Transcript Introduction to Electron Diffraction

Quantitative X-ray
Spectrometry
in TEM/STEM
Charles Lyman
Lehigh University
Bethlehem, PA
Based on presentations developed for Lehigh University
semester courses and for the Lehigh Microscopy School
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Quantitative X-ray Analysis of Thin Specimens
How much of each element is present?

Aim of quantitative analysis: to transform the intensities in the X-ray
spectrum into compositional values, with known precision and accuracy

Cliff-Lorimer method:
CA
I
 kAB A
CB
IB
and CA  CB  1
CA = concentration of element A
IA = x-ray intensity from element A
kAB= Cliff-Lorimer sensitivity factor


Precision: collect at least 10,000 counts in the smallest peak to obtain a
counting error of less than 3%
Accuracy: measure kAB on a known standard and find a way to handle x-ray
absorption effects
What could be simpler?
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Assumptions in Cliff-Lorimer Method

Basic assumptions
» X-ray intensities for each element are measured simultaneously
» Ratio of intensities accounts for thickness variations
» Specimen is thin enough that absorption and fluorescence can be ignored
– the “thin-film criterion”
– We would like to handle absorption in a better way!

Cliff-Lorimer equation:
CA
I
 kAB A
CB
IB
and CA  CB  1
» CA and CB are weight fractions or atomic fractions (choose one, be consistent)
» kAB depends on the particular TEM/EDS system and kV (use highest kV)
– k-factor is most
closely related to the atomic number correction
» Can expand to measure ternaries, etc. by measuring more k-factors
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Steps in Quantitative Analysis




Remove background intensity under peaks
Integrate counts in peaks
Determine k-factors (or z-factors)
Correct for absorption (if necessary)
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Calculate Background, the Subtract

Gross-Net Method
»
»
»

Three-Window Method
»
»
»
»

Draw line at ends of window
covering full width of peak
Impossible with peak overlap
Should work better above 2
keV where background
changes slowly
Set window with FWHM (or even better
1.2 FWHM)
Average backgrounds B1 and B2
Subtract Bave from peak
Requires well-separated peaks
Background Modeling
»
»
Mathematical model of background
as function of Z and E
Useful when peaks are close
together
from Williams and Carter, Transmission
Electron Microscopy, Springer, 1996
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Digital Filtering

Convolute spectrum
with “top-hat” filter
» Multiply channels of
top-hat filter times
each spectrum
channel
» Place result in central
channel
» Step filter over each
spectrum channel

Background becomes
zero
from Williams and Carter, Transmission
Electron Microscopy, Springer, 1996
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Digital Filtering
Spectrum before filtering
Note MgK, AlK, and SiK
Spectrum after filtering
Positive lobes are
proportional to peak
intensities
from Williams and Carter, Transmission
Electron Microscopy, Springer, 1996
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Obtaining k-factors

Requirements for standard specimen for k-factor measurement
»
Single phase (stoichiometric composition helpful)
» Homogeneous at the nanometer scale
» Thinned to electron transparency without composition change (microtome)
» Insensitive to beam damage

Measure k-factors on a known standard: k  CA IB
AB
CB IA
»
Usually kASi or kAFe
» Measure k-factors at various thicknesses and extrapolate to zero thickness

Other ways

»
Calculate k-factors (when standards are not available)
» Use literature values at same kV for x-rays 5-15keV (not recommended)
» Use kAB = kAC/kBC (use only when necessary - errors add)
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Why Collect 10,000 Counts?

There is a 99% chance that a single measurement is within
3N1/2 of the true value

3N 1/ 2
100%
The relative counting error 
=
N

Thus, for 10,000 counts the relative counting error =

310,0001/ 2
 

 10,000

x100%  3%



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Experimental k-factors
from Williams and Carter, Transmission Electron Microscopy, Springer, 1996
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Calculated k-factors

When suitable standard is not available
» When a modestly accurate analysis is acceptable

Most EDS system software can calculate k-factors
» But errors can be up to 20%

Simple expression:
QaB AA A
kAB 
QaA AB B

but Q not known well
which leads to error
Q = ionization cross-section
 = fluorescence yield
Intensity K
a = relative transition probability = Intensity (K  K )
A = atomic weight
 = detector efficiency

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Calculated k-factors
Calculated kAFe-factors using different ionization cross-sections
from Williams and Carter, Transmission Electron Microscopy, Springer, 1996
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kAFe for K-series

Errors of calculated versus standards ~ 4%
from Williams and Carter, Transmission Electron Microscopy, Springer, 1996
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kAFe for L-series

Errors of calcuated versus standards up to 20%
from Williams and Carter, Transmission Electron Microscopy, Springer, 1996
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The Absorption Problem

k-factors measured at different
specimen thicknesses will be
different
»
»
»

