Worked Examples in X-Ray Analysis

Moseley’s law relates the wavelength X of a characteristic line in a given series with atomic number Z. The usual expression of the law is 1<math display='block'> <mrow> <mfrac> <mn>1</mn> <mrow> <msqrt> <mi>&#x03BB;</mi> </msqrt> </mrow> </mfrac> <mtext>&#x2009;</mtext><mo>=</mo><mtext>&#x2009;</mtext><mi>K</mi><mtext>&#x2009;</mtext><mo stretchy='false'>(</mo><mi>Z</mi><mo>&#x2212;</mo><mi>&#x03C3;</mi><mo stretchy='false'>)</mo></mrow> </math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\frac{1}{{\sqrt \lambda }}\; = \;K\;(Z - \sigma )$$ where K is a constant for a given spectral series and σ is a shielding constant.

The electronic structure of Cu (Z = 29) is denoted 1s2, 2s22p6, 3s23p63d9, 4s2.

Plot the approximate distribution of intensity that would be obtained from a chromium anode X-ray tube operated at 50 kV. Use Kramers’ formula4) to give the relative intensity in the number of photons (I) over the wave­length range 0–5 Å.

A The continuous radiation from an X-ray tube is scattered by the sample, diffracted by the analyzing crystal and measured by the detector.

The resolution R of a proportional counter is expressed in terms of the peak width at half height multiplied by 100, and divided by the value of the maximum pulse amplitude.

LiF is commonly used as an analysing crystal. The (200) reflection planes are commonly used, having a “2d” value of 4.028 Å. It is, however, possible to use reflecting planes with other Miller indices (hkl) for example the (220), the (420) and the (422) planes.

In the measurement of a mixture of elements a problem of the overlap of two lines is encountered. The lines can be separated by use of the fol­lowing combinations: (i)topaz crystal (having good resolution but poor reflecting power) plus a coarse collimator, placed between sample and crystal.(ii)A LiF (200) crystal (having poor resolution but good reflectivity) plus a fine collimator, placed between sample and crystal.

It is necessary to set up an X-ray spectrometer for the measurement of sodium in a series of rock specimens. An argon/methane flow count­er is being employed and a KAP (potassium hydrogen phthalate) analysing crystal. The rock samples contain quite a lot of calcium and this leads to strong crystal fluorescence.

A mixture of sodium silicates and sodium phosphates is being analysed for phosphorus. The experimental conditions used are: tungsten anode X-ray tube, 50 kV-40 mA; coarse collimator; gypsum analysing crystal; gas flow proportional counter (Ar/10% methane) with pulse height selection.

A series of measurements are made in which counting rate is plotted as a function of counter voltage. The counter in this case is an argon/methane gas flow proportional counter. A single plateau is found for a sample of pure sulphur, using the S Kα line, but pure samples of iron and chromium are both found to give a second plateau, again when measuring their Kα lines.

A choice has to be made between the scintillation counter and the flow proportional counter for the measurement of a certain wavelength.

A silicon specimen is being examined using nickel filtered Cu Kα radiation. The intensities of two reflections are measured with a Geiger-Müller counter and the following counting rates obtained:

The dispersion of the diffractometer (dθ/dλ) can be expressed in the given form obtained by differentiating the Bragg law nλ = 2dsinθ5<math display='block'> <mrow> <mfrac> <mrow> <mi>d</mi><mi>&#x03B8;</mi></mrow> <mrow> <mi>d</mi><mi>&#x03BB;</mi></mrow> </mfrac> <mtext>&#x2009;</mtext><mo>=</mo><mtext>&#x2009;</mtext><mfrac> <mi>n</mi> <mrow> <mn>2</mn><mi>d</mi></mrow> </mfrac> <mtext>&#x2009;</mtext><mo>.</mo><mtext>&#x2009;</mtext><mfrac> <mn>1</mn> <mrow> <mi>cos</mi><mi>&#x03B8;</mi></mrow> </mfrac> </mrow> </math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\frac{{d\theta }}{{d\lambda }}\; = \;\frac{n}{{2d}}\;.\;\frac{1}{{\cos \theta }}$$where the angular separation (dθ) of two wavelengths separated by (dλ) is expressed in terms of the diffraction angle θ for a certain spacing d.

