Quantitative X-ray analysis—matrix thickness correction

Abstract

Quantitative X-ray analysis is carried out by making X-ray fluorescence measurements to determine the elemental composition of a sample and a correction measurement by measuring the transmitted intensity of X-rays at an energy E transmitted directly through the sample without deviation. An X-ray diffraction measurement is made in transmission by directing X-rays from an X-ray source at the energy E onto a sample at an incident angle ψ.sub.1 to the surface of the sample and measuring a measured intensity I.sub.d(θ.sub.fl) of the diffracted X-rays at the energy E with an X-ray detector at an exit angle ψ.sub.2 corresponding to an X-ray diffraction peak of a predetermined component. A matrix corrected X-ray intensity is obtained using the measured X-ray intensity in the X-ray diffraction measurement, the correction measurement and the mass attenuation coefficient of the sample calculated from the elemental composition and the mass attenuation coefficients of the elements.

Claims

1. A method of X-ray analysis comprising: making an X-ray diffraction measurement in transmission by directing X-rays from an X-ray source at the energy E onto a sample at an incident angle ψ.sub.1 to the surface of the sample and measuring a measured intensity I.sub.d(θ.sub.fl) of the diffracted X-rays at the energy E with an X-ray detector at an exit angle ψ.sub.2 corresponding to an X-ray diffraction peak of a predetermined component making a correction measurement of X-rays by measuring the transmitted intensity of X-rays at an energy E transmitted directly through the sample without deviation; making X-ray fluorescence measurements to determine the elemental composition of a sample; and calculating a matrix corrected X-ray intensity using the measured X-ray intensity in the X-ray diffraction measurement, the correction measurement and the mass attenuation coefficient of the sample calculated from the elemental composition and the mass attenuation coefficients of the elements, and wherein the matrix corrected measurement intensity I.sub.dc is calculated from $I_{\mathrm{dc}}=\frac{I_d\left({\theta }_{\mathrm{fl}}\right)}{M_{\mathrm{fl}}}$ with the matrix correction factor M.sub.fl calculated by: $M_{\mathrm{fl}}=\frac{e^{-\mu .Math.\rho .Math.d.Math.\cos \mathrm{ec}{\psi }_2}-e^{-\mu .Math.\rho .Math.d.Math.\cos \mathrm{ec}{\psi }_1}}{\mu .Math.\left(\cos \mathrm{ec}{\psi }_1-\cos \mathrm{ec}{\psi }_2\right)}$ using the product μρd calculated from the correction measurement and the mass attenuation coefficient μ of the sample calculated from the elemental composition determined from the X-ray fluorescence measurements.

2. The method according to claim 1, wherein the step of making a correction measurement and the step of making X-ray fluorescence measurements take place at the same time.

3. The method according to claim 1, comprising calculating the product μρd from the correction measurement using the Beer-Lambert law, where μ is the mass attenuation coefficient at the energy E, ρ the sample density and d the sample thickness.

4. The method according to claim 1 comprising calculating the mass attenuation coefficient μ(E) of the sample from the sum for all sample components: $\mu \left(E\right)=\underset{i=1}{\overset{i=\mathrm{nel}}{.Math.}}{w}_i{\mu }_i\left(E\right)$ where w.sub.i is the fraction of a sample component, and μ.sub.i is the mass attenuation coefficient of the respective sample components.

5. The method according to claim 1 wherein the predetermined component is free lime.

6. A method comprising: obtaining a calibration line by carrying out a method according to claim 1 for a plurality of samples having a known concentration of the predetermined component; and measuring the quantity of a predetermined component in an unknown sample by carrying out a method according to claim 1 for the unknown sample.

7. The method according to claim 6 wherein obtaining a calibration line comprises fitting a straight line to the corrected intensity as a function of the concentration of the plurality of samples having a known concentration of the predetermined component.

8. An X-ray apparatus, comprising: a sample stage for supporting a sample extending substantially horizontally; an X-ray source located on one side of the sample stage; an X-ray fluorescence X-ray detector; an diffraction X-ray detector located on the other side of the sample stage for carrying out X-ray diffraction in a transmission geometry; and a controller; wherein the controller is arranged to cause the X-ray apparatus to carry out a method according to any preceding claim.

