EPMA: Quantitative analysis
WDS spectrum:
Intensity is proportional
to concentration
Ci
Ii

C( i ) I ( i )
where,
Ii
 ki
I (i )
Ci
 ki  [ZAF ]i
C( i )
Ci and C(i): concentration of element ‘i’ in sample and standard
Ii and I(i): measured X-ray intensities of element ‘i’ in sample and standard
ki : k-ratio of element ‘i’
ZAF : matrix corrections
Matrix (ZAF) corrections
Z : atomic number correction
A : absorption correction
F : fluorescence correction
Atomic number (Z) correction
R =  CjRj
Ri
Si
Zi  *
Ri
*
Si
* sample
[ R = #X-ray photons
generated / #photons if
there were no back-scatter]
S =  CjSj
[ S = -(1/)(dE/ds),
stopping power]
Z, a function of E0 and composition
Duncumb-Reed-Yakowitz method:
Ri = j CjRij
Rij = R'1 - R'2ln(R'3Zj+25)
R'1 = 8.73x10-3 U3 - 0.1669 U2 + 0.9662 U + 0.4523
R'2 = 2.703x10-3 U3 - 5.182x10-2 U2 + 0.302 U - 0.1836
R'3 = (0.887 U3 - 3.44 U2 + 9.33 U - 6.43)/U3
Si = j CjSij
Sij = (const) [(2Zj/Aj )/(E0+Ec)]ln[583(E0+Ec)/Jj ]
where, J (keV) = (9.76Z + 58.82Z-0.19)x10-3
Z, a function of E0 and composition
Al-Cu alloy
1.2
1.15
1.1
1.1
1.05
1.05
1
CCu
0.95
0.1
0.3
0.5
0.7
0.9
0.9
0.85
0.8
10
15
Pure metal standards
1.15
ZCuK
ZAlK
1.2
Pure metal standards
1
CCu
0.95
0.1
0.3
0.5
0.7
0.9
0.9
0.85
0.8
20
10
E0 (keV)
1.2
CuAl2 standard
1.15
1.05
1.05
ZCuK
1.1
CCu
0.95
0.1
0.3
0.5
0.7
0.9
0.9
0.85
0.8
10
15
E0 (keV)
CuAl2 standard
1.15
1.1
1
20
E0 (keV)
1.2
ZAlK
15
1
CCu
0.95
0.1
0.3
0.5
0.7
0.9
0.9
0.85
0.8
20
10
15
E0 (keV)
20
X-ray absorption
I = I0 exp-(/)(x) = I0 exp-(/)( z cosec)
I: Intensity emitted; I0: Intensity generated
/: mass absorption coefficient
: density; z: depth; : take-off angle
Mass absorption
 )
(
coefficient, 
energy
absorber
A function of energy being absorbed
Energy
(keV)
CoK
6.925
NiK
7.472
CuK
8.041
ZnK
8.632
Ec(K-shell)
(keV)
8.331

(
)
(cm 2/g)
53
60
49
311
ZnK is absorbed by Ni
For compounds,


energy
(  )compound   (  )energy
element 'j' C j
j
energy
Ni
Absorption (A) correction
f ( i )
A i 
*
f ( i )
* sample
Absorption function,
f(i) = Ii(emitted)/Ii(generated)
A, a function of E0,  and
composition
Philibert method:

i

f (  i )  1 
  i
where,
i =
(  )
i  energy
specimen

 i hi 
1 

  i 1  hi 
cosec 
hi = 1.2Ai/Zi2
i = 4.5x105 / (E01.65-Ei(c)1.65)
For compounds:
hi   h j C j
j
(  )
i  energy
specimen
  (
j
)
i  energy
element 'j'
Cj
1
A, a function of E0,  and
composition
ANiK in Fe-Ni alloy
1.7
1.6
1.6
CFe
0.1
0.3
0.3
0.5
0.5
0.7
1.4
0.1
1.5
ANiK at 40o
ANiK at 15.5o
1.5
0.9
1.3
CFe
1.4
0.3
0.7
0.9
1.3
0.1
1.5
ANiK at 52.5o
1.6
CFe
1.2
0.5
1.4
0.7
0.9
1.3
1.2
1.2
1.1
1.1
1
1.1
1
10
15
20
E0 (keV)
25
30
1
10
15
20
E0 (keV)
25
30
10
15
20
E0 (keV)
25
30
Total X-ray intensity:
generated versus emitted
Generated intensity

