-Sections
part of
MSE672: Introduction to Transmission Electron Microscopy
Spring 2024
| Gerd Duscher | Khalid Hattar |
| Microscopy Facilities | Tennessee Ion Beam Materials Laboratory |
| Materials Science & Engineering | Nuclear Engineering |
| Institute of Advanced Materials & Manufacturing | |
4.4.1. Core –Loss Spectroscopy#
As we can see in figure below the energies of the core shells are well defined and can be viewed as delta functions, unlike in the low loss region where the broad valence bands are the initial state. In both cases, however, we excite electrons into the conduction band.
*Excitation from a core-shell state up into the conduction band above the Fermi level.*
If we look at the transition between two states \(< \Phi_f | H | \Phi_i >\) the transition should be quite sharp. In the case of the low-loss spectrum, we have many initial (the valence) states and many final (the conduction) states. The spectrum will be a convolution of these states.
The features of the core–loss edges are, therefore, much sharper than any details in the low–loss region. Because only the final states contribute to the features. These sharp features enable a wide variety of analysis to determine the chemical compositions and chemical bonding, probing the local conduction band of the sample.
4.4.1.1. Chemical Composition#
In this chapter we use the area under the ionization edge to determine the chemical composition of a (small) sample volume. The equation used to determine the number of atoms per unit volume \(N\) (also called areal density) is: \begin{equation} I_{edge}(\beta, \Delta E) = N I_{0}(\beta) \sigma_{edge}(\beta, \Delta E) \end{equation}
\(I_0\) is the number of electrons hitting the sample, and so directly comparable to the beam current.
The equation can be approximated assuming that the spectrum has not been corrected for single scattering: \begin{equation} I_{edge}(\beta, \Delta E) = N I_{low-loss}(\beta,\Delta E) \sigma_{edge}(\beta, \Delta E) \end{equation} where \(\beta\) is the collection angle and \(\sigma_{edge}\) is the partial cross–section (for energy window \(\Delta E\)) for the core–loss excitation.
It is this cross-section \( \sigma_{edge}\) that we want to explorein this notebook.
We will do the chemical composition in the next notebook
4.4.2. Load important packages#
4.4.2.1. Check Installed Packages#
import sys
import importlib.metadata
def test_package(package_name):
"""Test if package exists and returns version or -1"""
try:
version = importlib.metadata.version(package_name)
except importlib.metadata.PackageNotFoundError:
version = '-1'
return version
if test_package('pyTEMlib') < '0.2024.2.3':
print('installing pyTEMlib')
!{sys.executable} -m pip install --upgrade pyTEMlib -q
print('done')
4.4.2.2. Import all relevant libraries#
Please note that the EELS_tools package from pyTEMlib is essential.
import sys
%matplotlib ipympl
if 'google.colab' in sys.modules:
from google.colab import output
from google.colab import drive
output.enable_custom_widget_manager()
import matplotlib.pylab as plt
import numpy as np
## We need to import a few important additional function from matplotlib,
## because we want to demonstrate a few more hidden functionalities of the EELS_tools of pytTEMlib.
from matplotlib.widgets import Cursor
from matplotlib.patches import Rectangle
from matplotlib.widgets import SpanSelector
## import the configuration files of pyTEMlib (we need access to the data folder)
import pyTEMlib
import pyTEMlib.eels_tools as eels
import pyTEMlib.eels_dialog_utilities as ieels
# For archiving reasons it is a good idea to print the version numbers out at this point
print('pyTEM version: ',pyTEMlib.__version__)
You don't have igor2 installed. If you wish to open igor files, you will need to install it (pip install igor2) before attempting.
You don't have gwyfile installed. If you wish to open .gwy files, you will need to install it (pip install gwyfile) before attempting.
Symmetry functions of spglib enabled
pyTEM version: 0.2024.02.2
4.4.3. Load Cross-Sections#
The form factors are from: X-Ray Form Factor, Attenuation, and Scattering Tables NIST Standard Reference Database 66
DOI: https://dx.doi.org/10.18434/T4HS32
Detailed Tabulation of Atomic Form Factors, Photoelectric Absorption and Scattering Cross Section, and Mass Attenuation Coefficients for Z = 1-92 from E = 1-10 eV to E = 0.4-1.0 MeV C.T. Chantler,1 K. Olsen, R.A. Dragoset, J. Chang, A.R. Kishore, S.A. Kotochigova, and D.S. Zucker NIST, Physical Measurement Laboratory
The cross sections are part of the pyTEMlib package and are stored as a pickled dictionary in the package data directory.
