Chapter 4 Spectroscopy¶
Quantify Spectrum¶
part of
MSE672: Introduction to Transmission Electron Microscopy
by Gerd Duscher, Spring 2023
Microscopy Facilities
Institute of Advanced Materials & Manufacturing
Materials Science & Engineering
The University of Tennessee, Knoxville
Background and methods to analysis and quantification of data acquired with transmission electron microscopes.
First we import the essential libraries¶
All we need here should come with the annaconda or any other package
The xml library will enable us to read the Bruker file.
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')C:\Users\gduscher\AppData\Local\Temp\ipykernel_9792\3399850281.py:2: UserWarning: pkg_resources is deprecated as an API. See https://setuptools.pypa.io/en/latest/pkg_resources.html. The pkg_resources package is slated for removal as early as 2025-11-30. Refrain from using this package or pin to Setuptools<81.
from pkg_resources import get_distribution, DistributionNotFound
done
%matplotlib widget
import matplotlib.pyplot as plt
import numpy as np
import sys
sys.path.insert(0, '../../pyTEMlib/')
## import the configuration files of pyTEMlib (we need access to the data folder)
import pyTEMlib
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.
Using kinematic_scattering library version 0.2022.1.0 by G.Duscher
'0.2025.09.0'Takeoff angle and Absorption¶
After an X-ray is emitted it may get absorbed by an other atom with a line of lower energy. In NiO the Ni-K X-ray photon can be absobed by the O-K edge and thus the oxygen signal is more prominent than the chemical composition would allow. Therefore, the path-length of the X-rays within the sample to the detector is important With a sample-thickness and take-off angle , the path-length in the sample is:
NOTE
Absorption is not important for thin specimens (< ~50nm)
The absorption coefficient is defined as :
with: : depth distribution of Xray production
which is the ratio of the X-ray emission from a layer of element A/B of thickness at depth in the specimen with density to the X-ray emission from an identical, but isolated film.
: the mass-absorption coefficient of X-rays from element A in the specimen ( concentration of element with )
: the detector take-off angle.
import os
def get_bote_salvat_dict(z=0):
filename = os.path.join(pyTEMlib.config_path, 'Bote_Salvat.json')
x_sections = json.load(open(filename))
if z > 0:
return x_sections[str(z)]
return x_sections
def bote_salvat_xsection(x_section, subshell, acceleration_energy):
"""
Using Tabulated Values for electrons of:
- D. Bote and F. Salvat, "Calculations of inner-shell ionization by electron
impact with the distorted-wave and plane-wave Born approximations",
Phys. Rev. A77, 042701 (2008).
- Bote, David, et al. "Cross sections for ionization of K, L and M shells of
atoms by impact of electrons and positrons with energies up to 1 GeV:
Analytical formulas."
Atomic Data and Nuclear Data Tables 95.6 (2009): 871-909.
Computes the inner sub-shell ionization cross section for energetic electrons.
`subshell` is 1->K, 2->L₁, 3->L₂, ..., 9->M₅
Parameters
----------
z : int
The atomic number z in the range 1:99
subshell : int
The atomic sub-shell being ionized 1->K, 2->L₁, 3->L₂, ..., 9->M₅
acceleration_energy : float
The kinetic energy of the incident electron in eV
edge_energy : float
The edge energy of the sub-shell in eV
Returns
-------
float
The ionization cross section in barns
"""
edge_energy = x_section['edge'][subshell]
over_voltage = acceleration_energy / edge_energy
print(edge_energy , acceleration_energy)
if over_voltage < 1.0:
return
if over_voltage <= 16:
print('low')
a = x_section['A'][subshell]
opu = 1.0 / (1.0 + over_voltage)
ffitlo = a[0] + a[1] * over_voltage + opu*(a[2] + opu**2*(a[3] + opu**2*a[4]))
x_ion_e = (over_voltage - 1.0) * (ffitlo / over_voltage)**2
else:
REV = 5.10998918e5 # electron rest energy in eV
e_0 = acceleration_energy
beta2 = (e_0 * (e_0 + 2.0 * REV)) / ((e_0 + REV)**2)
x = np.sqrt(e_0 * (e_0 + 2.0 * REV)) / REV
g = x_section['G'][subshell]
print(g)
ffitup = (((2.0 * np.log(x)) - beta2) * (1.0 + g[0] / x)) + g[1] + g[2] * np.sqrt(REV / (e_0 + REV)) + g[3] / x
factr = x_section['Anlj'][subshell] / beta2
print(x_section['Anlj'][subshell])
x_ion_e = ((factr * over_voltage) / (over_voltage + x_section['Be'][subshell])) * ffitup
print(x_section['Be'][subshell])
return 4.0 * np.pi * 5.291772108e-9**2 * x_ion_e # in barns
x_section_dict = get_bote_salvat_dict(29)
for i in range(1,4):
x = bote_salvat_xsection(x_section_dict, i, 200000)*1e20
print(x)
1093.7 200000
[0.0494, 8.23, -0.173, 0.0823]
4.59e-07
1.02
0.2521903806012185
966.028 200000
[0.0401, 7.26, -0.151, 0.0813]
8.76e-07
0.959
0.42164825891918994
944.886 200000
[0.0361, 7.25, -0.0956, 0.0668]
1.8e-06
0.959
0.8696508772323364
import xraylib as xr
import re
shell_list = []
for key, item in xr.__dict__.items():
if '_SHELL' in key:
shell_list.append(key.split('_')[0])
line_list = []
for key, item in xr.__dict__.items():
if '_LINE' in key:
line_list.append(key.split('_')[0])
transition_dict = {}
regex = shell + r"[H-Z][0-9]"
for value, line in enumerate(line_list):
if line[0] == shell:
transition_dict[line] = -value
transition_dict
{'KL1': 0,
'KL2': -1,
'KL3': -2,
'KM1': -3,
'KM2': -4,
'KM3': -5,
'KM4': -6,
'KM5': -7,
'KN1': -8,
'KN2': -9,
'KN3': -10,
'KN4': -11,
'KN5': -12,
'KN6': -13,
'KN7': -14,
'KO': -15,
'KO1': -16,
'KO2': -17,
'KO3': -18,
'KO4': -19,
'KO5': -20,
'KO6': -21,
'KO7': -22,
'KP': -23,
'KP1': -24,
'KP2': -25,
'KP3': -26,
'KP4': -27,
'KP5': -28,
'KA': -383,
'KB': -384,
'KA1': -387,
'KA2': -388,
'KA3': -389,
'KB1': -390,
'KB2': -391,
'KB3': -392,
'KB4': -393,
'KB5': -394}dir (re)['A',
'ASCII',
'DEBUG',
'DOTALL',
'I',
'IGNORECASE',
'L',
'LOCALE',
'M',
'MULTILINE',
'Match',
'NOFLAG',
'Pattern',
'PatternError',
'RegexFlag',
'S',
'Scanner',
'U',
'UNICODE',
'VERBOSE',
'X',
'_MAXCACHE',
'_MAXCACHE2',
'_ZeroSentinel',
'__all__',
'__builtins__',
'__cached__',
'__doc__',
'__file__',
'__loader__',
'__name__',
'__package__',
'__path__',
'__spec__',
'__version__',
'_cache',
'_cache2',
'_casefix',
'_compile',
'_compile_template',
'_compiler',
'_constants',
'_parser',
'_pickle',
'_special_chars_map',
'_sre',
'_zero_sentinel',
'compile',
'copyreg',
'enum',
'error',
'escape',
'findall',
'finditer',
'fullmatch',
'functools',
'match',
'purge',
'search',
'split',
'sub',
'subn']In a TEM we can assume that the film is homogeneous and can set $ \varphi_B(\rho t) = 1$
and we get:
$$
A_{AB} = \frac{\left( \left. \frac{\mu}{\rho} \right]^A_{\rm spec} \right)}
{ \left( \left. \frac{\mu}{\rho} \right]^B_{\rm spec} \right)}
\left(\frac{1 - \exp\left( \left. \frac{\mu}{\rho} \right]^B_{\rm spec}\frac{\rho t}{\sin \alpha} \right)}
{ 1-\exp\left( \left. \frac{\mu}{\rho} \right]^A_{\rm spec} \frac{\rho t}{\sin \alpha} \right)}\right)
$$
> For absorption correction, we need to know desnity $\rho$ and thickness $t$ of our specimen.
Cell In[6], line 1
In a TEM we can assume that the film is homogeneous and can set $ \varphi_B(\rho t) = 1$
^
SyntaxError: invalid syntax
def get_abs_corr_cross_section(
composition, number_of_atoms, take_off_angle, probe_area
):
"""
Calculate absorption correction terms.
Parameters
----------
number_of_atoms: list of signal
Stack of maps with number of atoms per pixel.
take_off_angle: float
X-ray take-off angle in degrees.
"""
from exspy._misc import material
toa_rad = np.radians(take_off_angle)
Av = constants.Avogadro
elements = [intensity.metadata.Sample.elements[0] for intensity in number_of_atoms]
atomic_weights = np.array(
[
elements_db[element]["General_properties"]["atomic_weight"]
for element in elements
]
)
number_of_atoms = stack(number_of_atoms, show_progressbar=False).data
# calculate the total_mass in kg/m^2, or mass thickness.
total_mass = np.zeros_like(number_of_atoms[0], dtype="float")
for i, (weight) in enumerate(atomic_weights):
total_mass += number_of_atoms[i] * weight / Av / 1e3 / probe_area / 1e-18
# determine mass absorption coefficients and convert from cm^2/g to m^2/kg.
to_stack = material.mass_absorption_mixture(
weight_percent=material.atomic_to_weight(composition)
)
mac = stack(to_stack, show_progressbar=False) * 0.1
acf = np.zeros_like(number_of_atoms)
csc_toa = 1 / math.sin(toa_rad)
# determine an absorption coeficient per element per pixel.
for i, (weight) in enumerate(atomic_weights):
expo = mac.data[i] * total_mass * csc_toa
acf[i] = expo / (1 - np.exp(-expo))
return acfCross-section for EDS in Transmission¶
For thin samples such as used in transmission electron microscopy absorption, and fluorescence, can be neglected Chapter 2: Pennycook and Nellist, 2011. If multiple scattering and channelling is avoided the EDS partial scattering cross-section for a single atom of element x is given by Macarthur et al., 2015 and Macarthur et al., 2017:
Where:
: the volumetric number density of the elemental species being detected,
: the sample thickness (same length unit as volume in N_x),
: is the raw x-ray counts detected from the sample (a.u.),
: is the probe current (A),
is the exposure time (s),
is the electronic charge (C).
