φ(ρz) Equations for Quantitative Analysis

Abstract

The idea of φ(ρz) equations and the depth distribution of x-ray intensities as a basis for the quantitative correction of measured x-ray intensities goes back to the origins of electron probe microanalysis. φ(ρz) is the characteristic x-ray intensity generated in a thin layer dρz at depth ρz in the specimen relative to intensity generated in an identical layer dρz, isolated in space. Castaing [1] first suggested that you could write the measured k-ratio in terms of φ(ρz) equations as:

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AamaaBa % aaleaacaWGbbaabeaakiabg2da9maalaaabaGaamysamaaDaaaleaa % caWG4baabaGaamytaaaaaOqaaiaadMeadaqhaaWcbaGaamyqaaqaai % aad2eaaaaaaOGaeyypa0ZaaSaaaeaadaWdXaqaaiabew9aMnaaBaaa % leaacaWGZbaabeaakmaabmaabaGaeqyWdiNaamOEaaGaayjkaiaawM % caaiaadwgadaahaaWcbeqaaiabgkHiTiabeY7aTnaaDaaameaacaWG % bbaabaGaam4Caaaaliabeg8aYjaadQhaciGGJbGaai4Caiaacogacq % aHipqEaaGccaWGKbGaeqyWdiNaamOEaaWcbaGaaGimaaqaaiabg6Hi % LcqdcqGHRiI8aaGcbaWaa8qmaeaacqaHvpGzdaWgaaWcbaGaamyqaa % qabaGcdaqadaqaaiabeg8aYjaadQhaaiaawIcacaGLPaaacaWGLbWa % aWbaaSqabeaacqGHsislcqaH8oqBdaqhaaadbaGaamyqaaqaaiaadg % eaaaWccqaHbpGCcaWG6bGaci4yaiaacohacaGGJbGaeqiYdKhaaOGa % amizaiabeg8aYjaadQhaaSqaaiaaicdaaeaacqGHEisPa0Gaey4kIi % paaaaaaa!77B5! ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${k_A} = \frac{{I_x^M}}{{I_A^M}} = \frac{{\int_0^\infty {{\phi _s}\left( {\rho z} \right){e^{ - \mu _A^s\rho z\csc \psi }}d\rho z} }}{{\int_0^\infty {{\phi _A}\left( {\rho z} \right){e^{ - \mu _A^A\rho z\csc \psi }}d\rho z} }}$$

(1)

where subscripts s and A refer to specimen and pure element A, respectively, μ is the mass absorption coefficient, with subscript referring to characteristic line and superscript the absorber. ρz represents the mass depth in the specimen. Ψ is the x-ray take-off angle. Castaing (with Descamps) [2] also demonstrated that is was possible to measure φ(ρz) curves using a sandwich sample technique (fig. 1). The advantage of the φ(ρz) equation is its relative simplicity in concept and the fact that the major corrections of absorption, atomic number and characteristic fluorescence can be explicitly written. The equation for the fraction of x rays which escape from the specimen, f(χ), the absorption correction is

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa % aaleaacaWGbbaabeaakmaabmaabaGaeq4Xdm2aaSbaaSqaaiaadgea % aeqaaaGccaGLOaGaayzkaaGaeyypa0ZaaSaaaeaadaWdXaqaaiabew % 9aMnaaBaaaleaacaWGZbaabeaakmaabmaabaGaeqyWdiNaamOEaaGa % ayjkaiaawMcaaiaadwgadaahaaWcbeqaaiabgkHiTiabeY7aTnaaDa % aameaacaWGbbaabaGaam4Caaaaliabeg8aYjaadQhaciGGJbGaai4C % aiaacogacqaHipqEaaaabaGaaGimaaqaaiabg6HiLcqdcqGHRiI8aO % Gaamizaiabeg8aYjaadQhaaeaadaWdXaqaaiabew9aMnaaBaaaleaa % caWGZbaabeaakmaabmaabaGaeqyWdiNaamOEaaGaayjkaiaawMcaai % aadsgacqaHbpGCcaWG6baaleaacaaIWaaabaGaeyOhIukaniabgUIi % Ydaaaaaa!68A5!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${f_A}\left( {{\chi _A}} \right) = \frac{{\int_0^\infty {{\phi _s}\left( {\rho z} \right){e^{ - \mu _A^s\rho z\csc \psi }}} d\rho z}}{{\int_0^\infty {{\phi _s}\left( {\rho z} \right)d\rho z} }}$$

(2)

where μ ^s_A is the mass absorption coefficient for the characteristic x rays of element A in the specimen. The combined factor μ cscΨ is the so-called absorption parameter χ. The numerator is simply the number of x rays which escape from the specimen while the denominator represents the total number of x rays generated in the specimen.