The diffraction pattern of an iron specimen is being recorded with a copper target X-ray tube operating at 40 kV-50 mA. A Ni-filter is placed between the X-ray tube and specimen. A Xe proportional counter is used to record counts on one of the haematite reflections but even though this is one of the strongest reflections usable, the peak to background is only about 1.5 to 1.

Adiffractogram of quartz exhibited an additional spurious peak at approximately 6° (2θ).The effect was observed both in specimens containing only low atomic number elements and specimens containing high atomic number elements. The patterns were always taken using Cu radiation, proportional counter (Xe-filled) and pulse height selection. The pulse height selector was set at the 45% acceptance level since the resolution of the counter (peak width at half height) was 18% for Cu Kα radiation.

A diffractometer trace is recorded of a compound known to contain a high concentration of Ba. The pattern is characterized by a very strong line with a d -spacing of 3.08 Å, plus many others. A weak, broad line was observed above a rather high background at approximately 6.5 o(2θ). It was suspected from the shape of this line that it did not belong to the regular diffraction pattern. Can there be an alternative explanation for this line.

A certain analysis line gave 800 c/s under certain instrumental conditions. Given that the standard deviation (σ) of a given number of counts (N) is equal to √ N, calculate the standard deviation expected in a counting time of 20 seconds.

Whenever X-ray intensity measurements are made on a selected peak, occuring at a certain setting of the goniometer, some fraction of the measured counting rate is always due to background. The significance of this background is dependent on its relative magnitude, and it is often difficult to judge whether or not it can be ignored in the calculation of the counting error. The true counting error ɛ % is related to the peak R _p and background R _b counting rates in the following way: 6<math display='block'> <mrow> <mi>&#x03B5;</mi><mi>&#x0025;</mi><mo>=</mo><mtext>&#x2009;</mtext><mfrac> <mrow> <mn>100</mn></mrow> <mrow> <msqrt> <mi>T</mi> </msqrt> </mrow> </mfrac> <mo>.</mo><mfrac> <mn>1</mn> <mrow> <msqrt> <mrow> <msub> <mi>R</mi> <mi>p</mi> </msub> </mrow> </msqrt> <mtext>&#x2009;</mtext><mo>&#x2212;</mo><msqrt> <mrow> <msub> <mi>R</mi> <mi>b</mi> </msub> </mrow> </msqrt> </mrow> </mfrac> </mrow> </math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\varepsilon \% = \;\frac{{100}}{{\sqrt T }}.\frac{1}{{\sqrt {{R_p}} \; - \sqrt {{R_b}} }}$$ where T is the total analysis time.

The lower limit of detection is often defined as that concentration equal to two standard deviations of the background count rate. Derive an ex­pression which defines the lower limit of detection in terms of the anal­ysis time T, the background counting rate R _b and the response m of an element given in c/s per %. Remember that trace analysis always re­quires two measurements, i.e. peak plus background.

A series of samples of alloyed steels were found to give 300 c/s/% for the Si Kα line. A calibration procedure has to be prepared over the range 0.20 – 0.60% Si and the choice has to be made between fixed count and fixed time operation.

The choice has to be made between the ratio and absolute method of counting in a counting programme which already involves collection of counts on a standard sample (i.e. whether the absolute or ratio method is chosen the standard is counted anyway, so no time is to be saved by not counting the standard).

A series of measurements of fluorine in cement sample yielded a peak counting rate of 4 c/s/% for fluorine (F Kα) over a background of 6.5 c/s.

A series of measurements are being made to determine the concentration of BaO in a mixture of BaO and BaO.6Fe₂O₃.