Below are the lines for accessing the cross sections with eels_tools of pyTEMlib.
Xsections = eels.get_x_sections()
4.4.4. Plot Cross Sections#
Please add your favourite element ot the list of atomic numbers.
With the code cell above we made the whole database of cross secitons available for this notebook.
# -----Input ------------ #
atomic_numbers = [58, 28]
# ----------------------- #
fig, ax = plt.subplots()
for Z in atomic_numbers:
ax.plot(Xsections[str(Z)]['ene'], Xsections[str(Z)]['dat'], label = Xsections[str(Z)]['name'])
ax.set_xlim(0,1500)
ax.set_ylim(0,2.5e17)
ax.set_xlabel('energy_loss (eV)')
ax.set_ylabel('cross section (atoms/nm$^2$)')
plt.legend();
4.4.4.1. List All Edges of an Element#
element = str(58)
for key in Xsections[element]:
if isinstance(Xsections[element][key], dict):
if 'onset' in Xsections[element][key]:
print(f"{Xsections[element]['name']}-{key}: {Xsections[element][key]['onset']:8.1f} eV ")
Ce-O3: 19.8 eV
Ce-O2: 19.8 eV
Ce-O1: 37.8 eV
Ce-N6: 85.9 eV
Ce-N5: 110.0 eV
Ce-N4: 110.0 eV
Ce-N3: 207.2 eV
Ce-N2: 223.3 eV
Ce-N1: 289.6 eV
Ce-M5: 883.3 eV
Ce-M4: 901.3 eV
Ce-M3: 1185.4 eV
Ce-M2: 1272.8 eV
Ce-M1: 1434.6 eV
Ce-L3: 5723.4 eV
Ce-L2: 6164.2 eV
Ce-L1: 6548.8 eV
Ce-K1: 40443.0 eV
Or ordered
major_edges = ['K1', 'L3', 'M5', 'N5']
all_edges = ['K1','L1','L2','L3','M1','M2','M3','M4','M5','N1', 'N2','N3','N4','N5','N6','N7','O1','O2','O3','O4','O5','O6','O7', 'P1', 'P2', 'P3']
first_close_edges = ['K1', 'L3', 'M5', 'M3', 'N5', 'N3']
element = str(28)
def list_all_edges(Z):
element = str(Z)
print('All edges')
for key in all_edges:
if key in Xsections[element]:
if 'onset' in Xsections[element][key]:
print(f" {Xsections[element]['name']}-{key}: {Xsections[element][key]['onset']:8.1f} eV ")
def list_major_edges(Z):
element = str(Z)
print('Major edges')
for key in major_edges:
if key in Xsections[element]:
if 'onset' in Xsections[element][key]:
print(f" {Xsections[element]['name']}-{key}: {Xsections[element][key]['onset']:8.1f} eV ")
## Here with the function of the EELS_tools package
list_all_edges(element)
list_major_edges(element)
All edges
Ni-K1: 8332.8 eV
Ni-L1: 1008.1 eV
Ni-L2: 871.9 eV
Ni-L3: 854.7 eV
Ni-M1: 111.8 eV
Ni-M2: 68.1 eV
Ni-M3: 68.1 eV
Ni-M4: 3.6 eV
Ni-M5: 3.6 eV
Major edges
Ni-K1: 8332.8 eV
Ni-L3: 854.7 eV
Ni-M5: 3.6 eV
4.4.4.2. Plotting all edges of an element in view#
Now, let’s do it graphically
major_edges = ['K1', 'L3', 'M5', 'N5']
all_edges = ['K1','L1','L2','L3','M1','M2','M3','M4','M5','N1', 'N2','N3','N4','N5','N6','N7','O1','O2','O3','O4','O5','O6','O7', 'P1', 'P2', 'P3']
first_close_edges = ['K1', 'L3', 'M5', 'M3', 'N5', 'N3']
def get_Z(Z):
"""
returns the atomic number independent of input as a string or number
input:
Z: atomic number of chemical symbol (0 if not valid)
"""
Xsections = eels.get_x_sections()
Z_out = 0
if str(Z).isdigit():
Z_out = Z
elif isinstance(Z, str):
for key in Xsections:
if Xsections[key]['name'].lower() == Z.lower(): ## Well one really should know how to write elemental
Z_out = int(key)
return Z_out
class ElementalEdges(object):
def __init__(self, ax, Z):
self.ax = ax
self.labels = None
self.lines = None
self.Z = get_Z(Z)
self.color = 'black'
self.Xsections = eels.get_x_sections()
self.cid = ax.figure.canvas.mpl_connect('draw_event', self.onresize)
#self.update()
def set_edge(self,Z):
self.Z = get_Z(Z)
self.update()
def onresize(self, event):
self.update()
def update(self):
if self.labels != None:
for label in self.labels:
label.remove()
if self.lines != None:
for line in self.lines:
line.remove()
if self.Z>0:
self.labels = [] ; self.lines =[]
x_min, x_max = self.ax.get_xlim()
y_min, y_max = self.ax.get_ylim()
x_bounds = ax.get_xlim()
element = str(self.Z)
Xsections = self.Xsections
for key in all_edges:
if key in Xsections[element]:
if 'onset' in Xsections[element][key]:
x = Xsections[element][key]['onset']
if x > x_min and x < x_max:
if key in first_close_edges:
label2 = self.ax.text(x, y_max,f"{Xsections[element]['name']}-{key}",
verticalalignment='top', rotation = 0, color = self.color)
else:
label2 = self.ax.text(x, y_max,f"\n{Xsections[element]['name']}-{key}",
verticalalignment='top', color = self.color)
line2 = self.ax.axvline(x,ymin = 0,ymax = 1,color=self.color)
self.labels.append(label2)
self.lines.append(line2)
def disconnect(self):
if self.labels != None:
for label in self.labels:
label.remove()
if self.lines != None:
for line in self.lines:
line.remove()
self.labels = None
self.lines = None
self.ax.figure.canvas.mpl_disconnect(self.cid)
def reconnect(self):
self.cid = ax.figure.canvas.mpl_connect('draw_event', self.onresize)
ax.figure.canvas.draw_idle()
fig, ax_Xsec = plt.subplots()
for Z in atomic_numbers:
ax_Xsec.plot(Xsections[str(Z)]['ene'], Xsections[str(Z)]['dat'], label = Xsections[str(Z)]['name'])
ax_Xsec.set_xlim(100,1450)
ax_Xsec.set_ylim(0,1e17)
plt.legend();
Z = 58
edges = ElementalEdges(ax_Xsec, 'Ce')
Let’s make the lines disappear
edges.disconnect()
and reappear in the plot above
edges.set_edge(Z)
Let’s set another edge
edges.set_edge(28)
4.4.4.3. Find Edges Listed in Xsection Dictionary#
please note that the two functions below are as ususal available in the EELS_tools of pyTEMlib
# --- Input ----
edge_onset = 284
maximal_chemical_shift = 7
# -------------
print(f'Major Edges within {maximal_chemical_shift:.1f} eV of {edge_onset:.1f}')
print(eels.find_all_edges(edge_onset, maximal_chemical_shift, major_edges_only=True))
print(f'\nAll Edges within {maximal_chemical_shift:.1f} eV of {edge_onset:.1f}')
print(eels.find_all_edges(edge_onset, maximal_chemical_shift))
Major Edges within 7.0 eV of 284.0
C -K1: 283.8 eV
Ru-M5: 279.4 eV
All Edges within 7.0 eV of 284.0
C -K1: 283.8 eV
Kr-M1: 288.3 eV
Sr-M2: 279.8 eV
Ru-M5: 279.4 eV
Ru-M4: 283.6 eV
Ce-N1: 289.6 eV
Eu-N2: 283.9 eV
Gd-N2: 288.5 eV
Tb-N3: 285.0 eV
Os-N4: 289.4 eV
4.4.4.4. Find Edges Depending on Cursor Postion#
# We are using
# from matplotlib.widgets import Cursor
maximal_chemical_shift = 5
fig, ax = plt.subplots()
plt.title(f'Click with left for major and right mouse button for all \n ionization edges within {maximal_chemical_shift:.1f} eV of cursor')
maximal_chemical_shift = 5
cursor = ieels.EdgesAtCursor(ax, Xsections['42']['ene'], Xsections['42']['dat'],maximal_chemical_shift)
cursor.maximal_chemical_shift =maximal_chemical_shift
cid = plt.connect('motion_notify_event', cursor.mouse_move)
ax.plot(Xsections['16']['ene'], Xsections['16']['dat']*2, label = 'S')
ax.plot(Xsections['42']['ene'], Xsections['42']['dat'], 'r', label = 'Mo')
ax.set_xlim(0,500)
ax.set_ylim(0,2.5e17);
4.4.5. EELS cross sections#
4.4.5.1. Determine Effective Collection Angle#
EELS cross sections are dependent on the momentum transfer (angle dependence), while photons cannot transfer any momentum. The angle dependence is given by the experimental set-up and can be calculated by the convolution of collection and convergence angle.