A more consistent way would be to relate the cross-section with areal density and to use incident flux like in the case of EELS earlier.
Where:
: the areal density of the elemental species being detected,
: incident flux: total number of electrons the sample is exposed to during the experiment
Comparison with EELS¶
Above formula is the same as derived for EELS earlier. The difference is that is the number of electrons that scattered inelastically per energy transfer (and momentum transfer).
While is the raw counts of X-rays that originate in the relaxation of excitation from a specific core level of element . This implies that the competing relaxation process of generation of Auger electrons is the same for all elements.
This is a fundamental requirement for quantification of both EDS and Auger spectroscopy.
The relationship between and is, therfore, a linear one. And we need to determine the efficiency of X-ray detection (not generation) per EDS system, which also has an take-off angle dependence.
However, the linear dependency is not between the EELS and EDS cross section as defined here, but the cross section of the core excitation, as introduced by Egerton. In that approach the inelastic scattering of lower-energy core-losses are approximated together with the background caused by plasmon losses.
Therefore, in the code cell below we need to subtract the background from lower-level excitations from the core-level in question.
z = 38
def get_eds_xsection(Xsection, energy_scale, start_bgd, end_bgd):
background = pyTEMlib.eels_tools.power_law_background(Xsection, energy_scale, [start_bgd, end_bgd], verbose=False)
cross_section_core = Xsection- background[0]
cross_section_core[cross_section_core < 0] = 0.0
cross_section_core[energy_scale < end_bgd] = 0.0
return cross_section_core
energy_scale = np.arange(10,2000)
Xsection = pyTEMlib.eels_tools.xsec_xrpa(energy_scale, 200, z, 3000. )
edge_info = pyTEMlib.eels_tools.get_x_sections(z)
if 'K1' in edge_info:
start_bgd = edge_info['K1']['onset']*.8
end_bgd = edge_info['K1']['onset']-5
K_eds_xsection = get_eds_xsection(Xsection, energy_scale, start_bgd, end_bgd)
if 'L3' in edge_info:
start_bgd = edge_info['L3']['onset']*.8
end_bgd = edge_info['L3']['onset']-5
L_eds_xsection = get_eds_xsection(Xsection, energy_scale, start_bgd, end_bgd)
if 'M5' in edge_info:
start_bgd = edge_info['L3']['onset']*.8
end_bgd = edge_info['L3']['onset']-5
M5_eds_xsection = get_eds_xsection(Xsection, energy_scale, start_bgd, end_bgd)
background = pyTEMlib.eels_tools.power_law_background(Xsection, energy_scale, [80, 95], verbose=False)
plt.figure()
plt.plot(energy_scale, np.log(1+Xsection) , label='EELS X-section')
plt.plot(energy_scale, np.log(1+background[0]), label='L-core background')
plt.plot(energy_scale, np.log(1+L_eds_xsection), label='L-level X-section')
plt.plot(energy_scale, np.log(1+K_eds_xsection), label='K-level X-section')
plt.xlim(20,30000)
# plt.ylim(0, 10)
plt.xlabel ('energy (eV)')
plt.ylabel('cross section (barns)')
plt.legend();---------------------------------------------------------------------------
TypeError Traceback (most recent call last)
Cell In[2], line 15
13 start_bgd = edge_info['K1']['onset']*.8
14 end_bgd = edge_info['K1']['onset']-5
---> 15 K_eds_xsection = get_eds_xsection(Xsection, energy_scale, start_bgd, end_bgd)
17 if 'L3' in edge_info:
18 start_bgd = edge_info['L3']['onset']*.8
Cell In[2], line 4, in get_eds_xsection(Xsection, energy_scale, start_bgd, end_bgd)
3 def get_eds_xsection(Xsection, energy_scale, start_bgd, end_bgd):
----> 4 background = pyTEMlib.eels_tools.power_law_background(Xsection, energy_scale, [start_bgd, end_bgd], verbose=False)
5 cross_section_core = Xsection- background[0]
6 cross_section_core[cross_section_core < 0] = 0.0
File ~\AppData\Local\anaconda3\Lib\site-packages\pyTEMlib\eels_tools.py:1625, in power_law_background(spectrum, energy_scale, fit_area, verbose)
1622 err = yy - power_law(xx, pp[0], pp[1])
1623 return err
-> 1625 [p, _] = leastsq(bgdfit, p0, args=(y, x), maxfev=2000)
1627 background_difference = y - power_law(x, p[0], p[1])
1628 background_noise_level = std_dev = np.std(background_difference)
File ~\AppData\Local\anaconda3\Lib\site-packages\scipy\optimize\_minpack_py.py:430, in leastsq(func, x0, args, Dfun, full_output, col_deriv, ftol, xtol, gtol, maxfev, epsfcn, factor, diag)
427 m = shape[0]
429 if n > m:
--> 430 raise TypeError(f"Improper input: func input vector length N={n} must"
431 f" not exceed func output vector length M={m}")
433 if epsfcn is None:
434 epsfcn = finfo(dtype).eps
TypeError: Improper input: func input vector length N=2 must not exceed func output vector length M=0Cross-section for EDS¶
Of course we do not need the dependence of the cross section on energy but the sum.
The relevant functions will then be:
def get_eds_xsection(Xsection, energy_scale, start_bgd, end_bgd):
background = pyTEMlib.eels_tools.power_law_background(Xsection, energy_scale, [start_bgd, end_bgd], verbose=False)
cross_section_core = Xsection- background[0]
cross_section_core[cross_section_core < 0] = 0.0
cross_section_core[energy_scale < end_bgd] = 0.0
return cross_section_core
def get_eds_line_strength(z, acceleration_voltage, max_kV=60000 ):
keV = acceleration_voltage /1000.
energy_scale = np.arange(10, max_kV, 1)
edge_info = pyTEMlib.eels_tools.get_x_sections(z)
eds_cross_sections = {'_element': {'atomic_weight': edge_info['atomic_weight'],
'name': edge_info['name'],
'nominal_density': edge_info['nominal_density']}}
Xsection = pyTEMlib.eels_tools.xsec_xrpa(energy_scale, keV, z, 3000. )
if 'K1' in edge_info:
start_bgd = edge_info['K1']['onset'] * 0.8
if edge_info['K1']['onset'] - start_bgd >100:
start_bgd = edge_info['K1']['onset'] - 100
end_bgd = edge_info['K1']['onset'] - 5
K_eds_xsection = get_eds_xsection(Xsection, energy_scale, start_bgd, end_bgd)
eds_cross_sections['K'] = {'x-section': K_eds_xsection[int(end_bgd) : int(end_bgd)+200].sum(),
'strength': K_eds_xsection[int(end_bgd) : int(end_bgd)+200].sum()}
if 'fluorescent_yield' in edge_info:
eds_cross_sections['K']['fluorescent_yield'] = edge_info['fluorescent_yield']['K']
eds_cross_sections['K']['strength'] *= edge_info['fluorescent_yield']['K']
if 'L3' in edge_info:
start_bgd = edge_info['L3']['onset'] * 0.8
if edge_info['L3']['onset'] - start_bgd >100:
start_bgd = edge_info['L3']['onset'] - 100
end_bgd = edge_info['L3']['onset'] - 5
L_eds_xsection = get_eds_xsection(Xsection, energy_scale, start_bgd, end_bgd)
eds_cross_sections['L'] = {'x-section': L_eds_xsection[int(end_bgd) : int(end_bgd)+200].sum(),
'strength': L_eds_xsection[int(end_bgd) : int(end_bgd)+200].sum()}
if 'fluorescent_yield' in edge_info:
if 'L' in edge_info['fluorescent_yield']:
eds_cross_sections['L']['fluorescent_yield'] = edge_info['fluorescent_yield']['L']
eds_cross_sections['L']['strength'] *= edge_info['fluorescent_yield']['L']
else:
eds_cross_sections['L']['strength'] *= 0.
if 'M5' in edge_info:
if(edge_info['M5']['onset']) >100:
start_bgd = edge_info['M5']['onset'] * 0.8
if edge_info['M5']['onset'] - start_bgd >100:
start_bgd = edge_info['M5']['onset'] - 100
end_bgd = edge_info['M5']['onset'] - 5
M_eds_xsection = get_eds_xsection(Xsection, energy_scale, start_bgd, end_bgd)
eds_cross_sections['M'] = {'x-section': M_eds_xsection[int(end_bgd) : int(end_bgd)+300].sum(),
'strength': M_eds_xsection[int(end_bgd) : int(end_bgd)+300].sum()}
if 'fluorescent_yield' in edge_info:
if 'M' in edge_info['fluorescent_yield']:
eds_cross_sections['M']['fluorescent_yield'] = edge_info['fluorescent_yield']['M']
# eds_cross_sections['M']['strength'] *= edge_info['fluorescent_yield']['M']
else:
eds_cross_sections['M']['strength'] *= 0.
