Quantitative X-ray analysis—ratio correction

Abstract

A method of X-ray analysis measures X-ray diffraction in transmission. In order to carry out quantitative measurements, a background measurement is taken slightly away from the diffraction peak and the ratio of measured intensities used to correct for variations in sample composition.

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 an energy E onto a sample at an incident angle ψ.sub.1 to the surface of the sample and measuring a diffraction intensity I.sub.d(θ.sub.d) of the diffracted X-rays at the energy E with a X-ray detector at an exit angle ψ.sub.2 to the surface of the sample, a difference 2θ.sub.d between the incident angle and the exit angle corresponding to an X-ray diffraction peak of a predetermined component; and making a correction measurement of X-rays by measuring a background intensity of X-rays I.sub.d(θ.sub.bg) at the energy E, with the difference 2θ.sub.bg between the incident angle and the exit angle deviating by 0.2 to 5° from the difference 2θ.sub.d corresponding to the X-ray diffraction peak; and calculating the quantity of the predetermined component from the intensity ratio I.sub.d(θ.sub.fl)/ I.sub.d(θ.sub.bg) of the diffraction intensity and the background intensity.

2. The method according to claim 1, wherein the step of making a correction measurement of X-rays uses incident X-rays directed onto the sample at the same incident angle ψ.sub.1 to the angle used in the step of making an X-ray diffraction measurement and measures the background intensity of X-rays at an exit angle ψ.sub.3 to the surface of the sample, where ψ.sub.3=ψ.sub.2±Δψ and Δψ is in the range 0.2 to 5°.

3. The method according to claim 1, wherein calculating the quantity of the predetermined component comprises obtaining the quantity of the predetermined component from a calibration curve linking the intensity ratio to the quantity of the predetermined component.

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

5. A method comprising obtaining a calibration curve relating the intensity ratio to concentration of the predetermined component by carrying out a method according to claim 1 for a plurality of samples each 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.

6. An X-ray apparatus, comprising: a sample stage for supporting a sample extending substantially horizontally; an X-ray source located to direct X-rays to the sample stage; an X-ray detector located on the other side of the sample stage for measuring X-ray intensity of diffracted X-rays 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 claim 1 on a sample mounted on the sample stage.

7. The method according to claim 2, wherein calculating the quantity of the predetermined component comprises using a linear relation between the intensity ratio and a calculated $W=\frac{w_{\mathrm{fl}}}{1-w_{\mathrm{fl}}}$ where w.sub.fl is the weight fraction of the component of interest.

8. The method according to claim 5 further comprising fitting a straight line to the intensity ratio as a function of the concentration of the plurality of samples having a known concentration of the predetermined component.

9. The X-ray apparatus according to claim 6, further comprising a further X-ray detector located on the same side of the sample stage as the X-ray source for making X-ray fluorescence measurements of the sample.

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 the ratio of Matrix correction terms as a function of sample thickness d;

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

(7) FIG. 6 shows a calibration curve of the percentage of free lime against intensity ratio.

DETAILED DESCRIPTION

(8) 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.

(9) Theory

(10) 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.

(11) The absorption of X-rays pass through a sample is determined by:

(12) $\mu \left(E\right)=\underset{i=1}{\overset{i=\mathrm{nel}}{.Math.}}{w}_i{\mu }_i\left(E\right)$

(13) 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/diffraction energy E.

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

(15) TABLE-US-00001 w.sub.fl The weight fraction of the predetermined component I.sub.d The intensity (kcps) of scattered photons recorded.by the detector $M_{\mathrm{fl}}=\frac{e^{-\mu .Math.\rho .Math.d.Math.\cos \mathrm{ec}{\psi }_2}-e^{-\mu .Math.\rho .Math.d.Math.\cos \mathrm{ec}{\psi }_1}}{\mu .Math.\left(\cos \mathrm{ec}{\psi }_1-\cos \mathrm{ec}{\psi }_2\right)}$ 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 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 channel). ψ.sub.3 The angle formed between the direction of the scattered photons and the surface of the sample for a shifted detector angle (exit angle - background channel). G.sub.fl The geometry factor for the diffraction channel G.sub.bg The geometry factor for the background channel θ.sub.d Diffraction angle σ Scattering cross section

(16) 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.

(17) 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.

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

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

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

(21) Note that σ is the scattering cross section of the predetermined component measured at a diffraction peak (free lime in this case) and σ.sub.oth the scattering cross section of all other components.

