EPMA: Quantitative analysis WDS spectrum: Intensity is proportional to concentration
Ci I i Ii where, ki C i I i )()( 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 [ R = #X-ray photons S i generated / #photons if Z i * there were no back-scatter] Ri * S = C S S i j j [ S = -(1/)(dE/ds), * sample stopping power] Z, a function of E0 and composition
Duncumb-Reed-Yakowitz method:
Ri = j CjRij
Rij = R' 1 - R'2 ln(R'3 Zj+25)
-3 3 2 R'1 = 8.73x10 U - 0.1669 U + 0.9662 U + 0.4523 -3 3 -2 2 R'2 = 2.703x10 U - 5.182x10 U + 0.302 U - 0.1836 3 2 3 R'3 = (0.887 U - 3.44 U + 9.33 U - 6.43)/U
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 1.2 Pure metal standards Pure metal standards 1.15 1.15 1.1 1.1 1.05 1.05 1 1 AlK
C CuK C 0.95 Cu 0.95 Cu Z 0.1 0.1 Z 0.9 0.3 0.9 0.3 0.5 0.5 0.85 0.7 0.85 0.7 0.9 0.9 0.8 0.8 10 15 20 10 15 20
E0 (keV) E0 (keV) 1.2 1.2 CuAl standard CuAl standard 1.15 2 1.15 2 1.1 1.1 1.05 1.05 1 1 AlK
C CuK C 0.95 Cu 0.95 Cu Z 0.1 0.1 Z 0.9 0.3 0.9 0.3 0.5 0.5 0.85 0.7 0.85 0.7 0.9 0.9 0.8 0.8 10 15 20 10 15 20
E0 (keV) E0 (keV) X-ray absorption
-(/)(x) -(/)( z cosec) I = I0 exp = I0 exp
I: Intensity emitted; I0: Intensity generated /: mass absorption coefficient : density; z: depth; : take-off angle energy Mass absorption coefficient, )( absorber
A function of energy being absorbed
energy )( Ni Energy Ec(K- shell) (keV) (keV) (cm 2/g) CoK 6.925 53 NiK 7.472 8.331 60 CuK 8.041 49 ZnK 8.632 311
ZnK is absorbed by Ni For compounds, energy energy C compound )()( element 'j' j j Absorption (A) correction
Absorption function, f i )( Ai * f i )( f(i) = Ii(emitted)/Ii(generated) * sample A, a function of E0, and composition
Philibert method: 1 h i i i f i 11)( i i 1 hi
i energy )( specimen where, i = cosec 2 hi = 1.2Ai/Zi
5 1.65 1.65 i = 4.5x10 / (E0 -Ei(c) )
For compounds:
i Chh jj j
i energy i energy C specimen )()( element 'j' j j A, a function of E0, and composition
ANiK in Fe-Ni alloy
1.7 1.6 1.6
CFe CFe CFe 1.6 0.1 1.5 0.1 1.5 0.1 0.3 0.3 0.3 1.5 0.5 0.5 0.5 o o 1.4 1.4 o 0.7 0.7 0.7 1.4 0.9 0.9 0.9 at 40 1.3 1.3 at 52.5 at 15.5 1.3 NiK NiK NiK A A A 1.2 1.2 1.2
1.1 1.1 1.1
1 1 1 10 15 20 25 30 10 15 20 25 30 10 15 20 25 30
E0 (keV) E0 (keV) E0 (keV) Total X-ray intensity: generated versus emitted
Generated intensity I gen 0 zdzz )()()(
Emitted intensity z II emit II gen exp z 0 )()( exp zdzz )( where, )( 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 2 )( Cu = 4837.5 cm /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 I emit (z)0 (z )exp zd )(
* * z Iemit z )( 0 )( exp zdz )(
i z 0 i ( z )exp zd )( AZ ii * i z 0 i z )( exp zd )( (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)
NiK )( Absorber Absorber EK EK Ec(K) (Atomic No.) (keV) (keV) (keV) (cm2/g) Mn(25)* 5.895 6.492 6.537 344 Fe(26)* 6.4 7.059 7.111 380 Co(27) 6.925 7.649 7.709 53 Ni(28) 7.472 8.265 8.331 59 Cu(29) 8.041 8.907 8.98 65.5
* NiK fluoresces Mn and Fe
Element Radiation causing fluorescen ce Mn FeK, CoK, CoK, NiK, NiK, CuK, CuK Fe CoK, NiK, NiK, CuK, CuK Co NiK, CuK, CuK Ni CuK Cu none Characteristic fluorescence (F) correction
f f I ij : intensity of X-ray ‘i’ 1 ij II i j fluoresced by element j Fi * f 1 ij II i Ii : intensity of X-ray ‘i’ j generated by electron * sample 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
1.67 Y1 = [(Uj-1)/(Ui-1)]
j j Y2 = (/) i/(/) spec
Y3 = [ln(1+u)]/u + [ln(1+v)]/v i j u = [(/) spec/(/) spec] cosec 5 1.65 1.65 j v = 3.3x10 /[(E0 -Ec )(/) spec]
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 FFeK in Fe-Ni alloy
1.1 1.1 1.1
1 1 1 o o o
0.9 0.9 0.9 at 40 CFe CFe at 52.5 at 15.5 CFe 0.1 0.1 0.1 FeK FeK FeK 0.3 F 0.3 F F 0.3 0.8 0.8 0.8 0.5 0.5 0.5 0.7 0.7 0.7 0.9 0.9 0.9 0.7 0.7 0.7 10 15 20 25 30 10 15 20 25 30 10 15 20 25 30
E0 (keV) E0 (keV) E0 (keV) 1.7 1.6 1.6 CFe CFe CFe 1.6 0.1 1.5 0.1 1.5 0.1 0.3 0.3 0.3 1.5 0.5 0.5 0.5 o o 1.4 1.4 0.7 o 0.7 0.7 1.4 0.9 0.9 1.3 1.3 0.9 at 40 at 52.5 at 15.5 1.3 NiK NiK NiK 1.2 1.2 A A A 1.2
1.1 1.1 1.1
1 1 1 10 15 20 25 30 10 15 20 25 30 10 15 20 25 30
E0 (keV) E0 (keV) E0 (keV)
ANiK in Fe-Ni alloy 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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