return eds_cross_sections
Comparison to k-factors to Cliff Lorimer
k-factors a weight% cross-sections in atom%
energy_scale = np.linspace(10, 20000, 20000)
keV = 200000
ref_z = 14
z = 29
def get_line_weight(lines, line_family='K'):
line_weight = 0
for key, item in lines.items():
if line_family in key:
line_weight += item['weight']
return line_weight
x_section_dict = get_bote_salvat_dict(ref_z)
Si_K1 = bote_salvat_xsection(x_section_dict, 0, keV)*1e20
edge_info = pyTEMlib.eels_tools.get_x_sections(ref_z)
Si_yield = edge_info['fluorescent_yield']['K']
Si_amu = edge_info['atomic_weight']
Si_line_weight = get_line_weight(edge_info['lines'])
edge_info = pyTEMlib.eels_tools.get_x_sections(z)
x_section_dict = get_bote_salvat_dict(z)
x_K1 = bote_salvat_xsection(x_section_dict, 0, keV)*1e23
yield_K1 = edge_info['fluorescent_yield']['K']
xK_line_weight = get_line_weight(edge_info['lines'], 'K')
if 'L' in edge_info['fluorescent_yield'].keys():
xL = 0
for i in range (1,4):
x_L = bote_salvat_xsection(x_section_dict, i, keV)*1e23
yield_L3 = edge_info['fluorescent_yield']['L']
xL_line_weight = get_line_weight(edge_info['lines'], 'L')
x_amu = edge_info['atomic_weight']
edge_info['name']1828.5 200000
[0.0621, 7.14, -0.207, 0.0953]
3.74e-07
1.14
8950.25 200000
[0.302, 6.26, -1.02, 0.454]
6.49e-08
0.765
1093.7 200000
[0.0494, 8.23, -0.173, 0.0823]
4.59e-07
1.02
966.028 200000
[0.0401, 7.26, -0.151, 0.0813]
8.76e-07
0.959
944.886 200000
[0.0361, 7.25, -0.0956, 0.0668]
1.8e-06
0.959
'Cu'for i in range(1,4):
print(i)1
2
3
k_a = x_K1 * yield_K1 * xK_line_weight / x_amu
l_a = x_L * yield_L3 *
/ x_amu
ref_k = Si_K1 * Si_yield * Si_line_weight / Si_amu
print(f"{k_a:.4f}, {l_a:.4f}, {ref_k:.4f}")
print(f" k-factor {k_a/ref_k:.4f} L: {l_a/ref_k:.4f}")
print('k_a/l_a' , k_a/l_a)
print(f"{yield_K1:.2f}, {yield_L3:.3f}, {ref_yield:.3f} ")
print(f"{xK_line_weight:.2f}, {xL_line_weight:.3f}, {Si_line_weight:.3f} ")
print(f"x-sec {x_K1*1000:.2f}, {x_L3*1000:.2f}, {ref_K1*1000:.2f}")
print( x_K1/yield_K1, x_L3/yield_L3)
'k_a/l_a', (5.13247e-01/ 7.07008e-01), 'k-factor', (5.13247e-01 / 3.24726e-01), (x_amu/Si_amu) * 4.7841 /5, 5/8.2614550.2958, 0.3304, 0.0004
k-factor 772.1603 L: 862.3140
k_a/l_a 0.895451408808201
0.42, 0.008, 0.040
1.89, 2.951, 1.532
x-sec 23492.02, 869.65, 174.53
55.623490445748104 106.31428817021227
('k_a/l_a',
0.725942280709695,
'k-factor',
1.5805540671212037,
2.164892336614979,
0.6052202668900333)'K': np.float64(2.192963366430227e-22),
'L1': np.float64(2.5054476181469093e-21),
'L2': np.float64(4.181630548806971e-21),
'L3': np.float64(8.647677139867e-21), Cell In[434], line 1
'K': np.float64(2.192963366430227e-22),
^
SyntaxError: illegal target for annotation
"KL2": {
"energy": 1.7394,
"cs": 2.4079062124253643e-23
},
"KL3": {
"energy": 1.74,
"cs": 4.7841761325214666e-23
"29": {
"KL2": {
"energy": 8.0279,
"cs": 2.5681370026132807e-23
},
"KL3": {
"energy": 8.0478,
"cs": 5.003801989880983e-23
},edge_info['total_fluorescent_yield']{'K': 0.43381, 'L1': 0.009138, 'L2': 0.009302, 'L3': 0.008804}print(x_K1/yield_K1, x_L3/yield_L3)
print( (yield_K1/x_K1) / (yield_L3/ x_L3))
print(1/((yield_K1/x_K1) / (ref_yield/ reference))*(fe_amu/ref_weight) )
print((fe_amu/ref_weight))
k_a = 1/x_K1 * yield_K1 / ref_weight * xK_line_weight
k_b = ref_K1 * ref_yield / fe_amu * ref_line_weight
l_a = x_L3* yield_L3/ref_weight * xL_line_weight
print(xL_line_weight, xK_line_weight)
print(k_a/k_b , k_a, k_b, l_a/k_a, k_a/ l_a)
print((x_K1 * yield_K1 * xK_line_weight) / (ref_K1* ref_yield * ref_line_weight) * (ref_weight /fe_amu ))
1(5.13247e-01 / 3.24726e-01) 0.05562349044574811 106.31428817021227
1911.319971440933
2.1241189051165168e+17
2.262591016716811
0.33886075083341954 0.5277930544545206
4681.15961157362 0.33784949408754095 7.217217999835929e-05 0.0002540470048098375 3936.27943281021
0.5046406596868588
0.632689523757567weight = 0
for key, item in edge_info['lines'].items():
if "K" in key:
weight (1.894682, 0.5277930544545206)def get_k_factors(z, keV):
Si_eds_cross_sections = pyTEMlib.eds_tools.get_eds_line_strength(14, keV)
z_eds_cross_sections = pyTEMlib.eds_tools.get_eds_line_strength(z, keV)
if 'K' in z_eds_cross_sections:
z_eds_cross_sections['K']['k_factor'] = z_eds_cross_sections['K']['strength'] / Si_eds_cross_sections['K'] ['strength']
if 'L' in z_eds_cross_sections:
z_eds_cross_sections['L']['k_factor'] = z_eds_cross_sections['L']['strength'] / Si_eds_cross_sections['K'] ['strength']
if 'M' in z_eds_cross_sections:
z_eds_cross_sections['M']['k_factor'] = z_eds_cross_sections['M']['strength'] / Si_eds_cross_sections['K'] ['strength']
return z_eds_cross_sections
k_factors = pyTEMlib.eds_tools.load_k_factors()
z = 47
x = []
y = []
y1 = []
y2 = []
y3 = []
mult = 17
added = .5
print(1/3/11.4)
mult = 11.4
added = 0.02915
nx_sections = get_k_factors(14, 200)
si_k = nx_sections['K']['fluorescent_yield']/nx_sections['K']['x-section'] + added
si_kk = float(k_factors.get('Si', {}).get('Ka1', 1))
for z in range(10, 40, 2):
nx_sections = get_k_factors(z, 200)
# print(As_eds_cross_sections)
# print(f"ration of Si to {z} for K lines intensities {nx_sections['K']['k_factor']}")
#print(f"ration of Si to {z} for L lines intensities {nx_sections['L']['k_factor']}")
yy = nx_sections['K']['x-section'] * nx_sections['K']['fluorescent_yield']
if yy > 0:
x.append(z)
y.append(nx_sections['K']['fluorescent_yield']/nx_sections['K']['x-section'] + added)
y1.append(nx_sections['K']['x-section'])
y2.append(nx_sections['K']['fluorescent_yield'])
key = nx_sections['_element']['name']
y3.append(float(k_factors.get(key, {}).get('Ka1', 1)))
# print(f"{nx_sections['K']['x-section']:.2f}, {nx_sections['K']['fluorescent_yield']:.3f}, {nx_sections['K']['x-section'] * nx_sections['K']['fluorescent_yield']:.3f}")
plt.figure()
plt.plot(x, np.array(y)*10, label='kk')
plt.plot(x, np.array(y1)/20000, label='X-sec')
plt.plot(x, np.array(y2)/100, label = 'flu')
plt.plot(x, np.array(y3), label = 'k')
plt.plot(x, np.array(y3)/si_kk- np.array(y/si_k))
plt.legend()
p = np.array(y3)/si_kk- np.array(y/si_k)
print(f"{p.min():2f}, {p.max():.3f}, {p.std():.5f}")
Loading...
added * np.array(y2)
array([0.0003729, 0.0006999, 0.0012075, 0.0029451, 0.004233 , 0.005799 ,
0.0076032, 0.0095775, 0.0116367, 0.0136923, 0.0156675, 0.0175038,
0.0191658, 0.0206376])x = []
y = []
y1 = []
y2 = []
y3 = []
added = 40448*0
mult = 1e3
nx_sections = get_k_factors(14, 200)
si_k = nx_sections['K']['x-section'] /nx_sections['K']['fluorescent_yield'] + 0.02915
si_kk = float(k_factors.get('Si', {}).get('Ka1', 1))
for z in range(26, 60, 2):
nx_sections = get_k_factors(z, 200)
# print(As_eds_cross_sections)
# print(f"ration of Si to {z} for K lines intensities {nx_sections['K']['k_factor']}")
#print(f"ration of Si to {z} for L lines intensities {nx_sections['L']['k_factor']}")
yy = nx_sections['L']['fluorescent_yield'] / nx_sections['L']['x-section']
if yy > 0:
x.append(z)
y.append(((nx_sections['L']['fluorescent_yield'] / nx_sections['L']['x-section'] )*mult) )
y1.append(nx_sections['L']['x-section'])
y2.append(nx_sections['L']['fluorescent_yield'])
key = nx_sections['_element']['name']
y3.append(float(k_factors.get(key, {}).get('La1', 1)))
# print(f"{nx_sections['K']['x-section']:.2f}, {nx_sections['K']['fluorescent_yield']:.3f}, {nx_sections['K']['x-section'] * nx_sections['K']['fluorescent_yield']:.3f}")
plt.figure()
plt.plot(x, np.array(y), label='kk')
#plt.plot(x, np.array(y1)/20000, label='X-sec')
#plt.plot(x, np.array(y2)/100, label = 'flu')
plt.plot(x, np.array(y3), label = 'k')
#plt.plot(x, np.array(y3)/si_kk- np.array(y/si_k))
p = np.array(y3)/si_kk- np.array(y/si_k)
print(f"{p.min():2f}, {p.max():.3f}, {p.std():.5f}")
plt.legend()
y, si_k, y3, addedLoading...