(22) If the detector and collimator is now rotated by Δψ in the range 0.5° to 5°, for example 1° we will be observing the scattered intensity at an angle 2θ.sub.bg=2θ.sub.fl+1° which will be expressed as follows:

(23) $\begin{array}{cc}{I}_d\left({\theta }_{\mathrm{bg}}\right)=G_{\mathrm{bg}}.Math.\mathrm{cosec}{\psi }_1.Math..Math.w_{\mathrm{fl}}.Math.{\sigma }_{\mathrm{fl}}\left({\theta }_{\mathrm{bg}}\right)+\left(1-w_{\mathrm{fl}}\right).Math.{\sigma }_{\mathrm{oth}}\left({\theta }_{\mathrm{bg}}\right).Math..Math.M_{\mathrm{bgl}}& \left(2\right)\\ \mathrm{where}\text{:}& \\ M_{\mathrm{fl}}=\frac{e^{-\mu .Math.\rho .Math.d.Math.\mathrm{cosec}{\psi }_3}-e^{-\mu .Math.\rho .Math.d.Math.\mathrm{cosec}{\psi }_1}}{\mu .Math.\left(\mathrm{cosec}{\psi }_1-\mathrm{cosec}{\psi }_2\right)}& \left(2a\right)\end{array}$

(24) Notice, that the matrix absorption term is slightly modified since it contains now the exit angle ψ.sub.3=ψ.sub.2+1°.

(25) For the evaluated experimental arrangement ψ.sub.1=57° and the exit angles ψ.sub.2=57°+13.3°=70.3° and ψ.sub.3=71.3°.

(26) Forming the ratio between the measured intensity in the two channels we obtain:

(27) $\begin{array}{cc}\frac{I_d\left({\theta }_{\mathrm{fl}}\right)}{I_d\left({\theta }_{\mathrm{bg}}\right)}=\frac{G_{\mathrm{fl}}}{G_{\mathrm{bg}}}.Math.\frac{.Math.w_{\mathrm{fl}}.Math.{\sigma }_{\mathrm{fl}}\left({\theta }_{\mathrm{fl}}\right)+\left(1-w_{\mathrm{fl}}\right).Math.{\sigma }_{\mathrm{oth}}\left({\theta }_{\mathrm{fl}}\right).Math.}{\left[w_{\mathrm{fl}}.Math.{\sigma }_{\mathrm{fl}}\left({\theta }_{\mathrm{bg}}\right)+\left(1-w_{\mathrm{fl}}\right).Math.{\sigma }_{\mathrm{oth}}\left({\theta }_{\mathrm{bg}}\right)\right]}.Math.\frac{M_{\mathrm{fl}}}{M_{\mathrm{bg}}}& \left(3\right)\end{array}$

(28) or equivalently:

(29) $\begin{array}{cc}\frac{I_d\left({\theta }_{\mathrm{fl}}\right)}{I_d\left({\theta }_{\mathrm{bg}}\right)}=\frac{G_{\mathrm{fl}}}{G_{\mathrm{bg}}}.Math.\frac{{\sigma }_{\mathrm{oth}}\left({\theta }_{\mathrm{fl}}\right)}{{\sigma }_{\mathrm{oth}}\left({\theta }_{\mathrm{bg}}\right)}.Math.\frac{\frac{w_{\mathrm{fl}}}{1-w_{\mathrm{fl}}}\frac{{\sigma }_{\mathrm{fl}}\left({\theta }_{\mathrm{fl}}\right)}{{\sigma }_{\mathrm{oth}}\left({\theta }_{\mathrm{fl}}\right)}+1}{\frac{w_{\mathrm{fl}}}{1-w_{\mathrm{fl}}}\frac{{\sigma }_{\mathrm{fl}}\left({\theta }_{\mathrm{bg}}\right)}{{\sigma }_{\mathrm{oth}}\left({\theta }_{\mathrm{bg}}\right)}+1}.Math.\frac{M_{\mathrm{fl}}}{M_{\mathrm{bg}}}& \left(3a\right)\end{array}$

(30) However, assuming the first measured intensity is a diffraction peak then the intensity corresponding to this free lime peak at an exit angle ψ.sub.3 i.e. θ.sub.bg is small. In this case, therefore, the first term in equation (3b) below is small and we may write:

(31) $\begin{array}{cc}\frac{w_{\mathrm{fl}}}{1-w_{\mathrm{fl}}}\frac{{\sigma }_{\mathrm{fl}}\left({\theta }_{\mathrm{bg}}\right)}{{\sigma }_{\mathrm{oth}}\left({\theta }_{\mathrm{bg}}\right)}+1\approx 1& \left(3b\right)\end{array}$

(32) This applies in particular for the example of free lime peak. For normal free lime ranges (0.1%-2.0%) the first term of the sum in the denominator should be significantly lower than 10.sup.−3. This originates from the value of the weight fraction term and the fact that the probability of scattering for the free lime crystals far from their corresponding diffraction angle for Ag-Ka is at least one order of magnitude lower.

(33) Under these considerations the ratio now becomes:

(34) $\begin{array}{cc}\frac{I_d\left({\theta }_{\mathrm{fl}}\right)}{I_d\left({\theta }_{\mathrm{bg}}\right)}=G.Math.\frac{{\sigma }_{\mathrm{oth}}\left({\theta }_{\mathrm{fl}}\right)}{{\sigma }_{\mathrm{oth}}\left({\theta }_{\mathrm{bg}}\right)}.Math.\left[\frac{w_{\mathrm{fl}}}{1-w_{\mathrm{fl}}}\frac{{\sigma }_{\mathrm{fl}}\left({\theta }_{\mathrm{fl}}\right)}{{\sigma }_{\mathrm{oth}}\left({\theta }_{\mathrm{fl}}\right)}+1\right].Math.\frac{M_{\mathrm{fl}}}{M_{\mathrm{bg}}}& \left(4\right)\end{array}$

(35) where G=G.sub.fl/G.sub.bg is the ratio of the geometry factors.

(36) The formula is still far from a simple linear relation and includes the ratio of the self-absorption terms M. FIG. 4 illustrates the ratio of the self-absorption terms M in the example free lime case as function of thickness d for a typical clinker composition with μ≈5.0 cm.sup.2/g, ρ≈2.0 g/cm.sup.3 and for thickness values d ranging from 2.8 mm to 3.2 mm.

(37) As can be seen in FIG. 4, a change of 300 μm in the sample thickness will affect the ratio of the mass absorption coefficients by about 0.1%. In this sense, the intensity ratio becomes much more insensitive to sample thickness variations.

(38) Thus, the inventors have shown that it is a reasonable approximation to rewrite equation (4) as:

(39) 0$\begin{array}{cc}\frac{I_d\left({\theta }_{\mathrm{fl}}\right)}{I_d\left({\theta }_{\mathrm{bg}}\right)}=G.Math.\frac{{\sigma }_{\mathrm{oth}}\left({\theta }_{\mathrm{fl}}\right)}{{\sigma }_{\mathrm{oth}}\left({\theta }_{\mathrm{bg}}\right)}.Math.\frac{w_{\mathrm{fl}}}{1-w_{\mathrm{fl}}}+G.Math.\frac{{\sigma }_{\mathrm{oth}}\left({\theta }_{\mathrm{fl}}\right)}{{\sigma }_{\mathrm{oth}}\left({\theta }_{\mathrm{bg}}\right)}& \left(5\right)\end{array}$

(40) If the composition of the sample remains nearly constant then the ratios of equivalent cross sections σ.sub.fl(θ.sub.fl)/σ.sub.oth(θ.sub.bg) and σ.sub.oth(θ.sub.fl)/σ.sub.oth(θ.sub.bg) will also remain constant. In such case the intensity ratio can be written only as function of the free lime weight fraction:

(41) $\begin{array}{cc}\frac{I_d\left({\theta }_{\mathrm{fl}}\right)}{I_d\left({\theta }_{\mathrm{bg}}\right)}=A.Math.\frac{w_{\mathrm{fl}}}{1-w_{\mathrm{fl}}}+B& \left(5a\right)\end{array}$

(42) Generally speaking the ratio of two linear functions is not a linear but a rational one. With the variable change

(43) $\begin{array}{cc}W=\frac{w_{\mathrm{fl}}}{1-w_{\mathrm{fl}}}& \left(6\right)\end{array}$

(44) we may rewrite:

(45) $\begin{array}{cc}\frac{I_d\left({\theta }_{\mathrm{fl}}\right)}{I_d\left({\theta }_{\mathrm{bg}}\right)}=A.Math.W+B& \left(7\right)\end{array}$

(46) Thus, the ratio of intensities can be considered to be a linear function of the ratio W where

(47) $W=\frac{w_{\mathrm{fl}}}{1-w_{\mathrm{fl}}}$
and w.sub.fl is the weight fraction of the component of interest.