9. The X-ray apparatus according to claim 8, further comprising a transmission X-ray detector located above the sample stage for measuring the intensity of X-rays emitted by the X-ray source and passing through the sample without deflection.

10. The X-ray apparatus according to claim 8 wherein the X-ray source is a source of Ag Ka radiation.

11. The X-ray apparatus according to claim 8 in which the X-ray source and the X-ray fluorescence X-ray detector are provided under the sample stage and the diffraction X-ray detector is provided above the sample stage.

12. The X-ray apparatus according to claim 10 further comprising a filter between the sample stage and the diffraction X-ray detector for carrying out X-ray diffraction in a transmission geometry, the filter filtering out Ag Kb radiation.

13. The X-ray apparatus according to claim 12 wherein the filter is a stack of (a) Rh or Pd and (b) Ag or other element with an atomic number higher than 47 for filtering out continuum radiation of the tube spectrum.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) An example of the invention will now be described with reference to the accompanying diagrams, in which:

(2) FIG. 1 is a graph of measurement intensity against thickness for samples of different concentrations of powder in wax;

(3) FIG. 2 is likewise a graph of measurement intensity against thickness for samples of different composition;

(4) FIG. 3 is a schematic indicating incident and exit X-rays in a configuration considered below in the “theory” section;

(5) FIG. 4 shows apparatus used to take measurements in a first embodiment of the invention; and

(6) FIG. 5 shows some results measured using the invention.

DETAILED DESCRIPTION

(7) The invention relates to a method which can be applied for the correction of measured photon intensities in X-ray diffraction in transmission geometry without explicit knowledge of the sample's thickness.

(8) Theory

(9) XRD measurements performed in transmission geometry require that the measured specimens have finite thickness in order to allow the generated photons to escape the specimen from the back side and at a certain exit angle. Theoretical calculations predict that the measured photon intensity will depend both on sample thickness as well as on composition. In that sense the repeatability of the measurements with respect to sample preparation can be highly affected even for specimens prepared from a single sample assuming that a different dilution ratio (binder/material) is applied during sample preparation.

(10) The way in which X-rays pass through the sample is determined by:

(11) $\begin{array}{cc}\mu \left(E\right)=\underset{i=1}{\overset{i=\mathrm{nel}}{.Math.}}{w}_i{\mu }_i\left(E\right)& \left(2\right)\end{array}$

(12) which is the mass attenuation coefficient of the specimen (typically expressed in cm.sup.2/g) which is directly related to the composition of the specimen since it contains the weight fractions of all elements in the specimen w.sub.i and the mass absorption coefficient of each element μ.sub.i(E) at the excitation energy E.

(13) Other relevant definitions used in this and other equations in this document are collected below for convenience:

(14) TABLE-US-00001 w.sub.fl The weight fraction of the predetermined component I.sub.d The intensity (kcps) of scattered/diffracted photons recorded by the detector I.sub.dc The intensity (kcps) of scattered/diffracted photons recorded by the detector after matrix and thickness correction $M=\frac{e^{-{\mu }^{″}.Math.\rho .Math.d}-e^{-{\mu }^{\prime }.Math.\rho .Math.d}}{\mu .Math.\left(\cos \mathrm{ec}{\psi }_1+\cos \mathrm{ec}{\psi }_2\right)}$ Matrix/thickness correction term μ′ = μ .Math. cos ec ψ.sub.1 Effective mass absorption coefficient (cm.sup.2/g) for the incident photons μ″ = μ .Math. cos ec ψ.sub.2 Effective mass absorption coefficient (cm.sup.2/g) for the scattered/diffracted photons ψ.sub.1 The angle formed between the direction of the incident photons and the surface of the sample (incident angle). ψ.sub.2 The angle formed between the direction of the diffracted photons and the surface of the sample (exit angle - diffraction angle). G.sub.fl The geometry factor for the diffraction channel θ.sub.d Diffraction angle σ Scattering cross section

(15) Consider X-rays incident on a sample at an incident angle ψ.sub.1 and further consider the X-rays diffracted by a particular component at an exit angle ψ.sub.2 as illustrated in FIG. 1.