added/mult0.029239766081871343{'_element': {'atomic_weight': 140.12, 'name': 'Ce', 'nominal_density': 6.637},
'K': {'x-section': np.float64(0.5937873827996667),
'strength': np.float64(0.541540030987124),
'fluorescent_yield': 0.91201,
'k_factor': np.float64(0.01323890388139376)},
'L': {'x-section': np.float64(174.80969013623059),
'strength': np.float64(20.88800987437819),
'fluorescent_yield': 0.11949,
'k_factor': np.float64(0.5106443460817243)},
'M': {'x-section': np.float64(31818.63679637004),
'strength': np.float64(144.77479742348368),
'fluorescent_yield': 0.00455,
'k_factor': np.float64(3.539275986752169)}}k_factors = pyTEMlib.eds_tools.load_k_factors()
k_factors{'': {},
'B': {'Ka1': '1.00000e+00'},
'C': {'Ka1': '5.34908e-01'},
'N': {'Ka1': '4.23635e-01'},
'O': {'Ka1': '3.71908e-01'},
'F': {'Ka1': '3.85149e-01'},
'Ne': {'Ka1': '3.46539e-01'},
'Na': {'Ka1': '3.45547e-01'},
'Mg': {'Ka1': '3.27781e-01'},
'Al': {'Ka1': '3.30728e-01'},
'Si': {'Ka1': '3.24726e-01'},
'P': {'Ka1': '3.34291e-01'},
'S': {'Ka1': '3.21669e-01'},
'Cl': {'Ka1': '3.34284e-01'},
'Ar': {'Ka1': '3.58291e-01', 'Kb1': '3.58291e-01'},
'K': {'Ka1': '3.36238e-01', 'Kb1': '3.36238e-01'},
'Ca': {'Ka1': '3.33218e-01', 'Kb1': '3.33218e-01'},
'Sc': {'Ka1': '3.63890e-01', 'Kb1': '3.63890e-01', 'La1': '1.63109e+00'},
'Ti': {'Ka1': '3.79229e-01',
'Kb1': '3.79229e-01',
'La1': '1.30000e+00',
'Lb1': '1.31997e+00'},
'V': {'Ka1': '3.97724e-01',
'Kb1': '3.97724e-01',
'La1': '1.15585e+00',
'Lb1': '1.17588e+00'},
'Cr': {'Ka1': '4.02650e-01',
'Kb1': '4.02650e-01',
'La1': '1.05303e+00',
'Lb1': '1.07281e+00'},
'Mn': {'Ka1': '4.24546e-01',
'Kb1': '4.24546e-01',
'La1': '9.95024e-01',
'Lb1': '1.01531e+00'},
'Fe': {'Ka1': '4.32916e-01',
'Kb1': '4.32916e-01',
'La1': '8.76707e-01',
'Lb1': '8.95690e-01'},
'Co': {'Ka1': '4.60642e-01',
'Kb1': '4.60642e-01',
'La1': '8.14528e-01',
'Lb1': '8.33085e-01'},
'Ni': {'Ka1': '4.65132e-01',
'Kb1': '4.65132e-01',
'La1': '7.25953e-01',
'Lb1': '7.43277e-01'},
'Cu': {'Ka1': '5.13247e-01',
'Kb1': '5.13247e-01',
'La1': '7.07008e-01',
'Lb1': '7.24974e-01'},
'Zn': {'Ka1': '5.42706e-01',
'Kb1': '5.42706e-01',
'La1': '6.67212e-01',
'Lb1': '6.85234e-01'},
'Ga': {'Ka1': '5.99894e-01',
'Kb1': '5.99894e-01',
'La1': '6.58445e-01',
'Lb1': '6.77436e-01'},
'Ge': {'Ka1': '6.55539e-01',
'Kb1': '6.55539e-01',
'La1': '6.39731e-01',
'Lb1': '6.59345e-01'},
'As': {'Ka1': '7.18892e-01',
'Kb1': '7.18892e-01',
'La1': '6.17577e-01',
'Lb1': '6.37511e-01'},
'Se': {'Ka1': '8.17924e-01',
'Kb1': '8.17924e-01',
'La1': '6.13103e-01',
'Lb1': '6.33918e-01'},
'Br': {'Ka1': '9.05321e-01',
'Kb1': '9.05321e-01',
'La1': '5.85224e-01',
'Lb1': '6.06287e-01'},
'Kr': {'Ka1': '1.05287e+00',
'Kb1': '1.05287e+00',
'La1': '5.99310e-01',
'Lb1': '6.22023e-01'},
'Rb': {'Ka1': '1.20424e+00',
'Kb1': '1.20424e+00',
'La1': '5.80512e-01',
'Lb1': '6.03815e-01'},
'Sr': {'Ka1': '1.39900e+00',
'Kb1': '1.39900e+00',
'La1': '5.69435e-01',
'Lb1': '5.93524e-01'},
'Y': {'Ka1': '1.62293e+00',
'Kb1': '1.62293e+00',
'La1': '5.57761e-01',
'Lb1': '5.82556e-01',
'Mz1': '1.02442e+00'},
'Zr': {'Ka1': '1.91568e+00',
'Kb1': '1.91568e+00',
'La1': '5.45923e-01',
'Lb1': '5.71390e-01',
'Mz1': '8.61843e-01'},
'Nb': {'Ka1': '2.25478e+00',
'Kb1': '2.25478e+00',
'La1': '5.30983e-01',
'Lb1': '5.56977e-01',
'Mz1': '7.24052e-01'},
'Mo': {'Ka1': '2.69449e+00',
'Kb1': '2.69449e+00',
'La1': '5.25675e-01',
'Lb1': '5.52708e-01',
'Mz1': '6.16939e-01'},
'Tc': {'Ka1': '3.22822e+00',
'La1': '5.16718e-01',
'Lb1': '5.44536e-01',
'Mz1': '5.47504e-01'},
'Ru': {'Ka1': '3.88953e+00',
'La1': '5.13871e-01',
'Lb1': '5.42901e-01',
'Mz1': '5.05635e-01'},
'Rh': {'Ka1': '4.63368e+00',
'La1': '5.05700e-01',
'Lb1': '5.35561e-01',
'Mz1': '4.67148e-01'},
'Pd': {'La1': '5.06174e-01', 'Lb1': '5.37481e-01', 'Mz1': '4.44887e-01'},
'Ag': {'La1': '4.98260e-01', 'Lb1': '5.30439e-01', 'Mz1': '4.30077e-01'},
'Cd': {'La1': '5.05865e-01', 'Lb1': '5.39947e-01', 'Mz1': '4.25568e-01'},
'In': {'La1': '5.04319e-01',
'Lb1': '5.39780e-01',
'Mz1': '4.17840e-01',
'Mg1': '6.57558e-01'},
'Sn': {'La1': '5.09935e-01',
'Lb1': '5.47270e-01',
'Mz1': '4.21105e-01',
'Mg1': '6.49928e-01'},
'Sb': {'La1': '5.12226e-01',
'Lb1': '5.51281e-01',
'Mz1': '4.21279e-01',
'Mg1': '6.39031e-01'},
'Te': {'La1': '5.26569e-01',
'Lb1': '5.68362e-01',
'Mz1': '4.32137e-01',
'Mg1': '6.44967e-01'},
'I': {'La1': '5.14608e-01',
'Lb1': '5.57163e-01',
'Mz1': '4.22139e-01',
'Mg1': '6.19924e-01'},
'Xe': {'La1': '5.24463e-01',
'Lb1': '5.69649e-01',
'Mz1': '4.68719e-01',
'Mg1': '6.78135e-01'},
'Cs': {'La1': '5.23529e-01',
'Lb1': '5.70196e-01',
'Mz1': '4.64699e-01',
'Mg1': '6.62293e-01'},
'Ba': {'La1': '5.34069e-01',
'Lb1': '5.83511e-01',
'Mz1': '4.67297e-01',
'Mg1': '6.57819e-01'},
'La': {'La1': '5.33425e-01',
'Lb1': '5.84777e-01',
'Mz1': '4.58435e-01',
'Mg1': '6.39833e-01'},
'Ce': {'La1': '5.31928e-01', 'Lb1': '5.85111e-01'},
'Pr': {'La1': '5.28869e-01',
'Lb1': '5.83851e-01',
'Ma1': '4.38386e-01',
'Mz1': '4.49639e-01',
'Mg1': '6.18119e-01'},
'Nd': {'La1': '5.35635e-01',
'Lb1': '5.93536e-01',
'Ma1': '4.11091e-01',
'Mz1': '4.22006e-01',
'Mg1': '5.76148e-01'},
'Pm': {'La1': '5.33684e-01',
'Lb1': '5.93634e-01',
'Ma1': '3.99029e-01',
'Mz1': '4.10431e-01',
'Mg1': '5.56685e-01'},
'Sm': {'La1': '5.49178e-01',
'Lb1': '6.13232e-01',
'Ma1': '4.00913e-01',
'Mz1': '4.12353e-01'},
'Eu': {'La1': '5.51261e-01',
'Lb1': '6.18013e-01',
'Ma1': '3.91117e-01',
'Mz1': '4.03409e-01'},
'Gd': {'La1': '5.67189e-01',
'Lb1': '6.38476e-01',
'Ma1': '3.91704e-01',
'Mz1': '4.04381e-01'},
'Tb': {'La1': '5.70624e-01',
'Lb1': '6.45045e-01',
'Ma1': '3.83551e-01',
'Mz1': '3.96090e-01'},
'Dy': {'La1': '5.85460e-01',
'Lb1': '6.64663e-01',
'Ma1': '3.77862e-01',
'Mz1': '3.91055e-01'},
'Ho': {'La1': '5.88659e-01',
'Lb1': '6.71274e-01',
'Ma1': '3.71586e-01',
'Mz1': '3.84865e-01'},
'Er': {'La1': '5.96254e-01',
'Lb1': '6.83067e-01',
'Ma1': '3.64371e-01',
'Mz1': '3.78093e-01'},
'Tm': {'La1': '6.02152e-01',
'Lb1': '6.93047e-01',
'Ma1': '3.55756e-01',
'Mz1': '3.69489e-01'},
'Yb': {'La1': '6.17735e-01',
'Lb1': '7.14360e-01',
'Ma1': '3.55218e-01',
'Mz1': '3.68860e-01'},
'Lu': {'La1': '6.25766e-01', 'Lb1': '7.27232e-01', 'Ma1': '3.54027e-01'},
'Hf': {'La1': '6.42767e-01',
'Lb1': '7.50714e-01',
'Ma1': '3.51486e-01',
'Na1': '4.52786e+00'},
'Ta': {'La1': '6.56343e-01',
'Lb1': '7.70329e-01',
'Ma1': '3.46374e-01',
'Na1': '4.35880e+00'},
'W': {'La1': '6.73991e-01',
'Lb1': '7.95195e-01',
'Ma1': '3.41710e-01',
'Na1': '4.29405e+00'},
'Re': {'La1': '6.88316e-01', 'Lb1': '8.16463e-01', 'Ma1': '3.38218e-01'},
'Os': {'La1': '7.12581e-01', 'Lb1': '8.49883e-01', 'Ma1': '3.36600e-01'},
'Ir': {'La1': '7.32378e-01',
'Lb1': '8.78311e-01',
'Ma1': '3.30316e-01',
'Na1': '3.96275e+00'},
'Pt': {'La1': '7.59814e-01',
'Lb1': '9.16413e-01',
'Ma1': '3.25640e-01',
'Na1': '3.89910e+00'},
'Au': {'La1': '7.85243e-01',
'Lb1': '9.52694e-01',
'Ma1': '3.19403e-01',
'Na1': '3.83686e+00'},
'Hg': {'La1': '8.16157e-01',
'Lb1': '9.96008e-01',
'Ma1': '3.16574e-01',
'Na1': '3.85492e+00'},
'Tl': {'La1': '8.54296e-01', 'Lb1': '1.04880e+00', 'Ma1': '3.14438e-01'},
'Pb': {'La1': '8.94557e-01',
'Lb1': '1.10531e+00',
'Ma1': '3.10504e-01',
'Na1': '3.97231e+00'},
'Bi': {'La1': '9.30226e-01',
'Lb1': '1.15668e+00',
'Ma1': '3.05023e-01',
'Na1': '3.97923e+00'},
'Po': {'La1': '9.62629e-01', 'Lb1': '1.20508e+00', 'Ma1': '2.97722e-01'},
'At': {'La1': '1.00547e+00', 'Lb1': '1.26703e+00', 'Ma1': '2.91770e-01'},
'Rn': {'La1': '1.10732e+00', 'Lb1': '1.40473e+00', 'Ma1': '3.00835e-01'},