Here we use the method of Pierre Trebbia, Ultramicroscopy 24 (1988) pp.399-408
def effective_collection_angle(ene, alpha, beta, beam_kv):
""" effective collection angle for convergent beam setup
Original abstract of function y = effbeta(ene, alpha, beta, beamkV)
#
# This program computes etha(alpha,beta), that is the collection
# efficiency associated to the following geometry :
#
# alpha = half angle of illumination (0 -> pi/2)
# beta = half angle of collection (0 -> pi/2)
# (pi/2 = 1570.795 mrad)
#
# A constant angular distribution of incident electrons is assumed
# for any incident angle (-alpha,alpha). These electrons impige the
# target and a single energy loss event occurs, with a characteristic
# angle theta-e (relativistic). The angular distribution of the
# electrons after the target is analytically derived.
# This program integrates this distribution from theta=0 up to
# theta=beta with an adjustable angular step.
# This program also computes beta* which is the theoretical
# collection angle which would give the same value of etha(alpha,beta)
# with a parallel incident beam.
#
# subroutines and function subprograms required
# ---------------------------------------------
# none
#
# comments
# --------
#
# The following parameters are asked as input :
# accelerating voltage (kV), energy loss range (eV) for the study,
# energy loss step (eV) in this range, alpha (mrad), beta (mrad).
# The program returns for each energy loss step :
# alpha (mrad), beta (mrad), theta-e (relativistic) (mrad),
# energy loss (eV), etha (#), beta * (mrad)
#
# author :
# --------
# Pierre TREBBIA
# US 41 : "Microscopie Electronique Analytique Quantitative"
# Laboratoire de Physique des Solides, Bat. 510
# Universite Paris-Sud, F91405 ORSAY Cedex
# Phone : (33-1) 69 41 53 68
#
#
"""
z1 = beam_kv*1000. ; # eV
z2 = ene[0];
z3 = ene[-1]
z4 = 100.0
z5 = alpha*0.001 # rad
z6 = beta*0.001 # rad
z7 = 500 # number of integration steps to be modified at will
# main loop on energy loss
for zx in range(int(z2),int(z3),int(z4)): #! zx = current energy loss
eta=0.0;
x0=float(zx)*(z1+511060.)/(z1*(z1+1022120.)); # x0 = relativistic theta-e
x1 = np.pi/(2.*x0);
x2=x0*x0+z5*z5;
x3=z5/x0*z5/x0;
x4=0.1*np.sqrt(x2);
dtheta=(z6-x4)/z7;
#
# calculation of the analytical expression
#
for zi in range(1, int(z7)):
theta=x4+dtheta*float(zi);
x5=theta*theta;
x6=4.*x5*x0*x0;
x7=x2-x5;
x8=np.sqrt(x7*x7+x6);
x9=(x8+x7)/(2.*x0*x0);
x10=2.*theta*dtheta*np.log(x9);
eta=eta+x10;
eta=eta+x2/100.*np.log(1.+x3) ; # addition of the central contribution
x4=z5*z5*np.log(1.+x1*x1); # normalisation
eta=eta/x4;
#
# correction by geometrical factor (beta/alpha)**2
#
if (z6<z5):
x5=z5/z6;
eta=eta*x5*x5;
etha2=eta*100.;
#
# calculation of beta *
#
x6=np.power((1.+x1*x1),eta);
x7=x0*np.sqrt(x6-1.);
y=x7*1000.;
return y
energy_scale = np.linspace(100,500,1000)
alpha=30
beta=60
eff_beta = effective_collection_angle(energy_scale, alpha=alpha, beta=beta, beam_kv=200)
print(f'Convergence angle {alpha:.2f} mrad and collection angle {beta:.2f} mrad \n '
f'results in effective collection angle of is {eff_beta:.2f} mrad')
Convergence angle 30.00 mrad and collection angle 60.00 mrad
results in effective collection angle of is 55.90 mrad
4.4.5.2. Calculating EELS Cross Section#
EELS cross sections are dependent on the momentum transfer (angle dependence), while photons cannot transfer any momentum. The angle dependence of the cross-section is aproximated by the equation 4 in: Egerton Ultramicroscopy 50 (1993) 13-28
The units of atom density in electron microscopy is atoms per nm\(^2\). Cross sections are normally given in barns (a neutron scattering joke).