0
I gen   (z )   ( z )d ( z )
Emitted intensity
I emit  I gen exp
  z

0
  (z )   ( z ) exp   z d ( z )
whe re,   (   )cosec
Generated and emitted X-ray intensity
variation with depth: AlK in Al-Cu alloy
Effect of matrix (mainly Cu) on the emitted intensity of
AlK, which is highly absorbed in Cu;
(  )
AlKα
Cu
= 4837.5 cm2/g
(z) matrix correction
The combined atomic number and absorption corrections
is the ratio of emitted intensities
in standard (I emit) to sample (I *emit)

  z


(

z)

(

z
)
exp
d ( z )
I emit
0
*   (z )  * ( z ) exp   z d ( z )
I emit
0


0
  i z
d ( z )
 i ( z )exp
Zi Ai 
  i z
 *
d ( z )
0 i ( z )exp
(z) matrix correction
0 : the value of
(z) at z=0
Rm : the depth at
which (z) is
maximum (m)
Rx : the maximum
depth of X-ray
production (X-ray
range)
ZiAi is modeled in terms of 0, Rm, Rx and the integral of the (z)
function (Pouchou and Pichoir: PAP method)
(z) dependence on Z and E0

Rm and Rx
decrease with
increase in atomic
number, Z
• increase with
beam energy, E0
(0) increases with
beam energy, E0
•

X-ray fluorescence
A consequence of X-ray absorption when
Eabsorbed X-ray > Ec(absorber shell)
Absorber
EK
EK
Ec(K)
(  )
(Atomic No.)
Mn(25)*
Fe(26)*
Co(27)
Ni(28)
Cu(29)
(keV)
5.895
6.4
6.925
7.472
8.041
(keV)
6.492
7.059
7.649
8.265
8.907
(keV)
6.537
7.111
7.709
8.331
8.98
(cm2/g)
344
380
53
59
65.5
NiK
Absorber
* NiK fluoresces Mn and Fe
Element
Mn
Fe
Co
Ni
Cu
Radiation causing fluorescence
FeK, CoK, CoK, NiK, NiK, CuK, CuK
CoK, NiK, NiK, CuK, CuK
NiK, CuK, CuK
CuK
none
Characteristic fluorescence (F)
correction


Ifij : intensity of X-ray ‘i’


Ii : intensity of X-ray ‘i’


f
1   I ij I i 
j


Fi 
*


f
1   I ij I i 
j


* sample
fluoresced by element j
generated by electron
beam
Fluorescence correction for an element includes the
summation of fluoresced intensities by other
elements in the compound
F, a function of E0 and composition
Castaing-Reed method:
I fij /Ii = CjY0Y1Y2Y3Pij
Y0 = 0.5[(ri-1)/ri][ j Ai/Aj]
 j : fluorescent yield
Y1 = [(Uj-1)/(Ui-1)]1.67
Y2 = (/)ji/(/)jspec
Y3 = [ln(1+u)]/u + [ln(1+v)]/v
u = [(/)ispec/(/)jspec] cosec 
v = 3.3x105/[(E01.65-Ec1.65)(/)jspec]
Pij=1 for K fluorescing K;
4.76 for K fluorescing L;
0.24 for L fluorescing K
F, a function of E0 and composition
1.1
1.1
1
1
1
0.9
CFe
0.1
0.3
0.8
0.9
CFe
0.1
0.3
0.8
0.5
0.5
0.7
0.7
0.9
0.9
0.7
15
20
25
30
10
15
20
25
30
10
0.5
0.9
1.3
1.2
1
1.4
E0 (keV)
30
25
30
0.3
0.7
0.9
1.3
1.2
0.5
1.4
0.7
0.9
1.3
1.2
1.1
1
25
30
0.1
1.5
1.1
1.1
25
CFe
ANiK at 52.5o
0.5
ANiK at 40o
0.3
0.7
20
E0 (keV)
0.1
1.5
20
15
1.6
0.3
15
0.5
CFe
0.1
10
0.3
0.8
0.9
1.6
CFe
1.4
0.1
E0 (keV)
1.7
1.5
CFe
0.7
E0 (keV)
1.6
0.9
0.7
0.7
10
ANiK at 15.5o
FFeK at 52.5o
1.1
FFeK at 40o
FFeK at 15.5o
FFeK in Fe-Ni alloy
1
10
15
20
25
30
E0 (keV)
ANiK in Fe-Ni alloy
10
15
20
E0 (keV)
Matrix correction flowchart
Concentrations are calculated through an iterative procedure:
k ------------ ZAF1 ----------- C1 ( = k * ZAF1)
C1 ----------- ZAF2 ----------- C2 ( = k * ZAF2)
(if C2 = C1, stop here)
C2 ----------- ZAF3 ----------- C3 ( = k * ZAF3)
(if C3 = C2, stop here)
and so on....
MIT OpenCourseWare
http://ocw.mit.edu
12.141 Electron Microprobe Analysis
January (IAP) 2012
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