'Fr': {'La1': '1.16189e+00', 'Lb1': '1.48454e+00', 'Ma1': '2.94695e-01'},
'Ra': {'La1': '1.23236e+00', 'Lb1': '1.58621e+00', 'Ma1': '2.90890e-01'},
'Ac': {'La1': '1.29927e+00', 'Lb1': '1.68479e+00', 'Ma1': '2.85144e-01'},
'Th': {'La1': '1.39632e+00', 'Lb1': '1.82458e+00', 'Ma1': '2.84257e-01'},
'Pa': {'La1': '1.46450e+00', 'Lb1': '1.92861e+00', 'Ma1': '2.75479e-01'},
'U': {'La1': '1.56935e+00', 'Lb1': '2.08401e+00', 'Ma1': '2.76069e-01'},
'Np': {'La1': '1.67460e+00', 'Lb1': '2.24244e+00', 'Ma1': '2.68009e-01'},
'Pu': {'La1': '1.77254e+00', 'Lb1': '2.39411e+00', 'Ma1': '2.68562e-01'},
'Am': {'La1': '1.92571e+00', 'Lb1': '2.62418e+00', 'Ma1': '2.60301e-01'},
'Cm': {'La1': '2.07864e+00', 'Lb1': '2.85895e+00', 'Ma1': '2.57507e-01'}}(.02409*2850)/(.04025*1016.), (.68792*7)/ (.04025*1016)
(1.6788893236171565, 0.1177541937692571)from pyTEMlib.xrpa_x_sections import x_sections
{'name': 'Sr',
'barns': 1454970000000.0002,
'NumEdges': 12,
'atomic_weight': 87.62,
'nominal_density': 2.54,
'photoabs_to_sigma': 145.497,
'lines': {'K-L3': {'weight': 1.0, 'position': 14165.000000000002},
'K-L2': {'weight': 0.564, 'position': 14097.800000000003},
'K-M3': {'weight': 0.144, 'position': 15835.500000000002},
'K-N3': {'weight': 0.0328, 'position': 16084.700000000003},
'K-M2': {'weight': 0.0731, 'position': 15824.800000000003},
'K-M5': {'weight': 0.000368, 'position': 15971.500000000002},
'L3-M5': {'weight': 1.0, 'position': 1806.5},
'L3-M4': {'weight': 0.114, 'position': 1804.6},
'L3-M1': {'weight': 0.0412, 'position': 1582.1},
'L2-N3': {'weight': 0.00524, 'position': 1986.9},
'L2-M4': {'weight': 0.499, 'position': 1871.8000000000002},
'L2-M3': {'weight': 0.000116, 'position': 1737.7000000000003},
'L2-N1': {'weight': 0.00187, 'position': 1969.1000000000001},
'L2-M1': {'weight': 0.0261, 'position': 1649.3000000000002},
'L1-M3': {'weight': 0.0274, 'position': 1947.1999999999998},
'L1-M2': {'weight': 0.0394, 'position': 1936.4999999999998},
'L1-N2': {'weight': 0.000191, 'position': 2196.3999999999996},
'L1-N3': {'weight': 0.00341, 'position': 2196.3999999999996}},
'fluorescent_yield': {'K': 0.68792, 'L': 0.02409},
'total_fluorescent_yield': {'K': 0.68568,
'L1': 0.026337,
'L2': 0.028651,
'L3': 0.026561},
'N3': {'filename': 'None',
'excl before': 5,
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'N1': {'filename': 'None',
'excl before': 5,
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'M5': {'filename': 'Sr.M5',
'excl before': 5,
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'excl before': 5,
'excl after': 50,
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'excl after': 50,
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'excl after': 50,
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'L3': {'filename': 'Sr.L3',
'excl before': 5,
'excl after': 50,
'onset': 1939.6,
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'shape': 'white_line'},
'L2': {'filename': 'None',
'excl before': 5,
'excl after': 50,
'onset': 2006.8000000000002,
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'L1': {'filename': 'Sr.L1',
'excl before': 5,
'excl after': 50,
'onset': 2216.2999999999997,
'factor': 1.0},
'K1': {'filename': 'None',
'excl before': 5,
'excl after': 50,
'onset': 16104.600000000002,
'factor': 1.0},
'ene': array([1.069000e+01, 1.142761e+01, 1.221612e+01, 1.305903e+01,
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1.141345e+03, 1.220098e+03, 1.304285e+03, 1.394281e+03,
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1.64353411e+14, 1.37711456e+14, 1.14753484e+14, 9.47301868e+13,
7.79441979e+13, 6.41932764e+13, 5.29172589e+13, 4.36651047e+13,
3.60643414e+13, 2.98137903e+13, 2.46704713e+13, 2.30205353e+13,
2.20529803e+13, 2.18056354e+13, 1.60366793e+14, 1.53455686e+14,
1.52437207e+14, 1.26289941e+14, 1.05998929e+14, 8.92987838e+13,
7.50779070e+13, 6.28183298e+13, 5.24996825e+13, 4.38557057e+13,
3.65066523e+13, 3.03564941e+13, 2.52451845e+13, 2.08642698e+13,
1.72384846e+13, 1.42445928e+13, 1.17708528e+13, 9.72632895e+12,
8.03623580e+12, 6.63640916e+12, 5.48087199e+12, 4.52670266e+12,
3.73898191e+12, 3.08846482e+12, 2.54707048e+12, 2.09413832e+12,
1.72035653e+12, 1.41009873e+12, 1.15590092e+12, 9.47563762e+11])}I_a = (1)
k = As_eds_cross_sections['K']['k_factor']
weight = As_eds_cross_sections['_element']['atomic_weight']
I_a*k*weight/(1*28+I_a*k*weight), (k/I_a)*weight/(1*28+k/I_a*weight), 1*28/(1*28+(1/1.3)*0.2703*56)(np.float64(0.9335227933722225),
np.float64(0.06685662799596748),
0.706291426708682)The above cross sections are for the excitation of a specific core-level of an atom. The sum of the experimental detected X-ray lines of one family are a fraction of those cross-sections.
Given the large discrepancy of excitation probabilities of the different core-levels, we use the lower-energy cross-sections if possible. Adding up the different families will result in sup-percent change in the total percentage of excitation probabilties.
Varambhia et al. used the linear thickness dependence of the signal to correlate the cross sections of EDS and EELS. This is an excellent way to callibrate the EDS system. And the correlation factor between EELS and EDS should be same for different X-Ray peak families of all elements. The similar method can be used to callibrate the cross section of ADF intensities.
k-Factor for relative composition¶
the k-factor for elements A and B is defined as:
[ionizations/e (atoms / cm)): ionization cross section
[X-rays/ ionization]: fluorescent yield = number of X-rays generated per ionization
[g/moles]: atomic weight
relative transition probability (a) of the same series of X-rays ()
detector efficiencies for lines. If we look back to the Ch4_14 and the generation of X-rays, we see that we eliminated the variables of the sample (areal density) , because they are the same.
rewriting above equation we see that:
We do not worry about the units. Since we are fitting all lines we can omit the relative transition probability.
The main uncertainties are: The cross section, and the fluorescent yield.
There are three customary reference elements for the k-factor:
silicon (semiconductor industrie and literature )
iron (materials science)
carbon (ThermoFisher seems to be using that)
Using silicon we need the factors and
Only for thicker samples we need the density and thickness for absorption corrections.
Atomic-resolution EDS¶
For atomic-resolution studies, an integration of the counts over a Voronoi cell in the EDX map gives a partial cross-section for that column, by analogy with the approach for ADF cross-sections (E et al., 2013). For comparison, the popularly used ζ-factor method defined by (Watanabe et al., 2003) is:
Where
is the raw x-ray counts,
ρ is the sample density in kg/m3,
t is the sample thickness,
is the weight fraction of element x.
In Eq. (2), the definition of sample density in the zeta-factor approach is in kg/m3 which makes absolute quantification of nanoparticles, in terms of “number of atoms”, cumbersome. Nonetheless, partial cross-section is very similar to the ζ-factor and can be equated to it using Eq. (4). Where M is the molar mass (in kg/mol) and NA is Avogadro’s constant.
Open a EDS spectrum file¶
Here the EDS.rto file in the example_data folder.
fileWidget = pyTEMlib.file_tools.FileWidget()
Loading...
datasets = fileWidget.datasets
dataset = fileWidget.selected_dataset
v = dataset.plot()Loading...