Only those crystallites having the reflecting planes almost parallel to the specimen surface can contribute to a certain reflection. The intensity of the resulting diffraction is thus dependent on this number of crystallites. Intensities of different diffraction lines or between different specimens can only be compared if the number of particles contributing is the same fraction of the total number of particles. If this total number is too small to warrant a random distribution, an error in the intensity measurement is introduced8). This may occur for instance when the particles are too large.

A series of oil samples gave the following intensities on the Pb Lα line: Prepare a calibration curve and work out the equation of the line. What is the concentration of lead in sample 6.

The following data were obtained using a series of cement samples. Counting ratios were determined on the Ca Kα line using sample 5 as the ratio standard.

A whole rock specimen was being analyzed for trace concentrations of strontium. The method of Norrish and Chappell(10) was used to measure and correct for absorption and the following data were obtained.

The total secondary absorption of a matrix is given by 16<math display='block'> <mrow> <msub> <mi>&#x03BC;</mi> <mrow> <mi>m</mi><mi>a</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>x</mi><mtext>&#x2009;</mtext></mrow> </msub> <mo>=</mo><munder> <mrow> <mo>&#x2211;</mo><mtext>&#x2009;</mtext><mo stretchy='false'>(</mo><msub> <mi>&#x03BC;</mi> <mi>i</mi> </msub> <msub> <mi>w</mi> <mi>i</mi> </msub> <mo stretchy='false'>)</mo></mrow> <mi>i</mi> </munder> </mrow> </math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${\mu _{matrix\;}} = \mathop {\sum \;({\mu _i}{w_i})}\limits_i $$ where μ _i and w _i are the individual mass absorption coefficients and weight fractions for each element i making up the matrix.

Calculate the total secondary absorption of a cement sample for Ca Kα radiation. Use the following mass absorption coefficients for Ca Kα.

The slope of a calibration curve for a certain element is dependent upon the total absorption of the measured wavelength within the sample matrix. Although it is common practice to assume that only secondary absorption is of significance, in fact both primary and secondary absorption should be considered. The following expression relates the so-called “efficiency factor” C(λλ _j ) with the mass absorption coefficient µ _i of each element i for the measured wavelength λ _j and an effective primary wavelength λ which represents the primary spectrum.

An ore specimen consisting predominantly of calcium and barium carbonates contained traces of nickel. A specimen containing no barium sulphate with a mass absorption coefficient for Ni K

α

of 124 gave the following data

240 c/s per % for nickel

background 120 c/s

(a)

Estimate the lower limit of detection for nickel in an analysis time of 200 s.

(b)

What would be the effect on this detection limit of adding 20% of barium sulphate to the specimen, assuming that the c/s per % value is inversely proportional to the matrix absorption? Assume also that the background is reduced by 8% by the addition of the barium sulphate. The atomic weights of Ba = 137.4, S = 32.1 and O = 16.

In the analysis of tin ores it is found that a very curved calibration line is obtained due to the large difference in the mass absorption coefficients of the two matrix components Sn (µ = 13) and SiO₂ (µ = 1.80) for the analysis line (Sn Kα).

A series of eight samples were analysed for a certain element A. An element B, also present in major concentration, was found to strongly absorb element A. The following intensity data were obtained: The concentrations of element B are unknown, although it is suspected that the eight standard samples are almost pure binary mixtures.

A series of samples are being analysed for lead using the Pb Lα line. It is found that zinc which is also present strongly absorbs the Pb Lα wavelength resulting in a very poor calibration curve. The following data were obtained:

It is often thought that the wavelength in the primary spectrum most efficient in exciting the fluorescent radiation of an element A is that which is closest to the absorption edge of element A. The probability function of the primary spectrum should follow the absorption curve for A.