(48) In other words, by making two measurements as described, it is possible to avoid the effect of the term M which otherwise makes quantitative measurement very difficult indeed without significant efforts to calibrate.

(49) Note that the method does not simply subtract off a background correction, i.e. the method does not measure the background intensity near a peak to determine the peak height by subtraction. Instead, the method uses an intensity ratio of the free lime peak and a suitable neighboring position since by using the intensity ratio the effect of the matrix correction term M is effectively compensated for.

(50) Implementation

(51) Equations (6) and (7) can be used to make measurements in the system illustrated in FIG. 5.

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

(53) An X-ray source 10 is mounted below the sample stage 4. Conveniently, although not required in this method, a fluorescence X-ray detector 12 for measuring X-ray fluorescence is provided 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 system consisting of goniometer, masks, crystals, collimators and detectors for selecting X-rays only of a particular wavelength. A transmission X-ray detector 14 is mounted above the sample stage 4 on a goniometer so that it can measure diffracted X-rays as a function of angle.

(54) A number of other components are provided, including collimator 16 and filter 18. The collimator 16 is required for diffraction measurements only and may be removed for XRF measurements. Further, note collimator 15 for selecting only X-rays diffracted by the sample at a particular angle. Collimator 15 is mounted between the sample and the transmission X-ray detector and is likewise mounted on a goniometer for rotation to allow the selection of different angles. In practice, all that is required to measure the background intensity in the correction measurement (see below) is to rotate the collimator 15 if X-ray detector 14 has a sufficiently large X-ray input window.

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

(56) 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.

(57) 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.

(58) 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 and the transmission X-ray detector located to provide an exit angle ψ.sub.2=ψ.sub.1+13.3°. Then, the transmission X-ray detector 14 and collimator 15 are rotated by Δψ to a new exit angle ψ.sub.3 and a correction measurement of intensity made with the detector at an exit angle ψ.sub.3 without moving the X-ray source 2.

(59) The intensity ratio of the measured intensity divided by the correction intensity is then obtained and the controller determines the weight fraction of the component of interest, here free lime, from the linear relation between the ratio and W (equation 7) and then converting this to a free lime concentration using equation (6).

(60) In order to obtain the calibration curve calibration is carried out with a number of samples of known concentration. Each of these samples is measured using the above method and the calibration curve of concentration against intensity ratio is obtained by a fit of intensity ratio to the parameter W.

(61) 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.

EXAMPLES

(62) An experimental example was carried out to test the validity of the approximations made, as well as the applicability of the method.

(63) A set of pressed pellets were prepared. The set consisted of specimens originating 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%. Measurements were taken as described above with a view to testing whether the expected linear relationship of measured intensity ratio held. The measurements were performed with the diffraction intensity measured at an ψ.sub.2=70.3° and near XRD (XRDBG) at an angle ψ.sub.3=71.3° with an incident angle

(64) The calibration curve for the FL component is shown in FIG. 6. Note that the calibration curve using the method discussed above is well defined and linear.

(65) Further experiments of similar nature confirm the results above and show that the ratio between the measured intensity at the free lime channel and a small angular deviation from that channel is nearly constant for a range of thickness values, so the ratio method discussed here can be used.

(66) An advantage of this method is that sample inhomogeneities are not particularly important since the X-ray beam incident and diffracted interacts with the same sample volume for both measurements. Further, the method cancels out most instrumental drift by the use of the intensity ratio which can avoid the need for further complex calibration.

(67) Note that when measuring free lime in clinker samples, there is an interest in other components of those cement samples. XRF using fluorescence X-ray detector 12 may be carried out to determine the elemental composition of those samples without removing the samples from the apparatus. In particular, measurement of Ca, Fe, Al and Si with the assumption that these elements are present in the samples in their oxidised form allows for measurement of all components of clinker samples.

(68) Those skilled in the art will appreciate that the methods and apparatus described above can be varied as required.

(69) Although the measurements have been discussed above with reference to the measurement of the free lime diffraction peak in clinker samples, the method presented is not limited to that example and other examples where a matrix correction is required may also be measured in the same way.