#dataset.energy_scale+= 250
dataset.energy_scale[-1]20220.0dataset.view_original_metadata()Core :
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type : double
value : 120
Detectors[SuperXG22].PulsePairResolutionTime :
type : double
value : 4.9999999999999998e-07
Detectors[SuperXG22].SpectrumBeginEnergy :
type : long
value : 120
Detectors[SuperXG23].BilatThresholdHi :
type : double
value : 0.0050390000000000001
Detectors[SuperXG23].CommercialName :
type : string
value : Super-X G2
Detectors[SuperXG23].DetectorConfigID :
type : string
value : 3101e35a-08ec-4b3d-9108-6aa5d0e49281
Detectors[SuperXG23].DistanceToSample :
type : double
value : 12.42
Detectors[SuperXG23].IncidentAngle :
type : double
value : 0.094596840000000001
Detectors[SuperXG23].KMax :
type : double
value : 180
Detectors[SuperXG23].KMin :
type : double
value : 120
Detectors[SuperXG23].PulsePairResolutionTime :
type : double
value : 4.9999999999999998e-07
Detectors[SuperXG23].SpectrumBeginEnergy :
type : long
value : 120
Detectors[SuperXG24].BilatThresholdHi :
type : double
value : 0.0050390000000000001
Detectors[SuperXG24].CommercialName :
type : string
value : Super-X G2
Detectors[SuperXG24].DetectorConfigID :
type : string
value : 3101e35a-08ec-4b3d-9108-6aa5d0e49281
Detectors[SuperXG24].DistanceToSample :
type : double
value : 12.42
Detectors[SuperXG24].IncidentAngle :
type : double
value : 0.094596840000000001
Detectors[SuperXG24].KMax :
type : double
value : 180
Detectors[SuperXG24].KMin :
type : double
value : 120
Detectors[SuperXG24].PulsePairResolutionTime :
type : double
value : 4.9999999999999998e-07
Detectors[SuperXG24].SpectrumBeginEnergy :
type : long
value : 120
Optics.MonoSpotSize :
type : string
value : <=11
StemMagnification :
type : double
value : 10000000
AcquisitionSettings :
encoding : uint16
bincount : 4096
StreamEncoding : uint16
Size : 2097152
import codecs
def open_Bruker(fname):
tree = ET.parse(fname)
root = tree.getroot()
spectrum_number = 0
i=0
image_count = 0
o_count = 0
tags = {}
for neighbor in root.iter():
#print(neighbor.attrib.keys())
if 'Type' in neighbor.attrib:
if 'verlay3' in neighbor.attrib['Type'] :
semImage = neighbor
#print(neighbor.attrib['Type'])
if 'Name' in neighbor.attrib:
print('\t',neighbor)
print('\t',neighbor.attrib['Type'])
print('\t',neighbor.attrib['Name'])
print('\t',neighbor.find("./ClassInstance[@Type='TRTSpectrumList']"))
#if 'TRTImageOverlay' in neighbor.attrib['Type'] :
if 'TRTCrossOverlayElement'in neighbor.attrib['Type'] :
if 'Spectrum' in neighbor.attrib['Name']:
#print(o_count)
o_count+=1
if 'overlay' not in tags:
tags['overlay']= {}
if 'image'+str(image_count) not in tags['overlay']:
tags['overlay']['image'+str(image_count)] ={}
tags['overlay']['image'+str(image_count)][neighbor.attrib['Name']] ={}
over = tags['overlay']['image'+str(image_count)][neighbor.attrib['Name']]
for child in neighbor.iter():
if 'verlay' in child.tag:
#print(child.tag)
pos = child.find('./Pos')
if pos != None:
over['posX'] = int(pos.find('./PosX').text)
over['posY'] = int(pos.find('./PosY').text)
#dd = neighbor.find('Top')
#print('dd',dd)
#print(neighbor.attrib)
if 'TRTImageData' in neighbor.attrib['Type'] :
#print('found image ', image_count)
dd = neighbor.find("./ClassInstance[@Type='TRTCrossOverlayElement']")
if dd != None:
print('found in image')
image = neighbor
if 'image' not in tags:
tags['image']={}
tags['image'][str(image_count)]={}
im = tags['image'][str(image_count)]
im['width'] = int(image.find('./Width').text) # in pixels
im['height'] = int(image.find('./Height').text) # in pixels
im['dtype'] = 'u' + image.find('./ItemSize').text # in bytes ('u1','u2','u4')
im['plane_count'] = int(image.find('./PlaneCount').text)
im['data'] = {}
for j in range( im['plane_count']):
#print(i)
img = image.find("./Plane" + str(i))
raw = codecs.decode((img.find('./Data').text).encode('ascii'),'base64')
array1 = np.frombuffer(raw, dtype= im['dtype'])
#print(array1.shape)
im['data'][str(j)]= np.reshape(array1,(im['height'], im['width']))
image_count +=1
if 'TRTDetectorHeader' == neighbor.attrib['Type'] :
detector = neighbor
tags['detector'] ={}
for child in detector:
if child.tag == "WindowLayers":
tags['detector']['window']={}
for child2 in child:
tags['detector']['window'][child2.tag]={}
tags['detector']['window'][child2.tag]['Z'] = child2.attrib["Atom"]
tags['detector']['window'][child2.tag]['thickness'] = float(child2.attrib["Thickness"])*1e-5 # stupid units
if 'RelativeArea' in child2.attrib:
tags['detector']['window'][child2.tag]['relative_area'] = float(child2.attrib["RelativeArea"])
#print(child2.tag,child2.attrib)
else:
#print(child.tag , child.text)
if child.tag != 'ResponseFunction':
if child.text !=None:
tags['detector'][child.tag]=child.text
if child.tag == 'SiDeadLayerThickness':
tags['detector'][child.tag]=float(child.text)*1e-6
#print(child.tag)
if child.tag == 'DetectorThickness':
tags['detector'][child.tag]=float(child.text)*1e-1
# ESMA could stand for Electron Scanning Microscope Analysis
if 'TRTESMAHeader' == neighbor.attrib['Type'] :
esma = neighbor
tags['esma'] ={}
for child in esma:
if child.tag in ['PrimaryEnergy', 'ElevationAngle', 'AzimutAngle', 'Magnification', 'WorkingDistance' ]:
tags['esma'][child.tag]=float(child.text)
if 'TRTSpectrum' == neighbor.attrib['Type'] :
if 'Name' in neighbor.attrib:
spectrum = neighbor
TRTHeader = spectrum.find('./TRTHeaderedClass')
if TRTHeader != None:
hardware_header = TRTHeader.find("./ClassInstance[@Type='TRTSpectrumHardwareHeader']")
spectrum_header = spectrum.find("./ClassInstance[@Type='TRTSpectrumHeader']")
#print(i, TRTHeader)
tags[spectrum_number] = {}
tags[spectrum_number]['hardware_header'] ={}
if hardware_header != None:
for child in hardware_header:
tags[spectrum_number]['hardware_header'][child.tag]=child.text
tags[spectrum_number]['detector_header'] ={}
tags[spectrum_number]['spectrum_header'] ={}
for child in spectrum_header:
tags[spectrum_number]['spectrum_header'][child.tag]=child.text
tags[spectrum_number]['data'] = np.fromstring(spectrum.find('./Channels').text,dtype='Q', sep=",")
spectrum_number+=1
return tags
fname = '../example_data/EDS.rto'
tags = open_Bruker(fname)
datasets = fileWidget.datasets
dataset += datasets['Channel_001']+datasets['Channel_002']+datasets['Channel_003']
dataset.energy_scale/= 2
dataset.energy_scale-= 250
v = dataset.plot()Loading...
spectrum1 =tags[0]['data']
spectrum2 =tags[1]['data']
offset = float(tags[0]['spectrum_header']['CalibAbs'])
scale = float(tags[0]['spectrum_header']['CalibLin'])
energy_scale1 = np.arange(len(spectrum1))*scale+offset
offset = float(tags[1]['spectrum_header']['CalibAbs'])
scale = float(tags[1]['spectrum_header']['CalibLin'])
energy_scale2 = np.arange(len(spectrum2))*scale+offset
plt.figure(figsize=(10,4))
ax1 = plt.subplot(1,2,1)
ax1.imshow(tags['image']['0']['data']['0'])
for key in tags['overlay']['image1']:
d = tags['overlay']['image1'][key]
ax1.scatter ([d['posX']], [d['posY']], marker="x", color='r')
ax1.text(d['posX']+5, d['posY']-5, key, color='r')
ax2 = plt.subplot(1,2,2)
plt.plot(energy_scale1,spectrum1, label = 'spectrum 1')
plt.plot(energy_scale2,spectrum2, label = 'spectrum 2')
plt.xlabel('energy [keV]')
plt.xlim(0,5)
plt.ylim(0)
plt.tight_layout()
plt.legend();
Loading...
Peak Finding¶
We can also use the peak finding routine of the scipy.signal to find all the maxima.
import scipy as sp
import scipy.signal as sig
spectrum1 = dataset
energy_scale1 = dataset.energy_scale.values
start = np.searchsorted(energy_scale1, 50)
## we use half the width of the resolution for smearing
width = int(np.ceil(125*1e-3/2 /(energy_scale1[1]-energy_scale1[0])/2)*2+1)
new_spectrum = sp.signal.savgol_filter(spectrum1[start:], width, 2) ## we use half the width of the resolution for smearing
new_energy_scale = energy_scale1[start:]
major_peaks, _ = sp.signal.find_peaks(new_spectrum, prominence=1000)
minor_peaks, _ = sp.signal.find_peaks(new_spectrum, prominence=30)
peaks = major_peaks
spectrum1 = np.array(spectrum1)
plt.figure()
plt.plot(energy_scale1,spectrum1, label = 'spectrum 1')
#plt.plot(energy_scale1,gaussian_filter(spectrum1, sigma=1), label = 'filtered spectrum 1')
plt.plot(new_energy_scale,new_spectrum, label = 'filtered spectrum 1')
plt.scatter( new_energy_scale[minor_peaks], new_spectrum[minor_peaks], color = 'blue')
plt.scatter( new_energy_scale[major_peaks], new_spectrum[major_peaks], color = 'red')
Loading...
plt.xlim(0,20000);Peak identification¶
Here we look up all the elemetns and see whether the position of major line (K-L3, K-L2’ or ‘L3-M5’) coincides with a peak position as found above.
Then we plot all the lines of such an element with the appropriate weight.