A barn is 10\(^{-28}\) m\(^2\) which is then 1 barn = 10\(^{-10}\) nm\(^2\)
Therefore, to get a probability per nm\(^2\) we have to divide the cross-section in barns by 10\(^{10}\)
𝐼𝑒𝑑𝑔𝑒(𝛽,Δ𝐸)=𝑁𝐼𝑙𝑜𝑤−𝑙𝑜𝑠𝑠(𝛽,Δ𝐸)𝜎𝑒𝑑𝑔𝑒(𝛽,Δ𝐸)
from scipy.interpolate import splev,splrep,splint
#from scipy.integrate import quad
from scipy.interpolate import interp1d
from scipy.optimize import leastsq
def xsec_xrpa(energy_scale, E0, Z, beta, shift=0 ):
"""
Calculate momentum-integrated cross-section for EELS from X-ray photoaborption cross-sections.
Input:
------
energy_scale: energyscale of spectrum to be analyzed
E0: acceleration voltage in keV
Z: atomic number of element
beta: effective collection angle in mrad
shift: chemical shift of edge in eV
"""
beta = beta * 0.001; #% collection half angle theta [rad]
#thetamax = self.parent.spec[0].convAngle * 0.001; #% collection half angle theta [rad]
dispersion = energy_scale[1]-energy_scale[0]
Xsections = eels.get_Xsections(Z)
enexs = Xsections['ene']
datxs = Xsections['dat']
#####
## Cross Section according to Egerton Ultramicroscopy 50 (1993) 13-28 equation (4)
#####
# Relativistic correction factors
T = 511060.0*(1.0-1.0/(1.0+E0/(511.06))**2)/2.0;
gamma=1+E0/511.06;
A = 6.5#e-14 *10**14
b = beta
thetaE = enexs/(2*gamma*T)
G = 2*np.log(gamma)-np.log((b**2+thetaE**2)/(b**2+thetaE**2/gamma**2))-(gamma-1)*b**2/(b**2+thetaE**2/gamma**2)
datxs = datxs*(A/enexs/T)*(np.log(1+b**2/thetaE**2)+G)/1e8
datxs = datxs * dispersion # from per eV to per dispersion
coeff = splrep(enexs,datxs,s=0) # now in areal density atoms / m^2
xsec = np.zeros(len(energy_scale ))
#shift = 0# int(ek -onsetXRPS)#/dispersion
lin = interp1d(enexs,datxs,kind='linear') # Linear instead of spline interpolation to avoid oscillations.
xsec = lin(energy_scale-shift)
return xsec
energy_scale = np.arange(100,600)
B_Xsection = eels.xsec_xrpa(energy_scale, 200, 5, 10. )/1e10
N_Xsection = eels.xsec_xrpa(energy_scale, 200, 7, 10. )/1e10 # xsec is in barns = 10^28 m2 = 10^10 nm2
4.4.5.3. Plotting of Cross Sections#
please note that spectrum and cross sections are not on the same scale
S_Xsection = eels.xsec_xrpa(energy_scale, 200, 16, 10. )/1e10
Mo_Xsection = eels.xsec_xrpa(energy_scale, 200, 42, 10. ,shift=0)/1e10 # xsec is in barns = 10^28 m2 = 10^10 nm2
fig, ax1 = plt.subplots()
ax1.plot(energy_scale, S_Xsection, label='S X-section' )
ax1.plot(energy_scale, Mo_Xsection, label='Mo X-section' )
ax1.set_xlabel('energy_loss [eV]')
ax1.set_ylabel('probability [atoms/nm$^{2}$]')
plt.legend();
fig.tight_layout();
4.4.6. Summary#
The cross section is key to determine the chemical composition of an EELS spectrum. These cross sections are dependent on:
acceleration voltage
effective collection angle
element
So these experimental parameters have to be provided for a calculations of cross sections.
We will use these cross sections in the chemical compostions notebook