A sample is known to contain two crystalline modifications of TiO₂, namely rutile and anatase. Spectrographic analysis failed to reveal the presence of other elements, so the sample can be assumed to consist only of TiO₂. A scan is made with the diffractometer and integrated intensities measured for the (110) reflection of rutile (I _R ) and for the (101) reflection of anatase (I _A ). Background measurements were made from a suitable part of the background. The measured intensities are as follows: The following relationship is known to exist between the concentration of rutile (C _R ) and the concentration of anatase (C _A ): 21<math display='block'> <mrow> <mfrac> <mrow> <msub> <mi>C</mi> <mi>R</mi> </msub> </mrow> <mrow> <msub> <mi>C</mi> <mi>A</mi> </msub> </mrow> </mfrac> <mtext>&#x2009;</mtext><mo>=</mo><mtext>&#x2009;</mtext><mi>K</mi><mfrac> <mrow> <msub> <mi>I</mi> <mi>R</mi> </msub> </mrow> <mrow> <msub> <mi>I</mi> <mi>A</mi> </msub> </mrow> </mfrac> </mrow> </math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\frac{{{C_R}}}{{{C_A}}}\; = \;K\frac{{{I_R}}}{{{I_A}}}$$ where K is a proportionality constant. Previous experiments 9) indicated a value of 1.33 for K.

The weight fraction of a quartz (SiO₂) has to be determined in a natural specimen. To correct for the unknown absorption in this specimen, 200 mg KCl are added to 1000 mg of the sample, both very finely ground, and well mixed. Determine the weight fraction of α quartz, using the following data:

The intensity of the 3.34 Å line of α-quartz (SiO₂) has to be estimated in a mixture consisting of 10% α-quartz and 90% CaSiO₃. Nickel filtered Cu Kα radiation is used. The intensity of the 3.34 Å line in pure α quartz was 100 c/s, measured under the same conditions. What is the expected intensity from the mixture.

Very often the quantity of a compound has to be determined in a complex mineral, the composition and absorption coefficients of which are unknown. To correct for these unknown facts, the attenuation of a diffraction line arising from the bottom of the sample holder may be used. The following intensities were collected in measuring the percentage quartz in a mineral with a conventional diffractometer, utilizing a sample holder with a nickel bottom.

It is necessary to determine the dead time of an X-ray spectrometer using the Kβ/Kα counting ratio method. The following data were obtained on the Kα and Kβ lines of tin by varying the X-ray tube current and keeping the kV constant.

A pellet of pure sugar was placed in the spectrometer and a spectrum run between 10° – 130° 2

θ

in order to ascertain the ‘blank’ spectrum of the tube. The following conditions were employed:

Cr anode tube at 60 kV 32 mA (this tube had been run for about 5000 h).

LiF (200) crystal, 2

d

= 4.028Å. Angle between incident and take-off = 90°.

Flow plus scintillation detectors in tandem.

Fine primary collimator, vacuum condition.

The thickness of Cr plating on a piece of iron of rather irregular but flat shape has to be measured.

A scan is being made on a rock sample in the region of the barium K spectra.

It is necessary to measure low concentrations of chromium using a spectrometer equipped with a chromium anode X-ray tube. A titanium filter can be fitted over the tube window to minimize the intensity of the Cr Kα radiation arising from the scattered tube radiation. Two thicknesses of titanium are available, at 0.13 mm and 0.011 mm.

A Ni filter has been constructed in such a way that only 2% of the incident Cu Kβ radiation may pass. Given that the density of Ni is 8.92 g/cm3 and that the mass absorption of Ni for Cu Kα and Cu Kβ is 49.2 cm2/g and 286 cm2/g respectively. calculate: AThe thickness of the filterBThe percentage of incident Cu Kα radiation that can pass the filter.

A filter is very often used in X-ray diffractometry to reduce the background level and the intensity of β-radiation. This filter can be placed either between X-ray tube and specimen or between the specimen and detector.

The following 12 lines were obtained from a crystalline powder. Data were obtained using nickel filtered Cu Kα radiation. The powder is known to belong to the cubic system.

Table 1 lists the relative line intensities obtained on samples of sodium, potassium and rubidium chlorides, using identical instrumental conditions.

Measurements of the nickel (111) reflection are being made on a series of Raney Nickel catalysts. Severe line broadening problems are encountered due to the very small particle size of the powders (about 50 – 100 Å). An estimate is required of the magnitude of the line broadening.