The positions and the weight are tabulated in the ffast.pkl file introduced in the Characteristic X-Ray peaks notebook
edge_info = pyTEMlib.eels_tools.get_x_sections(38)
edge_info['lines']{'K-L3': {'weight': 1.0, 'position': 14165.000000000002},
'K-L2': {'weight': 0.564, 'position': 14097.800000000003},
'K-M3': {'weight': 0.144, 'position': 15835.500000000002},
'K-N3': {'weight': 0.0328, 'position': 16084.700000000003},
'K-M2': {'weight': 0.0731, 'position': 15824.800000000003},
'K-M5': {'weight': 0.000368, 'position': 15971.500000000002},
'L3-M5': {'weight': 1.0, 'position': 1806.5},
'L3-M4': {'weight': 0.114, 'position': 1804.6},
'L3-M1': {'weight': 0.0412, 'position': 1582.1},
'L2-N3': {'weight': 0.00524, 'position': 1986.9},
'L2-M4': {'weight': 0.499, 'position': 1871.8000000000002},
'L2-M3': {'weight': 0.000116, 'position': 1737.7000000000003},
'L2-N1': {'weight': 0.00187, 'position': 1969.1000000000001},
'L2-M1': {'weight': 0.0261, 'position': 1649.3000000000002},
'L1-M3': {'weight': 0.0274, 'position': 1947.1999999999998},
'L1-M2': {'weight': 0.0394, 'position': 1936.4999999999998},
'L1-N2': {'weight': 0.000191, 'position': 2196.3999999999996},
'L1-N3': {'weight': 0.00341, 'position': 2196.3999999999996}}
plt.figure()
#plt.plot(energy_scale1,spectrum2, label = 'spectrum 1')
#plt.plot(energy_scale1,gaussian_filter(spectrum1, sigma=1), label = 'filtered spectrum 1')
plt.plot(new_energy_scale,new_spectrum, label = 'filtered spectrum 1')
plt.gca().axhline(y=0,color='gray', linewidth = 0.5);
out_tags = {}
out_tags['spectra'] = {}
out_tags['spectra'][0] = {}
out_tags['spectra'][0]['data'] = spectrum1
out_tags['spectra'][0]['energy_scale'] = energy_scale1
out_tags['spectra'][0]['energy_scale_start'] = start
out_tags['spectra'][0]['smooth_spectrum'] = new_spectrum
out_tags['spectra'][0]['smooth_energy_scale'] = new_energy_scale
out_tags['spectra'][0]['elements'] ={}
#print(ffast[6])
number_of_elements = 0
for peak in minor_peaks:
print(peak)
for z in range(5,82):
edge_info = pyTEMlib.eels_tools.get_x_sections(z)
if 'lines' not in edge_info:
pass
elif 'K-L3' in edge_info['lines']:
if abs(edge_info['lines']['K-L3']['position']- new_energy_scale[peak]) <10:
out_tags['spectra'][0]['elements'][number_of_elements] = {}
out_tags['spectra'][0]['elements'][number_of_elements]['element'] = edge_info['name']
out_tags['spectra'][0]['elements'][number_of_elements]['found_lines'] = 'K-L3'
out_tags['spectra'][0]['elements'][number_of_elements]['lines'] = edge_info['lines']
out_tags['spectra'][0]['elements'][number_of_elements]['experimental_peak_index'] = peak
out_tags['spectra'][0]['elements'][number_of_elements]['experimental_peak_energy'] = new_energy_scale[peak]*1e3
number_of_elements += 1
plt.plot([edge_info['lines']['K-L3']['position'], edge_info['lines']['K-L3']['position']], [0,new_spectrum[peak]], color = 'red')
plt.text(new_energy_scale[peak],0, edge_info['name']+'\nK-L3', verticalalignment='top')
for line in edge_info['lines']:
if 'K' in line:
if abs(edge_info['lines'][line]['position']-edge_info['lines']['K-L3']['position'])> 20:
if edge_info['lines'][line]['weight']>0.07:
#print(element, ffast[element]['lines'][line],new_spectrum[peak]*ffast[element]['lines'][line]['weight'])
plt.plot([edge_info['lines'][line]['position']/1000.,edge_info['lines'][line]['position']/1000.], [0,new_spectrum[peak]*edge_info['lines'][line]['weight']], color = 'red')
plt.text(edge_info['lines'][line]['position']/1000.,0, edge_info['name']+'\n'+line, verticalalignment='top')
elif 'K-L2' in edge_info['lines']:
if abs(edge_info['lines']['K-L2']['position']- new_energy_scale[peak]) <10:
plt.plot([new_energy_scale[peak],new_energy_scale[peak]], [0,new_spectrum[peak]], color = 'orange')
plt.text(new_energy_scale[peak],0, ffast[element]['element']+'\nK-L2', verticalalignment='top')
out_tags['spectra'][0]['elements'][number_of_elements] = {}
out_tags['spectra'][0]['elements'][number_of_elements]['element'] = ffast[element]['element']
out_tags['spectra'][0]['elements'][number_of_elements]['found_lines'] = 'K-L2'
out_tags['spectra'][0]['elements'][number_of_elements]['lines'] = ffast[element]['lines']
out_tags['spectra'][0]['elements'][number_of_elements]['experimental_peak_index'] = peak
out_tags['spectra'][0]['elements'][number_of_elements]['experimental_peak_energy'] = new_energy_scale[peak]*1e3
number_of_elements += 1
if 'lines' not in edge_info:
pass
elif 'L3-M5' in edge_info['lines']:
if abs(edge_info['lines']['L3-M5']['position']- new_energy_scale[peak]*1e3) <10:
pass
print('found_element', element,ffast[element]['lines']['L3-M5']['position'], new_energy_scale[peak] )
#plt.scatter( new_energy_scale[peak], new_spectrum[peak], color = 'blue')
out_tags['spectra'][0]['elements'][number_of_elements] = {}
out_tags['spectra'][0]['elements'][number_of_elements]['element'] = ffast[element]['element']
out_tags['spectra'][0]['elements'][number_of_elements]['found_lines'] = 'L3-M5'
out_tags['spectra'][0]['elements'][number_of_elements]['lines'] = ffast[element]['lines']
out_tags['spectra'][0]['elements'][number_of_elements]['experimental_peak_index'] = peak
out_tags['spectra'][0]['elements'][number_of_elements]['experimental_peak_energy'] = new_energy_scale[peak]*1e3
number_of_elements += 1
for element in out_tags['spectra'][0]['elements']:
print(out_tags['spectra'][0]['elements'][element]['element'],out_tags['spectra'][0]['elements'][element]['found_lines'])
Loading...
Using that we get for all peaks in the low energy region:
from scipy.interpolate import interp1d
import scipy.constants as const
tags={'detector': { 'window': {'0': {'Z': 6,
'thickness': 100}
},
'SiDeadLayerThickness': 20
}
}
def detector_efficiency(tags, energy_scale):
detector_Efficiency1 = np.ones(len(energy_scale))
for key in tags['detector']['window']:
Z = int(tags['detector']['window'][key]['Z'])
if Z < 14:
t = tags['detector']['window'][key]['thickness']
## interpolate mass absorption coefficient to our energy scale
lin = interp1d(ffast[Z]['E']/1000.,ffast[Z]['photoabsorption'],kind='linear')
mu = lin(energy_scale) * ffast[Z]['nominal_density']*100. #1/cm -> 1/m
detector_Efficiency1 = detector_Efficiency1 * np.exp(-mu * t)
print(Z,t)
t = float(tags['detector']['SiDeadLayerThickness'])*1e-6
print(t)
t = .30*1e-7
print(t)
lin = interp1d(ffast[14]['E']/1000.,ffast[14]['photoabsorption'],kind='linear')
mu = lin(energy_scale) * ffast[14]['nominal_density']*100. #1/cm -> 1/m
detector_Efficiency1 = detector_Efficiency1 * np.exp(-mu * t)
detector_thickness = float(tags['detector']['DetectorThickness'])*1e-1
## interpolate mass absorption coefficient to our energy scale
mu_Si = lin(energy_scale) * ffast[14]['nominal_density']*100. #1/cm -> 1/m
print(detector_thickness)
detector_Efficiency2 = (1.0 - np.exp(-mu * detector_thickness))# * oo4pi;
return detector_Efficiency1*detector_Efficiency2
detector_Efficiency = detector_efficiency(tags, new_energy_scale)
plt.figure()
plt.plot(new_energy_scale*1000, detector_Efficiency)
plt.xlim(0,2000)---------------------------------------------------------------------------
NameError Traceback (most recent call last)
Cell In[9], line 42
38 detector_Efficiency2 = (1.0 - np.exp(-mu * detector_thickness))# * oo4pi;
41 return detector_Efficiency1*detector_Efficiency2
---> 42 detector_Efficiency = detector_efficiency(tags, new_energy_scale)
44 plt.figure()
45 plt.plot(new_energy_scale*1000, detector_Efficiency)
Cell In[9], line 21, in detector_efficiency(tags, energy_scale)
18 t = tags['detector']['window'][key]['thickness']
20 ## interpolate mass absorption coefficient to our energy scale
---> 21 lin = interp1d(ffast[Z]['E']/1000.,ffast[Z]['photoabsorption'],kind='linear')
22 mu = lin(energy_scale) * ffast[Z]['nominal_density']*100. #1/cm -> 1/m
23 detector_Efficiency1 = detector_Efficiency1 * np.exp(-mu * t)
NameError: name 'ffast' is not definedPeaks = []
elements_peaks = []
intensities = []
for element in out_tags['spectra'][0]['elements']:
el_dict = out_tags['spectra'][0]['elements'][element]
position = el_dict['lines'][el_dict['found_lines']]['position']
weight = 0
for line in el_dict['lines']:
if abs(position - el_dict['lines'][line]['position'])<20:
weight += el_dict['lines'][line]['weight']
index = np.searchsorted(new_energy_scale,el_dict['lines'][el_dict['found_lines']]['position']/1000.)
intensity = new_spectrum[index]/weight
added_peaks = np.zeros(len(new_energy_scale))
for line in el_dict['lines']:
if line[0] == el_dict['found_lines'][0]:
if el_dict['lines'][line]['weight']> 0.01:
p = getPeak(el_dict['lines'][line]['position']/1000.,new_energy_scale)*el_dict['lines'][line]['weight']
Peaks.append(p)
added_peaks = added_peaks + p
elements_peaks.append(added_peaks)
intensities.append(intensity)
plt.figure()
plt.plot(new_energy_scale,new_spectrum, label = 'filtered spectrum 1',color='red')
for i in range(len(elements_peaks)):
plt.plot(new_energy_scale,elements_peaks[i]*intensities[i], label = f'Peak{i}')
pass
peaks = np.array(Peaks)
plt.plot(new_energy_scale,peaks.sum(axis=0), label = f'Peaks', color = 'blue')
print(peaks.shape)
#plt.xlim(0,5)
offset = float(tags[0]['spectrum_header']['CalibAbs'])
scale = float(tags[0]['spectrum_header']['CalibLin'])
energy_scale1 = np.arange(len(spectrum1))*scale+offset
p = [1, 37, .3, offset, scale]
E_0= 20
E = new_energy_scale
N = detector_Efficiency * (p[0] + p[1]*(E_0-E)/E + p[2]*(E_0-E)**2/E)
N = de * (p[0] + p[1]*(E_0-E)/E + p[2]*(E_0-E)**2/E)
pp = p[0:5].copy()
for i in range(len(elements_peaks)):
pp.append(intensities[i])
plt.plot(new_energy_scale,N, color= 'orange')
plt.xlim(0,4)
detector_Efficiency = de---------------------------------------------------------------------------
NameError Traceback (most recent call last)
Cell In[10], line 18
16 if line[0] == el_dict['found_lines'][0]:
17 if el_dict['lines'][line]['weight']> 0.01:
---> 18 p = getPeak(el_dict['lines'][line]['position']/1000.,new_energy_scale)*el_dict['lines'][line]['weight']
19 Peaks.append(p)
20 added_peaks = added_peaks + p
NameError: name 'getPeak' is not defined
def model(p,energy_scale):
E = energy_scale
spectrum = detector_Efficiency * (p[0] + p[1]*(E_0-E)/E + p[2]*(E_0-E)**2/E)
for i in range(5,len(p)):
spectrum = spectrum+elements_peaks[i-5]*abs(p[i])
return spectrum
def getFWHM(E):
return np.sqrt(2.5*(E-E_ref)+FWHM_ref**2)
def gaussian(enrgy_scale, mu, FWHM):
sig = FWHM/2/np.sqrt(2*np.log(2))
return np.exp(-np.power(enrgy_scale - mu, 2.) / (2 * np.power(sig, 2.)))