(16) An example will be presented in this section for assistance in understanding the mathematics. In the example the particular (pre-determined) component is free lime though the method is equally applicable to other components. Assuming the incident radiation is the Ag-Ka line then the first order diffraction will be expected to be at a diffraction angle 2θ.sub.fl=13.3°. Therefore, in this example and assuming an incident angle ψ.sub.1=57° the exit angle ψ.sub.2=57°+13.3°=70.3° for the diffraction peak.

(17) At the exit angle corresponding to the diffraction peak, the intensity that will be observed by the scintillation detector will be given by:

(18) $\begin{array}{cc}{I}_d\left({\theta }_{\mathrm{fl}}\right)=I_0.Math.G_{\mathrm{fl}}.Math.\cos \mathrm{ec}{\psi }_1.Math..Math.w_{\mathrm{fl}}.Math.{\sigma }_{\mathrm{fl}}\left({\theta }_{\mathrm{fl}}\right)+\left(1-w_{\mathrm{fl}}\right).Math.{\sigma }_{\mathrm{oth}}\left({\theta }_{\mathrm{fl}}\right).Math..Math.M_{\mathrm{fl}}\mathrm{Where}\text{:}& \left(3\right)\\ M_{\mathrm{fl}}=\frac{e^{-\mu .Math.\rho .Math.d.Math.\cos \mathrm{ec}{\psi }_2}-e^{-\mu .Math.\rho .Math.d.Math.\cos \mathrm{ec}{\psi }_1}}{\mu .Math.\left(\cos \mathrm{ec}{\psi }_1-\cos \mathrm{ec}{\psi }_2\right)}& \left(4\right)\end{array}$

(19) is the self-absorption term at the angle θ.sub.fl.

(20) Note that σ.sub.fl is the scattering cross section of the predetermined component measured at a diffraction peak and σ.sub.othl the scattering cross section of all other components.

(21) This can be written as

(22) $\begin{array}{cc}{I}_{\mathrm{dc}}=\frac{I_d\left({\theta }_{\mathrm{fl}}\right)}{M_{\mathrm{fl}}}=I_0.Math.G_{\mathrm{fl}}.Math.\cos \mathrm{ec}{\psi }_1.Math.w_{\mathrm{fl}}.Math.{\sigma }_{\mathrm{fl}}\left({\theta }_{\mathrm{fl}}\right)+I_0.Math.G_{\mathrm{fl}}.Math.\cos \mathrm{ec}{\psi }_1.Math.\left(1-w_{\mathrm{fl}}\right).Math.{\sigma }_{\mathrm{oth}}\left({\theta }_{\mathrm{fl}}\right)& \left(5\right)\end{array}$

(23) The left hand side of this equation expresses the matrix and thickness corrected diffraction intensity I.sub.dc(θ.sub.fl).

(24) The second part is the sum of two terms: One term which accounts for the diffraction signal originating from the free lime phase and another term accounting for the influence of other crystalline and amorphous phases. Within a limited range of clinker types this factor can be considered as a constant background and equation (5) can be rewritten as:
I.sub.dc(θ.sub.fl)=K.Math.w.sub.fl+I.sub.bg(θ.sub.fl)   (6)
with K being a proportionality constant

(25) In general, evaluating equation (4) is far from straightforward.

(26) However, the inventors have realised that some samples can be sufficiently well characterised by XRF that the composition of elements can be determined. Then, equation (2) can be used since the absorption μ is simply a sum of the absorption and is determined by the known compounds present in the sample. This leads to a value μ.

(27) It is also possible to calculate the product μρd by carrying out measurements of direct transmission with and without the sample and using the Beer-Lambert law of equation (1). Alternatively, and instead of measuring the direct beam which can saturate the detector, the transmitted radiation can be measured through a well characterized standard of appropriate thickness (for example Quartz glass). Then, the ratio of the two transmission experiments is formed and the unknown μρd of the sample is determined relative to the corresponding one of the standard.

(28) With these two pieces of information it is possible to correct the measured intensity of an XRD measurement by a matrix intensity correction M, i.e. to correct the measured value for the effects of absorption by other elements in the sample by substituting these values in equation (4).

(29) Thus, after calibration of the instrument for an XRD line corresponding to a particular component, it is possible to correct for absorption in the sample and obtain I.sub.dc(θ.sub.fl) from equation (5). From this value and by using the calibration curve the weight fraction of a particular component can be determined.