X-rays from some elements will be
absorbed more than others
“Thin-film criterion” breaks down if high
accuracy required
We need a better way to handle
absorption effects
What to do:
Measure unknown and standard at the
same thickness (impractical)
2. Extrapolate all k-factors to zerothickness, then apply absorption
correction to each measurement (but
we need to know the specimen
thickness)
3. Use z-factors
1.
from Williams and Carter, Transmission
Electron Microscopy, Springer, 1996
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Extrapolate to the Zero-Thickness k-factor
Horita et al. (1987) and van Cappellan (1990) methods
Zero-thickness k-factor
from Williams and Carter, Transmission Electron Microscopy, Springer, 1996
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Obtaining the Zero-thickness k-factor
Thin standard of
known composition
Pt-13wt% Rh
thermocouple wire
CPt IRh 0.87 I Rh
kPtRh 

CRh I Pt 0.13 I Pt
kP tRh  1.079
Thickness measured by
EELS log-ratio method
R. E. Lakis, C. E. Lyman, and H. G. Stenge r, J. Catal. 154 (1995 ) 261-275.
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Absorption Correction

Effective sensitivity factor
kAB* = kAB(ACF)
Zero-thickness k-factor
Equation 35.29:
  B

 A 
  1 exp  t cosec 


 Spec 
  Spec

ACF   B 
  A

  

   1 exp


t
cosec





  



 Spec 
  Spec

from Williams and Carter, Transmission Electron Microscopy, Springer, 1996
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Original z-factor method
Absorption correction contains t

»
foil thickness t must be determined at analysis point
» specimen density  for composition at analysis point)
z-factor method

assume x-ray intensity  t
» then
IA
»
t  z A
»


CA
subsititute into absorption equation:
  B

 A 
I 
A
  1 exp  z A  cosec 
IA  Spec 


CA
  Spec IB 

 kAB   B 

A


CB
I


I 
 
 B  


A



1
exp

z
c
osec









A
  




IB 


 Spec 
Spec


We can determine both absorption-corrected compositions
and t if kAB and zA known from measurements on standard
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Modified z-factor method

Measure the z-factor for both elements:
IA
t  z A
CA



IB
t  z B
CB
Assume CA + CB = 1 for binary system and rearrange:
z AI A
z BI B
CA 
, CB 
, t  z AI A  z BI B
z AI A  z BI
z AI A  z BI B
B

Determine CA, CB, and t simultaneously from three equations in three
unknowns
t can be determined if density is known
M. Watanabe and D.B. Williams, Z. Metalkd. 94 (2003) 307-316
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z-factor
z factor is dependent on
• x-ray energy
• accelerating voltage
• beam current
z factor is independent of
• specimen thickness
• specimen composition
• specimen density
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Quantitative analysis by z factor method
Lucadamo et al. (1999)
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Effect of kV on Beam Spreading

Elastic scattering
broadens the beam as it
traverses the specimen

Beam broadening is less
for
» Higher kV
b
» Lighter materials
» Smaller thicknesses

Goldstein-Reed Eqn.
5 Z
b  7.21x10
E0
 t

1 /2
3
2
A
from Williams and Carter, Transmission
Electron Microscopy, Springer, 1996
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b
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Spatial Resolution vs. Analytical Sensitivity
Conditions that favor high spatial resolution (thinnest specimen) result in poorer analytical
sensitivity and vice versa. For example to obtain equivalent analytical sensitivity in an AEM to an
EPMA, the X-ray generation and detection efficiency would have to be improved by a factor of 10 8
from Williams and Carter, Transmission Electron Microscopy, Springer, 1996
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Composition Profiles Across an Interphase Interface
The change in Mo and Cr composition across the interface can be used to determine the
compositions of the phases either side of the interface which, in turn, give the tie lines on
the Ni-Cr-Mo phase diagram.
Courtesy R. Ayer
from Williams and Carter, Transmission Electron Microscopy, Springer, 1996
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Measurement of Low T Diffusion Data
Low-temp data
High-temp data
Measurement of composition profiles with high spatial resolution permits extraction of lowtemperature diffusion data because the small diffusion distances at low T are detectable by AEM X-ray
microanalysis. Here Zn profiles across a 200 nm wide precipitate-free zone in Al-Zn are used to
determine values of the Zn diffusivity at T = 100-200°C.
Courtesy A.W. Nicholls
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from Williams and Carter, Transmission Electron Microscopy, Springer, 1996
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Predicted Phase Separation Observed in Nanoparticles
Two phases observed
Pt-rich phase
Rh-rich phase
Dotted misibility gap was predicted
from other similar systems --> only
observed in nanoparticles
C. E. Lyman, R. E. Lakis, and H. G. Stenger, Ultramicroscopy 58 (1995) 25-34.
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Summary

Know the question you are trying to answer

Know the precision and accuracy required to answer the
question

Accumulate enough counts in the spectrum to achieve the
required precision (> 10,000 counts in the smallest peak)

Know the precision and accuracy of your k-factor

Measure zero-thickness k-factors and apply an absorption
correction (need t at analysis point) or use z-factors where t is not
needed

Spatial resolution vs. detectability:
» You cannot achieve the highest spatial resolution and the best
analytical sensitivity under the same experimental conditions
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