def getPeak(E, energy_scale):
E_ref = 5895.0
FWHM_ref = 136 #eV
FWHM = np.sqrt(2.5*(E*1e3-E_ref)+FWHM_ref**2)*1e-3
return gaussian(energy_scale, E, FWHM)
Peaks = []
elements_peaks = []
intensities = []
for element in out_tags['spectra'][0]['elements']:
el_dict = out_tags['spectra'][0]['elements'][element]
position = el_dict['lines'][el_dict['found_lines']]['position']
weight = 0
for line in el_dict['lines']:
if abs(position - el_dict['lines'][line]['position'])<20:
weight += el_dict['lines'][line]['weight']
index = np.searchsorted(new_energy_scale,el_dict['lines'][el_dict['found_lines']]['position']/1000.)
intensity = new_spectrum[index]/weight
added_peaks = np.zeros(len(new_energy_scale))
for line in el_dict['lines']:
if line[0] == el_dict['found_lines'][0]:
if el_dict['lines'][line]['weight']> 0.01:
p = getPeak(el_dict['lines'][line]['position']/1000.,new_energy_scale)*el_dict['lines'][line]['weight']
Peaks.append(p)
added_peaks = added_peaks + p
elements_peaks.append(added_peaks)
intensities.append(intensity)
p = [1, 37, .3, offset, scale]
E_0= 20
E = new_energy_scale
N = detector_Efficiency * (p[0] + p[1]*(E_0-E)/E + p[2]*(E_0-E)**2/E)
pp = p[0:5].copy()
for i in range(len(elements_peaks)):
pp.append(intensities[i])
spectrum3 = model(pp[:-1],new_energy_scale)
plt.figure()
plt.plot(new_energy_scale,new_spectrum, label = 'filtered spectrum 1',color='red')
plt.plot(new_energy_scale, spectrum3)
plt.xlim(0,4)---------------------------------------------------------------------------
NameError Traceback (most recent call last)
Cell In[11], line 47
43 elements_peaks.append(added_peaks)
44 intensities.append(intensity)
---> 47 p = [1, 37, .3, offset, scale]
48 E_0= 20
49 E = new_energy_scale
NameError: name 'offset' is not defined
from scipy.optimize import leastsq
## background fitting
def specfit(p, y, x):
err = y - model(p,x)
return err
p, lsq = leastsq(specfit, pp[:-1], args=(new_spectrum, new_energy_scale), maxfev=2000)
print( f'element\t end \t start ')
for i, element in enumerate(out_tags['spectra'][0]['elements']):
if i<len(p)-5:
el_dict = out_tags['spectra'][0]['elements'][element]
print(f"{el_dict['element']:2s}: \t {abs(p[i+5]):6.2f} \t {pp[i+5]:.2f}")
#print(p,pp)element end start
C : 16487.64 16444.49
O : 3267.00 3274.63
Mg: 2831.85 3127.80
Al: 84.40 334.63
P : 5267.50 5499.66
S : 82.35 234.96
K : 2526.57 2693.25
spectrum3 = model(p,new_energy_scale)
plt.figure()
plt.plot(new_energy_scale,new_spectrum, label = 'filtered spectrum 1',color='red')
plt.plot(new_energy_scale, spectrum3)
plt.plot(new_energy_scale, new_spectrum-spectrum3)
plt.gca().axhline(y=0,color='gray');
plt.xlim(0,4);Loading...
Energy Scale¶
What happened?
Composition Basis¶
Castaing’s first approximation:¶
= Concentration of element
= X-ray intensity of element in the sample
= X-ray intensity of a pure sample of the element
We call the ratio: k-ratio
Stainless to Fe-Ka ratio = 0.745
Stainless to Cr-Ka ratio = 0.208
Stainless to Ni-Ka ratio = 0.080
therefore:
Fe: 70.8 wt%
Cr: 17.2 wt%
Ni: 9.1 wt%
ZAF Correction¶
While the k-ratio is a good starting point, the basic approximation does not take absorption and interaction between the different elements into account. To increase the precision the following corrections are applied:
Z: Atomic number factor
A: sample Absorption
F: Fluorescence
Atomic Number Factor - Z¶
The stopping power is equal to the energy loss of the electron per unit length it travels in the material:
D.C. Joy, Microsc. Microanal. 7(2001): 159-169
: electron energy
: density of material
: Atomic number
: atomic weight of material
: Mean ionizaton potential
The backscatter correction is the relation between how many photons are generated with and without backscattering.
: backscatter correction
: Mean atomic number of sample
: electron energy
: critical ionization energy
Absorption Correction¶
The numerator describes how many X-rays are generated at a given depth and scales the number by the attenuation due to the escape path in the sample.
The denominator is the total number of X-rays generated without attenuation.
This means that A is a normalized value that describes how many of the generated X-rays actually escaped the sample.
Fluorescence Correction¶
The high energy X-ray can originate from characteristic X-rays from elements with higher energy lines or from the bremsstrahlung. This means that even the element with the highest energy line can have a fluorescence correction.
ZAF assumptions¶
Several assumptions are made either directly or indirectly in the equations used to calculate the composition of a sample.
The sample is homogenous.
The density 𝜌 and the mass attenuation 𝜇/𝜌 is assumed to be constant.
The sample is flat.
𝑧 is well defined for any given position and no additional absorption takes place after the X-ray reaches the sample surface.
The sample is infinitely thick as seen by the electron beam.
-The energy of the incident electron is deposited in the sample and only secondary and backscatter electrons escape.
Composition Standardless¶
For standard-less analysis the k-ratio is either calculated in the software or based on internal standards.
Following
to calculate the mass concentration from the intensity of a line (), we use:
: fluorescence yield
: Avogadro’s number
: density
: mass concentration of element
:atomic weight
: backscatter loss
: ionization cross section
: rate of energy loss
: incident beam energy
: excitation energy
: EDS efficiency
where: : volume density of element (atoms per unit volume)
What do we know at this point?
and
def BrowningEmpiricalCrossSection(elm , energy):
""" * Computes the elastic scattering cross section for electrons of energy between
* 0.1 and 30 keV for the specified element target. The algorithm comes from<br>
* Browning R, Li TZ, Chui B, Ye J, Pease FW, Czyzewski Z & Joy D; J Appl
* Phys 76 (4) 15-Aug-1994 2016-2022
* The implementation is designed to be similar to the implementation found in
* MONSEL.
* Copyright: Pursuant to title 17 Section 105 of the United States Code this
* software is not subject to copyright protection and is in the public domain
* Company: National Institute of Standards and Technology
* @author Nicholas W. M. Ritchie
* @version 1.0
*/
Modified by Gerd Duscher, UTK
"""
#/**
#* totalCrossSection - Computes the total cross section for an electron of
#* the specified energy.
#*
# @param energy double - In keV
# @return double - in square meters
#*/
e = energy #in keV
re = np.sqrt(e);
return (3.0e-22 * elm**1.7) / (e + (0.005 * elm**1.7 * re) + ((0.0007 * elm**2) / re));
Summary¶
EDS quantification is not magic and the results will highly depend on sample preparation and correct peak identification.
Correct quantification with ZAF relies on the sample being:
Flat
Homogenous
Infinitely thick to the electron beam
Use of standards is not required for good quantification but it will give additional information and increase quality of the results.
Peak-to-background based ZAF is good for rough samples but requires better spectra than regular ZAF.
The best way to good EDS results:
Good samples
Thorough sample preparation
Exact SEM Parameters
Back: Detector Response¶
List of Content: Front¶
- Scanning Transmission Electron Microscopy. (2011). Springer New York. 10.1007/978-1-4419-7200-2
- MacArthur, K. E., Slater, T. J. A., Haigh, S. J., Ozkaya, D., Nellist, P. D., & Lozano-Perez, S. (2016). Quantitative Energy-Dispersive X-Ray Analysis of Catalyst Nanoparticles Using a Partial Cross Section Approach. Microscopy and Microanalysis, 22(1), 71–81. 10.1017/s1431927615015494
- MacArthur, K. E., Brown, H. G., Findlay, S. D., & Allen, L. J. (2017). Probing the effect of electron channelling on atomic resolution energy dispersive X-ray quantification. Ultramicroscopy, 182, 264–275. 10.1016/j.ultramic.2017.07.020
- Varambhia, A., Jones, L., London, A., Ozkaya, D., Nellist, P. D., & Lozano-Perez, S. (2018). Determining EDS and EELS partial cross-sections from multiple calibration standards to accurately quantify bi-metallic nanoparticles using STEM. Micron, 113, 69–82. 10.1016/j.micron.2018.06.015
- Joy, D. C. (2001). Fundamental Constants for Quantitative X-ray Microanalysis. Microscopy and Microanalysis, 7(2), 159–167. 10.1007/s100050010070
- Newbury, D. E., Swyt, C. R., & Myklebust, R. L. (1995). “Standardless” Quantitative Electron Probe Microanalysis with Energy-Dispersive X-ray Spectrometry: Is It Worth the Risk? Analytical Chemistry, 67(11), 1866–1871. 10.1021/ac00107a017