(30) Implementation

(31) An X-ray apparatus 2 has a sample stage 4 for holding a sample 6.

(32) In practice, this apparatus 2 is a conventional XRF apparatus with an X-ray source 10 mounted below the sample stage 4. In this embodiment there is a fluorescence X-ray detector 12 for measuring X-ray fluorescence below the sample stage 4. The fluorescence X-ray detector may be an energy dispersive detectors measuring X-ray intensity as a function of energy or a wavelength dispersive X-ray detector with a crystal for selecting X-rays only of a particular wavelength. The crystal may be movable to allow selection of different wavelengths or fixed to allow measurement of a particular wavelength of interest.

(33) To the conventional XRF apparatus is added a transmission X-ray detector 14 which is mounted above the sample stage 4 on a goniometer so that it can measure diffracted X-rays as a function of angle. A correction X-ray detector 30 is provided adjacent to the transmission X-ray detector as will be described below.

(34) A number of other components are provided, including collimator 16 and filter 18.

(35) The apparatus is under control of controller 20 which includes a memory 22 and processor 24.

(36) In the embodiment shown, the X-ray source 10 is arranged to emit Ag-Ka radiation and the filter 18 is arranged to filter out the Ag-Kb line and possibly also to filter out continuum radiation. The filter may be a multilayered filter including layers of Rh or Pd to filter out the Ag-Kb line and other layers such as Ag to filter out the continuum. Other high atomic number (Z) layers may be used as well as the Ag or additional to the Ag.

(37) For the diffraction measurement a parallel plate collimator 16 placed between source and sample is used. For the correction measurement simple pinhole optics between sample and detector 30 may be used for collimation.

(38) In use, a sample 6 is prepared by the pressed powder method. A powder is compressed together with wax binder in a ring to form a sample which is mounted on the sample stage. In the specific example, the sample is a clinker sample and the predetermined component which is to be measured is free lime.

(39) In a first stage of measurement, the X-rays source is activated (by removing a shutter) and X-rays are incident on a sample. In this case, XRF measurements are made by the fluorescence X-ray detector. The correction X-ray detector 30 is arranged so that it is directly in line with the incident X-rays, i.e. it detects directly transmitted X-rays. The correction X-ray detector measures the intensity of X-rays passing through the sample and hence the absorption in the sample.

(40) In order to avoid detector overloading and to ensure that the radiation recorded by the detector 30 is identical to the one used for the diffraction measurements an Ag filter 34 of appropriate thickness is placed between the sample and the detector to filter out the Ag-Kb line and possibly also to filter out continuum radiation. The filter may be a multilayered filter including layers of Rh or Pd to filter out the Ag-Kb line and other layers such as Ag to filter out the continuum

(41) A measurement of the intensity at the diffraction peak 2θ.sub.fl=13.3° is then measured in transmission with the source located to provide an incident angle ψ.sub.1=57° and the transmission X-ray detector located to provide an exit angle ψ.sub.2=57°+13.3°=70.3°.

(42) This information is then put together using equations (1) to (4) to obtain not merely information about the specific elements (obtained by XRF) but also one or more phases (obtained only by XRD).

(43) These measurements and the calibration are all carried out controlled by code stored in memory 24 which controls processor 22 in controller 20 to control the apparatus 2 to carry out the measurements.

(44) Those skilled in the art will realise that the apparatus and method may be varied to suit the specific equipment available. For example, a different X-ray source with a different radiation spectrum may be used. It may be possible to use the same detector for the diffraction measurements and transmission measurements in place of detectors 14, 30.

EXAMPLES

(45) An experimental example was carried out to test the validity of the approximations made, as well as the applicability of the method. A set of pressed pellets was made from a clinker matrix spiked with appropriate free lime (FL) quantities to yield final FL concentrations equal to 0%, 0.5% 1%, 1.5%, 2%, 3%, 4% and 5%. According to equation (6) it is possible to construct a calibration line which relates the corrected diffraction intensities with the concentration of the FL. The results are shown in FIG. 5.

(46) For comparison the uncorrected intensities are also plotted as function of the FL concentration. Note that the application of the correction improves the quality of the calibration line considerably. There is much less variability and the gradient of the line is steeper which makes accurate intensity measurement possible.